Empirical Methods in Finance

Part 5

Henrique C. Martins

Panel Data

Panel Data

As explained previously, OVB is a significant source of “endogeneity” in empirical research.

OVB is a problem because of the considerable heterogeneity in many empirical settings.

Many of the omitted variables are unobservable to the researcher.

Panel data can sometimes offer a partial.

Panel Data

We start defining the following:

\[y_{i,t} = \alpha + \beta_1 x_{i,t} + \epsilon_{i,t}\]

Where:

  • \(i = 1, . . . , N\)
  • \(t = 1, . . . , T\)

Imagine that the residual can be decomposed in:

\[\epsilon_{i,t} = c_i + \mu_{i,t}\]

The term \(c_i\) is constant.

Panel Data

The term \(c_i\) is constant.

It captures the aggregate effect of all of the unobservable, time-invariant explanatory variables for \(y_{it}\).

To focus attention on the issues specific to panel data, we assume that \(e_{it}\) has a zero mean conditional on \(x_{it}\) and \(c_i\) for all \(t\).

The most important thing here is whether \(x_{it}\) and \(c_i\) are correlated.

Why?

Panel Data

The most important thing here is whether \(x_{it}\) and \(c_i\) are correlated.

  • If \(x_{it}\) and \(c_i\) are correlated, then \(c_i\) is referred to as a “fixed effect”.

    • It there is correlation, there is violation of the Conditional Mean Independence (CMI) assumption.
  • If \(x_{it}\) and \(c_i\) are not correlated, then \(c_i\) is referred to as a “random effect”.

    • Endogeneity is not a concern; however, the computation of standard errors is affected.

Panel Data

Why might fixed effects arise?

FE are any time-invariant unit characteristic that cannot be observed in the data.

  • education level,
  • firm’s culture,
  • technology,
  • managerial talent,
  • investment opportunities,
  • location (economic development, institutions, etc.),
  • etc.

Panel Data

We say things like (you have to understand that they refer to FE):

  • Time-invariant heterogeneity at the unit-level
  • Unobserved variation that occur at the unit-level that do not vary over time

Important: with FE, you are capturing all unobserved heterogeneity that do not vary over time.

Panel Data

Definition of Panel Data:

You have multiple observations per unit (individual, firm, etc.)

In datasets, it is “one panel below the other” not “one panel beside the other”.

Four main topics in Panel Data:

  1. Pooled cross-sectional

  2. Fixed Effect models (including multidimensional FE)

  3. Random Effects model

  4. First differences

  5. Lagged models

Panel Data

Formal definition

\[y_{i,t} = \alpha + \beta_1 x_{i,t} + \delta FE + \epsilon_{i,t}\]

  • \(E(\epsilon_{i,t}) = 0\)

  • \(corr(x_{i,t},FE) \neq 0\)

  • \(corr(FE, \epsilon_{i,t}) = 0\)

  • \(corr(x_{i,t},epsilon_{i,t}) = 0\), for all t

The last assumption is called strict exogeneity assumption and means that the residual of any t is uncorrelated with x of any t.

That is, under a strict exogeneity assumption on the explanatory variables, the fixed effects estimator is unbiased: the idiosyncratic error should be uncorrelated with each explanatory variable across all time periods.

Remember that if we ignore FE, we have OVB.

Panel Data

Before we continue…

Comment #1

The standard errors in this framework must be “clustered” by panel unit (e.g., individual) to allow for correlation in the residual for the same person over time. This yields valid inference as long as the number of clusters is “large.”

Comment #2

FE cannot solve reverse causality, it might help you with OVB.

Comment #3

Three main types of FE:

  • Pooled
  • Within-transformation (when someone says FE, it is usually this one)
  • Random Effects

Pooling Cross-sections

Pooling Cross-Sections

When you have two periods of the same unit, but the periods are not consecutive, you have a pooled cross-sectional data.

This is common in survey data.

If you use only one period, you might find biased results.

Let’s practice with the dataset CRIME2 from Wooldridge.

This dataset contains data (many cities) on the crime rate, unemployment rate and many other city-related variables.

There are two years, 82 and 87 (this is pooled cross-section).

Pooling Cross-Sections

If we estimate only using the year 87, we would interpret that unemployment leads to lower crime rate.

R
library(haven) 
data <- read_dta("files/CRIME2.dta")
data1 <- subset(data, year == 87)
model <- lm(crmrte ~ unem, data = data1)
summary(model)

Call:
lm(formula = crmrte ~ unem, data = data1)

Residuals:
   Min     1Q Median     3Q    Max 
-57.55 -27.01 -10.56  18.01  79.75 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  128.378     20.757   6.185  1.8e-07 ***
unem          -4.161      3.416  -1.218     0.23    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 34.6 on 44 degrees of freedom
Multiple R-squared:  0.03262,   Adjusted R-squared:  0.01063 
F-statistic: 1.483 on 1 and 44 DF,  p-value: 0.2297
Python
import pandas as pd
import statsmodels.api as sm

data = pd.read_stata("files/CRIME2.dta")
data1 = data[data['year'] == 87]
model = sm.OLS(data1['crmrte'], sm.add_constant(data1['unem'])).fit()
print(model.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                 crmrte   R-squared:                       0.033
Model:                            OLS   Adj. R-squared:                  0.011
Method:                 Least Squares   F-statistic:                     1.483
Date:                qua, 11 set 2024   Prob (F-statistic):              0.230
Time:                        17:09:26   Log-Likelihood:                -227.27
No. Observations:                  46   AIC:                             458.5
Df Residuals:                      44   BIC:                             462.2
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const        128.3781     20.757      6.185      0.000      86.546     170.210
unem          -4.1611      3.416     -1.218      0.230     -11.047       2.724
==============================================================================
Omnibus:                        3.902   Durbin-Watson:                   1.106
Prob(Omnibus):                  0.142   Jarque-Bera (JB):                3.681
Skew:                           0.641   Prob(JB):                        0.159
Kurtosis:                       2.473   Cond. No.                         25.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Stata
use "files/CRIME2.dta" , clear
reg crmrte une if year ==87
      Source |       SS           df       MS      Number of obs   =        46
-------------+----------------------------------   F(1, 44)        =      1.48
       Model |  1775.90928         1  1775.90928   Prob > F        =    0.2297
    Residual |  52674.6428        44  1197.15097   R-squared       =    0.0326
-------------+----------------------------------   Adj R-squared   =    0.0106
       Total |  54450.5521        45  1210.01227   Root MSE        =      34.6

------------------------------------------------------------------------------
      crmrte | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
        unem |  -4.161134   3.416456    -1.22   0.230    -11.04655     2.72428
       _cons |   128.3781   20.75663     6.18   0.000     86.54589    170.2104
------------------------------------------------------------------------------

Pooling Cross-Sections

When we consider a panel, we get the expected positive sign. This is evidence that the previous model suffered from OVB. Still, the coefficient of unem is not significant probably because of time-invariant unobserved heterogeneity in the cities.

R
library(haven) 
data <- read_dta("files/CRIME2.dta")
model <- lm(crmrte ~ d87+ unem, data = data)
summary(model)

Call:
lm(formula = crmrte ~ d87 + unem, data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-53.474 -21.794  -6.266  18.297  75.113 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  93.4203    12.7395   7.333 9.92e-11 ***
d87           7.9404     7.9753   0.996    0.322    
unem          0.4265     1.1883   0.359    0.720    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 29.99 on 89 degrees of freedom
Multiple R-squared:  0.01221,   Adjusted R-squared:  -0.009986 
F-statistic: 0.5501 on 2 and 89 DF,  p-value: 0.5788
Python
import pandas as pd
import statsmodels.api as sm

data = pd.read_stata("files/CRIME2.dta")
model = sm.OLS(data['crmrte'], sm.add_constant(data[['d87','unem']])).fit()
print(model.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                 crmrte   R-squared:                       0.012
Model:                            OLS   Adj. R-squared:                 -0.010
Method:                 Least Squares   F-statistic:                    0.5501
Date:                qua, 11 set 2024   Prob (F-statistic):              0.579
Time:                        17:09:29   Log-Likelihood:                -441.90
No. Observations:                  92   AIC:                             889.8
Df Residuals:                      89   BIC:                             897.4
Df Model:                           2                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const         93.4203     12.739      7.333      0.000      68.107     118.733
d87            7.9404      7.975      0.996      0.322      -7.906      23.787
unem           0.4265      1.188      0.359      0.720      -1.935       2.788
==============================================================================
Omnibus:                        8.350   Durbin-Watson:                   1.157
Prob(Omnibus):                  0.015   Jarque-Bera (JB):                8.771
Skew:                           0.756   Prob(JB):                       0.0125
Kurtosis:                       2.935   Cond. No.                         40.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Stata
use "files/CRIME2.dta" , clear
reg crmrte  d87 une 
      Source |       SS           df       MS      Number of obs   =        92
-------------+----------------------------------   F(2, 89)        =      0.55
       Model |  989.717314         2  494.858657   Prob > F        =    0.5788
    Residual |  80055.7864        89  899.503218   R-squared       =    0.0122
-------------+----------------------------------   Adj R-squared   =   -0.0100
       Total |  81045.5037        91  890.609931   Root MSE        =    29.992

------------------------------------------------------------------------------
      crmrte | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         d87 |   7.940413   7.975324     1.00   0.322    -7.906386    23.78721
        unem |   .4265461   1.188279     0.36   0.720    -1.934539    2.787631
       _cons |   93.42026   12.73947     7.33   0.000      68.1072    118.7333
------------------------------------------------------------------------------

Pooling Cross-Sections

This shows us that we should also control for the year variable.

We call this, Year Fixed Effects.

We still most likely have OVB due to the unobserved heterogeneity in cities, that is, we still would need to include cities FE.

Demeaned variables

Demeaned variables

A first way to eliminate the FE is by demeaning the data.

Consider the following:

\[\bar{y_i} = \alpha +\beta \bar{x_i} + \delta FE + \bar{\epsilon_i}\]

\[\frac{1}{T}\sum{y_{i,t}} = \alpha +\beta \frac{1}{T}\sum{x_{i,t}} + \delta FE + \frac{1}{T}\sum{\epsilon_{i,t}}\]

If we subtract the mean of each variable, we have:

\[(y_{i,t} - \bar{y_i}) = \beta (x_{i,t} - \bar{x_i}) + (\epsilon_{i,t} - \bar{\epsilon_i})\]

Because the FE does not vary over time, each value is equal to the mean.

Thus, when you demean, you eliminate the FE from the equation. You also eliminate the intercept \(\alpha\).

Takeaway: OLS will estimate unbiased coefficients if you demean the variables.

This is called within-transformation because you are demeaning “within” the group.

Demeaned variables

Let’s use the dataset WAGEPAN to estimate the following equation.

\[Ln(wage)=\alpha + \beta_1 exper^2 + \beta_2 married + \beta_3 union + \epsilon\]

Some variables in the dataset do not vary over time. These variables cannot be included in this equation.

Demeaned variables

See page 495 Wooldridge.

R
library(foreign)
library(stargazer)
library(sandwich)

data <- read.dta("files/WAGEPAN.dta")
# Calculate mean by nr for lwage, expersq, married, and union
data <- data[order(data$nr), ]  # Sort data by nr for by-group operations
data$lwage_mean <- ave(data$lwage, data$nr, FUN = mean)
data$expersq_mean <- ave(data$expersq, data$nr, FUN = mean)
data$married_mean <- ave(data$married, data$nr, FUN = mean)
data$union_mean <- ave(data$union, data$nr, FUN = mean)

data$lwage_demean <- data$lwage - data$lwage_mean
data$expersq_demean <- data$expersq - data$expersq_mean
data$married_demean <- data$married - data$married_mean
data$union_demean <- data$union - data$union_mean

model1 <- lm(lwage ~ educ + black + hisp + exper + expersq + married + union + d81 + d82 + d83 + d84 + d85 + d86 + d87, data = data)
model2 <- lm(lwage_demean ~ expersq_demean + married_demean + union_demean + d81 + d82 + d83 + d84 + d85 + d86 + d87, data = data)

stargazer(model1, model2 ,title = "Regression Results", column.labels=c("OLS","Demean"),  type = "text")

Regression Results
=======================================================================
                                    Dependent variable:                
                    ---------------------------------------------------
                              lwage                 lwage_demean       
                               OLS                     Demean          
                               (1)                       (2)           
-----------------------------------------------------------------------
educ                        0.091***                                   
                             (0.005)                                   
                                                                       
black                       -0.139***                                  
                             (0.024)                                   
                                                                       
hisp                          0.016                                    
                             (0.021)                                   
                                                                       
exper                       0.067***                                   
                             (0.014)                                   
                                                                       
expersq                     -0.002***                                  
                             (0.001)                                   
                                                                       
married                     0.108***                                   
                             (0.016)                                   
                                                                       
union                       0.182***                                   
                             (0.017)                                   
                                                                       
expersq_demean                                        -0.005***        
                                                       (0.001)         
                                                                       
married_demean                                        0.047***         
                                                       (0.017)         
                                                                       
union_demean                                          0.080***         
                                                       (0.018)         
                                                                       
d81                          0.058*                   0.151***         
                             (0.030)                   (0.021)         
                                                                       
d82                          0.063*                   0.253***         
                             (0.033)                   (0.023)         
                                                                       
d83                          0.062*                   0.354***         
                             (0.037)                   (0.027)         
                                                                       
d84                          0.090**                  0.490***         
                             (0.040)                   (0.034)         
                                                                       
d85                          0.109**                  0.617***         
                             (0.043)                   (0.042)         
                                                                       
d86                         0.142***                  0.765***         
                             (0.046)                   (0.053)         
                                                                       
d87                         0.174***                  0.925***         
                             (0.049)                   (0.064)         
                                                                       
Constant                      0.092                   -0.445***        
                             (0.078)                   (0.030)         
                                                                       
-----------------------------------------------------------------------
Observations                  4,360                     4,360          
R2                            0.189                     0.181          
Adjusted R2                   0.187                     0.179          
Residual Std. Error     0.480 (df = 4345)         0.328 (df = 4349)    
F Statistic         72.459*** (df = 14; 4345) 95.840*** (df = 10; 4349)
=======================================================================
Note:                                       *p<0.1; **p<0.05; ***p<0.01
Python
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
from statsmodels.iolib.summary2 import summary_col

data = pd.read_stata("files/WAGEPAN.dta")

data = data.sort_values(by='nr')  # Sort data by nr for by-group operations
data['lwage_mean'] = data.groupby('nr')['lwage'].transform('mean')
data['expersq_mean'] = data.groupby('nr')['expersq'].transform('mean')
data['married_mean'] = data.groupby('nr')['married'].transform('mean')
data['union_mean'] = data.groupby('nr')['union'].transform('mean')

data['lwage_demean'] = data['lwage'] - data['lwage_mean']
data['expersq_demean'] = data['expersq'] - data['expersq_mean']
data['married_demean'] = data['married'] - data['married_mean']
data['union_demean'] = data['union'] - data['union_mean']

model1 = sm.OLS(data['lwage'], sm.add_constant(data[['educ', 'black', 'hisp', 'exper', 'expersq', 'married', 'union', 'd81', 'd82', 'd83', 'd84', 'd85', 'd86', 'd87']])).fit()
model2 = sm.OLS(data['lwage_demean'], sm.add_constant(data[['expersq_demean', 'married_demean', 'union_demean', 'd81', 'd82', 'd83', 'd84', 'd85', 'd86', 'd87']])).fit()

# Display regression results using stargazer
summary = summary_col([model1, model2], stars=True)
print(summary)

======================================
                 lwage    lwage_demean
--------------------------------------
R-squared      0.1893     0.1806      
R-squared Adj. 0.1867     0.1787      
black          -0.1392***             
               (0.0236)               
const          0.0921     -0.4446***  
               (0.0783)   (0.0297)    
d81            0.0583*    0.1512***   
               (0.0304)   (0.0205)    
d82            0.0628*    0.2530***   
               (0.0332)   (0.0228)    
d83            0.0620*    0.3544***   
               (0.0367)   (0.0274)    
d84            0.0905**   0.4901***   
               (0.0401)   (0.0339)    
d85            0.1092**   0.6175***   
               (0.0434)   (0.0423)    
d86            0.1420***  0.7655***   
               (0.0464)   (0.0525)    
d87            0.1738***  0.9250***   
               (0.0494)   (0.0643)    
educ           0.0913***              
               (0.0052)               
exper          0.0672***              
               (0.0137)               
expersq        -0.0024***             
               (0.0008)               
expersq_demean            -0.0052***  
                          (0.0007)    
hisp           0.0160                 
               (0.0208)               
married        0.1083***              
               (0.0157)               
married_demean            0.0467***   
                          (0.0171)    
union          0.1825***              
               (0.0172)               
union_demean              0.0800***   
                          (0.0181)    
======================================
Standard errors in parentheses.
* p<.1, ** p<.05, ***p<.01
Stata
use "files/WAGEPAN.dta" , clear

bys nr:  egen lwage_mean = mean(lwage) 
bys nr:  egen expersq_mean = mean(expersq) 
bys nr:  egen married_mean = mean(married) 
bys nr:  egen union_mean = mean(union)

gen lwage_demean = lwage - lwage_mean
gen expersq_demean = expersq - expersq_mean
gen married_demean = married - married_mean
gen union_demean = union - union_mean

eststo: qui reg lwage        educ black hisp exper expersq       married        union d81 d82 d83 d84 d85 d86 d87
eststo: qui reg lwage_demean expersq_demean married_demean union_demean d81 d82 d83 d84 d85 d86 d87
esttab , mtitles("OLS" "Demean") compress
(est1 stored)

(est2 stored)


------------------------------------
                 (1)          (2)   
                 OLS       Demean   
------------------------------------
educ          0.0913***             
             (17.44)                

black         -0.139***             
             (-5.90)                

hisp          0.0160                
              (0.77)                

exper         0.0672***             
              (4.91)                

expersq     -0.00241**              
             (-2.94)                

married        0.108***             
              (6.90)                

union          0.182***             
             (10.63)                

d81           0.0583        0.151***
              (1.92)       (7.36)   

d82           0.0628        0.253***
              (1.89)      (11.08)   

d83           0.0620        0.354***
              (1.69)      (12.96)   

d84           0.0905*       0.490***
              (2.26)      (14.46)   

d85            0.109*       0.617***
              (2.52)      (14.59)   

d86            0.142**      0.765***
              (3.06)      (14.58)   

d87            0.174***     0.925***
              (3.52)      (14.38)   

expe~emean               -0.00519***
                          (-7.87)   

marr~emean                 0.0467** 
                           (2.73)   

union_de~n                 0.0800***
                           (4.43)   

_cons         0.0921       -0.445***
              (1.18)     (-14.96)   
------------------------------------
N               4360         4360   
------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001

Practical Tips

Practical Tips

You will not need to demean the variables every time you want to estimate a fixed effect models.

The statistical softwares have packages that do that.

You only need to know that Fixed effects model is a demeaned model, i.e., a within-transformation model.

But notice that you will have many different Fixed Effects together:

  • Firm Fixed Effects
  • Year Fixed Effects
  • Individual Fixed Effects (if individuals change between firms)

I am calling a multidimensional fixed effects design if you expand the FE to interactions of FE. Most common:

  • Year-Industry Fixed Effects.
  • CEO-Firm Fixed Effects.

Practical Tips

Notice the number of dummies in the last two columns.

R
library(foreign)
library(stargazer)
library(sandwich)
library(plm)

data <- read.dta("files/WAGEPAN.dta")
# Calculate mean by nr for lwage, expersq, married, and union
data <- data[order(data$nr), ]  # Sort data by nr for by-group operations
data$lwage_mean <- ave(data$lwage, data$nr, FUN = mean)
data$expersq_mean <- ave(data$expersq, data$nr, FUN = mean)
data$married_mean <- ave(data$married, data$nr, FUN = mean)
data$union_mean <- ave(data$union, data$nr, FUN = mean)

data$lwage_demean <- data$lwage - data$lwage_mean
data$expersq_demean <- data$expersq - data$expersq_mean
data$married_demean <- data$married - data$married_mean
data$union_demean <- data$union - data$union_mean

# set panel data
pdata <- pdata.frame(data, index = c("nr", "year"))

# Random effects regression using plm
model_de <- lm(lwage_demean ~  expersq_demean + married_demean + union_demean +  d81 +d82+ d83+ d84+ d85 +d86 +d87 , data = data)
model_fe <- plm(lwage ~  expersq + married + union + factor(year)              + educ + black + hisp + exper, data = pdata, model = "within")
model_du <- lm( lwage ~  expersq + married + union + factor(year) + factor(nr) + educ + black + hisp + exper, data = data)

# Display regression results using stargazer
#summary(model_de)
#summary(model_fe)
#summary(model_du)
stargazer(model_de, model_fe, model_du ,title = "Regression Results",  type = "text")

Regression Results
==================================================================================================
                                                 Dependent variable:                              
                    ------------------------------------------------------------------------------
                          lwage_demean                               lwage                        
                               OLS                      panel                      OLS            
                                                       linear                                     
                               (1)                       (2)                       (3)            
--------------------------------------------------------------------------------------------------
expersq_demean              -0.005***                                                             
                             (0.001)                                                              
                                                                                                  
married_demean              0.047***                                                              
                             (0.017)                                                              
                                                                                                  
union_demean                0.080***                                                              
                             (0.018)                                                              
                                                                                                  
d81                         0.151***                                                              
                             (0.021)                                                              
                                                                                                  
d82                         0.253***                                                              
                             (0.023)                                                              
                                                                                                  
d83                         0.354***                                                              
                             (0.027)                                                              
                                                                                                  
d84                         0.490***                                                              
                             (0.034)                                                              
                                                                                                  
d85                         0.617***                                                              
                             (0.042)                                                              
                                                                                                  
d86                         0.765***                                                              
                             (0.053)                                                              
                                                                                                  
d87                         0.925***                                                              
                             (0.064)                                                              
                                                                                                  
expersq                                               -0.005***                 -0.005***         
                                                       (0.001)                   (0.001)          
                                                                                                  
married                                                0.047**                   0.047**          
                                                       (0.018)                   (0.018)          
                                                                                                  
union                                                 0.080***                   0.080***         
                                                       (0.019)                   (0.019)          
                                                                                                  
factor(year)1981                                      0.151***                   0.151***         
                                                       (0.022)                   (0.022)          
                                                                                                  
factor(year)1982                                      0.253***                   0.253***         
                                                       (0.024)                   (0.024)          
                                                                                                  
factor(year)1983                                      0.354***                   0.354***         
                                                       (0.029)                   (0.029)          
                                                                                                  
factor(year)1984                                      0.490***                   0.490***         
                                                       (0.036)                   (0.036)          
                                                                                                  
factor(year)1985                                      0.617***                   0.617***         
                                                       (0.045)                   (0.045)          
                                                                                                  
factor(year)1986                                      0.765***                   0.765***         
                                                       (0.056)                   (0.056)          
                                                                                                  
factor(year)1987                                      0.925***                   0.925***         
                                                       (0.069)                   (0.069)          
                                                                                                  
factor(nr)17                                                                     0.579***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)18                                                                     0.929***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)45                                                                     0.554***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)110                                                                    1.046***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)120                                                                     0.239           
                                                                                 (0.176)          
                                                                                                  
factor(nr)126                                                                    0.783***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)150                                                                     -0.140          
                                                                                 (0.176)          
                                                                                                  
factor(nr)162                                                                     0.269           
                                                                                 (0.176)          
                                                                                                  
factor(nr)166                                                                     0.138           
                                                                                 (0.176)          
                                                                                                  
factor(nr)189                                                                     0.191           
                                                                                 (0.176)          
                                                                                                  
factor(nr)193                                                                    0.981***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)209                                                                    0.707***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)212                                                                    0.833***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)218                                                                    1.231***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)243                                                                    0.586***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)259                                                                    0.478***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)260                                                                    0.789***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)309                                                                    1.052***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)351                                                                    0.358**          
                                                                                 (0.177)          
                                                                                                  
factor(nr)353                                                                    0.583***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)383                                                                    0.564***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)408                                                                    0.801***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)424                                                                    1.345***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)464                                                                     0.080           
                                                                                 (0.176)          
                                                                                                  
factor(nr)483                                                                    0.713***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)556                                                                    1.089***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)560                                                                     0.205           
                                                                                 (0.176)          
                                                                                                  
factor(nr)569                                                                    -0.431**         
                                                                                 (0.176)          
                                                                                                  
factor(nr)583                                                                    0.852***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)647                                                                    0.360**          
                                                                                 (0.177)          
                                                                                                  
factor(nr)658                                                                    0.838***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)684                                                                    1.132***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)711                                                                     -0.097          
                                                                                 (0.176)          
                                                                                                  
factor(nr)729                                                                    0.837***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)731                                                                    0.574***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)732                                                                    0.531***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)793                                                                    0.498***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)797                                                                    0.841***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)800                                                                    0.380**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)813                                                                   -0.657***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)823                                                                    -0.393**         
                                                                                 (0.176)          
                                                                                                  
factor(nr)827                                                                    0.507***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)847                                                                    0.489***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)851                                                                    0.735***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)863                                                                    0.550***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)873                                                                     -0.127          
                                                                                 (0.176)          
                                                                                                  
factor(nr)891                                                                    0.735***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)908                                                                    -0.405**         
                                                                                 (0.176)          
                                                                                                  
factor(nr)910                                                                    0.679***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)916                                                                    0.564***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)919                                                                     0.238           
                                                                                 (0.178)          
                                                                                                  
factor(nr)922                                                                    1.011***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)924                                                                     -0.071          
                                                                                 (0.176)          
                                                                                                  
factor(nr)925                                                                    0.809***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)944                                                                     0.106           
                                                                                 (0.176)          
                                                                                                  
factor(nr)945                                                                     0.106           
                                                                                 (0.176)          
                                                                                                  
factor(nr)947                                                                    1.154***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)955                                                                    1.006***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)965                                                                    0.434**          
                                                                                 (0.180)          
                                                                                                  
factor(nr)996                                                                     0.007           
                                                                                 (0.177)          
                                                                                                  
factor(nr)1007                                                                   0.996***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)1054                                                                    0.133           
                                                                                 (0.177)          
                                                                                                  
factor(nr)1064                                                                   0.858***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)1081                                                                    0.179           
                                                                                 (0.176)          
                                                                                                  
factor(nr)1085                                                                    0.114           
                                                                                 (0.176)          
                                                                                                  
factor(nr)1091                                                                    0.226           
                                                                                 (0.183)          
                                                                                                  
factor(nr)1094                                                                   0.600***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)1096                                                                    0.251           
                                                                                 (0.182)          
                                                                                                  
factor(nr)1098                                                                   0.837***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1102                                                                   0.948***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1107                                                                   0.762***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)1142                                                                   0.715***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1156                                                                   0.569***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)1180                                                                   0.636***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1190                                                                    0.119           
                                                                                 (0.176)          
                                                                                                  
factor(nr)1204                                                                   0.888***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)1249                                                                   0.468***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)1272                                                                   0.612***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1311                                                                   0.481***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1316                                                                   0.921***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)1318                                                                   0.625***         
                                                                                 (0.192)          
                                                                                                  
factor(nr)1345                                                                   0.549***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)1397                                                                   0.653***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)1434                                                                    -0.082          
                                                                                 (0.177)          
                                                                                                  
factor(nr)1492                                                                   0.408**          
                                                                                 (0.178)          
                                                                                                  
factor(nr)1496                                                                    -0.219          
                                                                                 (0.176)          
                                                                                                  
factor(nr)1506                                                                    0.082           
                                                                                 (0.176)          
                                                                                                  
factor(nr)1515                                                                    0.281           
                                                                                 (0.176)          
                                                                                                  
factor(nr)1520                                                                   0.375**          
                                                                                 (0.181)          
                                                                                                  
factor(nr)1528                                                                    0.136           
                                                                                 (0.177)          
                                                                                                  
factor(nr)1554                                                                    0.312*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)1575                                                                   0.577***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)1576                                                                   0.504***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1628                                                                   0.440**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)1641                                                                   0.974***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)1644                                                                    0.263           
                                                                                 (0.176)          
                                                                                                  
factor(nr)1653                                                                   0.629***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1654                                                                   0.663***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)1721                                                                   1.160***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)1742                                                                   0.861***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1744                                                                   0.557***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1763                                                                    -0.194          
                                                                                 (0.176)          
                                                                                                  
factor(nr)1777                                                                    0.115           
                                                                                 (0.176)          
                                                                                                  
factor(nr)1843                                                                   1.310***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)1891                                                                    -0.090          
                                                                                 (0.176)          
                                                                                                  
factor(nr)1895                                                                   0.604***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)1899                                                                    0.171           
                                                                                 (0.177)          
                                                                                                  
factor(nr)1925                                                                   0.488***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1930                                                                    0.251           
                                                                                 (0.176)          
                                                                                                  
factor(nr)1961                                                                   0.745***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)1963                                                                   0.470***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)1979                                                                   0.828***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)1988                                                                    -0.062          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2000                                                                   0.637***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2014                                                                    0.310*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2025                                                                   0.432**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2038                                                                    0.180           
                                                                                 (0.176)          
                                                                                                  
factor(nr)2075                                                                   0.928***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)2101                                                                   1.158***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2106                                                                   0.595***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)2107                                                                   0.460***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2108                                                                   0.816***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2147                                                                  -0.860***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2157                                                                    0.055           
                                                                                 (0.180)          
                                                                                                  
factor(nr)2163                                                                   1.092***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)2173                                                                    0.148           
                                                                                 (0.176)          
                                                                                                  
factor(nr)2180                                                                   0.411**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2183                                                                   0.392**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2216                                                                   0.540***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2220                                                                   0.991***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2227                                                                    0.255           
                                                                                 (0.177)          
                                                                                                  
factor(nr)2264                                                                    -0.093          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2306                                                                    -0.052          
                                                                                 (0.177)          
                                                                                                  
factor(nr)2312                                                                   0.816***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2314                                                                   0.969***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)2329                                                                   0.545***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2335                                                                   0.506***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)2341                                                                   0.471***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2351                                                                    0.046           
                                                                                 (0.176)          
                                                                                                  
factor(nr)2386                                                                    -0.154          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2401                                                                   0.644***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)2413                                                                   0.999***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)2421                                                                    0.036           
                                                                                 (0.177)          
                                                                                                  
factor(nr)2445                                                                   0.463***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)2451                                                                   0.592***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2494                                                                    0.168           
                                                                                 (0.179)          
                                                                                                  
factor(nr)2508                                                                   0.848***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2535                                                                   0.486***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)2540                                                                    -0.059          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2685                                                                   1.031***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2711                                                                    0.052           
                                                                                 (0.176)          
                                                                                                  
factor(nr)2718                                                                   1.059***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2721                                                                    0.157           
                                                                                 (0.184)          
                                                                                                  
factor(nr)2733                                                                    -0.146          
                                                                                 (0.178)          
                                                                                                  
factor(nr)2741                                                                   0.419**          
                                                                                 (0.181)          
                                                                                                  
factor(nr)2745                                                                   0.556***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2751                                                                   0.456**          
                                                                                 (0.179)          
                                                                                                  
factor(nr)2774                                                                   0.903***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2801                                                                    0.050           
                                                                                 (0.176)          
                                                                                                  
factor(nr)2813                                                                    0.222           
                                                                                 (0.177)          
                                                                                                  
factor(nr)2833                                                                   0.621***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2839                                                                   0.646***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)2842                                                                   0.942***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2866                                                                   0.707***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2868                                                                   0.978***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2874                                                                    0.007           
                                                                                 (0.177)          
                                                                                                  
factor(nr)2916                                                                   0.345**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)2951                                                                    0.313*          
                                                                                 (0.182)          
                                                                                                  
factor(nr)2980                                                                    0.327*          
                                                                                 (0.177)          
                                                                                                  
factor(nr)2994                                                                   0.506***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)2997                                                                   0.601***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3017                                                                   0.936***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3037                                                                   1.040***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)3059                                                                   0.694***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)3062                                                                   1.166***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3100                                                                   0.637***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)3102                                                                    -0.259          
                                                                                 (0.176)          
                                                                                                  
factor(nr)3127                                                                   -0.407**         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3136                                                                   0.698***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3137                                                                   0.892***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3138                                                                   0.665***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)3140                                                                   0.843***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3193                                                                   0.686***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3196                                                                   0.681***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3200                                                                   0.771***         
                                                                                 (0.187)          
                                                                                                  
factor(nr)3202                                                                   0.593***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)3207                                                                   0.471***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)3208                                                                   0.855***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3210                                                                   0.708***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3215                                                                    0.324*          
                                                                                 (0.178)          
                                                                                                  
factor(nr)3219                                                                    0.237           
                                                                                 (0.176)          
                                                                                                  
factor(nr)3226                                                                   0.642***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3235                                                                    0.144           
                                                                                 (0.176)          
                                                                                                  
factor(nr)3239                                                                   -0.375**         
                                                                                 (0.175)          
                                                                                                  
factor(nr)3271                                                                   0.756***         
                                                                                 (0.182)          
                                                                                                  
factor(nr)3275                                                                   0.542***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3282                                                                   0.660***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3289                                                                   0.454***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3290                                                                   0.456***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3307                                                                   1.321***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3333                                                                    -0.078          
                                                                                 (0.176)          
                                                                                                  
factor(nr)3353                                                                    0.083           
                                                                                 (0.176)          
                                                                                                  
factor(nr)3380                                                                   0.522***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3381                                                                   0.587***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3389                                                                   0.672***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3394                                                                   0.656***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3401                                                                   0.397**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)3414                                                                   0.701***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3420                                                                    0.301*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)3440                                                                    -0.211          
                                                                                 (0.176)          
                                                                                                  
factor(nr)3461                                                                   0.711***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3468                                                                    0.195           
                                                                                 (0.176)          
                                                                                                  
factor(nr)3482                                                                   1.056***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3495                                                                   0.538***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)3503                                                                   0.965***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)3525                                                                   1.040***         
                                                                                 (0.175)          
                                                                                                  
factor(nr)3526                                                                   0.594***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)3538                                                                   0.962***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)3563                                                                   0.636***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3575                                                                   0.997***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)3580                                                                   0.361**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)3581                                                                   0.671***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3589                                                                    0.287           
                                                                                 (0.177)          
                                                                                                  
factor(nr)3591                                                                   0.674***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3598                                                                   0.864***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3602                                                                   1.082***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)3607                                                                    -0.019          
                                                                                 (0.187)          
                                                                                                  
factor(nr)3621                                                                   0.528***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)3628                                                                   0.984***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3653                                                                   0.723***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3706                                                                   0.650***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3707                                                                   0.556***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3708                                                                   0.617***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3743                                                                   0.607***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3777                                                                   0.732***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)3831                                                                    0.227           
                                                                                 (0.176)          
                                                                                                  
factor(nr)3844                                                                   0.510***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3847                                                                   1.104***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)3848                                                                   0.651***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)3882                                                                    0.047           
                                                                                 (0.175)          
                                                                                                  
factor(nr)3937                                                                   0.520***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)4000                                                                   1.072***         
                                                                                 (0.201)          
                                                                                                  
factor(nr)4004                                                                   0.976***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)4025                                                                    0.135           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4032                                                                    0.157           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4046                                                                   0.564***         
                                                                                 (0.184)          
                                                                                                  
factor(nr)4088                                                                   1.447***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)4091                                                                   1.271***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)4122                                                                   0.713***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)4127                                                                    0.093           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4128                                                                    0.297*          
                                                                                 (0.178)          
                                                                                                  
factor(nr)4159                                                                   0.605***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4204                                                                    0.324*          
                                                                                 (0.177)          
                                                                                                  
factor(nr)4229                                                                    0.314*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)4258                                                                   0.718***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4261                                                                   1.342***         
                                                                                 (0.200)          
                                                                                                  
factor(nr)4264                                                                    0.073           
                                                                                 (0.178)          
                                                                                                  
factor(nr)4278                                                                   0.594***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)4297                                                                   0.538***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4302                                                                    -0.129          
                                                                                 (0.176)          
                                                                                                  
factor(nr)4321                                                                   0.473***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4328                                                                    0.239           
                                                                                 (0.177)          
                                                                                                  
factor(nr)4332                                                                   -0.371**         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4335                                                                    0.342*          
                                                                                 (0.177)          
                                                                                                  
factor(nr)4357                                                                   0.527***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)4365                                                                    -0.039          
                                                                                 (0.176)          
                                                                                                  
factor(nr)4380                                                                   0.880***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)4394                                                                   0.724***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4510                                                                   0.643***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4559                                                                   0.374**          
                                                                                 (0.180)          
                                                                                                  
factor(nr)4563                                                                    0.019           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4569                                                                   0.791***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4603                                                                   0.734***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4607                                                                    0.166           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4633                                                                   0.505***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4676                                                                    0.125           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4701                                                                   0.952***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4716                                                                    0.122           
                                                                                 (0.178)          
                                                                                                  
factor(nr)4720                                                                   1.137***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)4759                                                                   0.596***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)4791                                                                   0.678***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)4811                                                                    0.096           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4828                                                                    0.108           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4857                                                                    -0.098          
                                                                                 (0.176)          
                                                                                                  
factor(nr)4858                                                                   0.976***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)4859                                                                    0.031           
                                                                                 (0.176)          
                                                                                                  
factor(nr)4866                                                                   0.818***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)4881                                                                   0.358**          
                                                                                 (0.177)          
                                                                                                  
factor(nr)4884                                                                   0.522***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)4888                                                                    0.096           
                                                                                 (0.177)          
                                                                                                  
factor(nr)4901                                                                    0.218           
                                                                                 (0.179)          
                                                                                                  
factor(nr)4917                                                                    0.059           
                                                                                 (0.177)          
                                                                                                  
factor(nr)4926                                                                   0.440**          
                                                                                 (0.178)          
                                                                                                  
factor(nr)4982                                                                    0.102           
                                                                                 (0.178)          
                                                                                                  
factor(nr)5017                                                                   1.088***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5033                                                                   1.061***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5048                                                                    0.315*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)5122                                                                   0.657***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)5141                                                                   0.711***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)5147                                                                   1.045***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)5158                                                                   0.700***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5221                                                                   0.538***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)5223                                                                   0.486***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5227                                                                   0.652***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)5248                                                                    0.179           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5252                                                                    0.088           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5263                                                                   0.494***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)5274                                                                   0.756***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5335                                                                    0.318*          
                                                                                 (0.177)          
                                                                                                  
factor(nr)5345                                                                    0.233           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5359                                                                    -0.082          
                                                                                 (0.177)          
                                                                                                  
factor(nr)5368                                                                    0.060           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5377                                                                   0.382**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)5390                                                                    0.288           
                                                                                 (0.177)          
                                                                                                  
factor(nr)5419                                                                    0.125           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5435                                                                    0.033           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5437                                                                    0.265           
                                                                                 (0.177)          
                                                                                                  
factor(nr)5497                                                                    0.195           
                                                                                 (0.177)          
                                                                                                  
factor(nr)5525                                                                    0.315*          
                                                                                 (0.180)          
                                                                                                  
factor(nr)5529                                                                   0.573***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)5531                                                                   0.927***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)5579                                                                    0.229           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5588                                                                   0.936***         
                                                                                 (0.194)          
                                                                                                  
factor(nr)5599                                                                   0.569***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5650                                                                    0.330*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)5660                                                                   0.724***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5665                                                                    0.155           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5666                                                                   0.495***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5698                                                                   0.686***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)5699                                                                   0.679***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)5731                                                                    0.318*          
                                                                                 (0.177)          
                                                                                                  
factor(nr)5750                                                                   0.661***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)5755                                                                   0.546***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)5772                                                                    0.126           
                                                                                 (0.179)          
                                                                                                  
factor(nr)5816                                                                    0.211           
                                                                                 (0.177)          
                                                                                                  
factor(nr)5823                                                                    0.107           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5851                                                                    0.061           
                                                                                 (0.176)          
                                                                                                  
factor(nr)5857                                                                   0.438**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)5859                                                                   0.427**          
                                                                                 (0.177)          
                                                                                                  
factor(nr)6016                                                                   0.794***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)6020                                                                    -0.282          
                                                                                 (0.177)          
                                                                                                  
factor(nr)6025                                                                   -0.425**         
                                                                                 (0.177)          
                                                                                                  
factor(nr)6056                                                                    -0.104          
                                                                                 (0.176)          
                                                                                                  
factor(nr)6094                                                                   0.484***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)6186                                                                   0.487***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)6395                                                                   0.979***         
                                                                                 (0.188)          
                                                                                                  
factor(nr)6430                                                                   0.403**          
                                                                                 (0.180)          
                                                                                                  
factor(nr)6446                                                                   0.555***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)6463                                                                    0.352*          
                                                                                 (0.187)          
                                                                                                  
factor(nr)6558                                                                   0.807***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)6559                                                                   0.581***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)6561                                                                   0.953***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)6574                                                                    0.098           
                                                                                 (0.178)          
                                                                                                  
factor(nr)6648                                                                   0.774***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)6813                                                                    0.274           
                                                                                 (0.177)          
                                                                                                  
factor(nr)6824                                                                   0.419**          
                                                                                 (0.179)          
                                                                                                  
factor(nr)6888                                                                   0.494***         
                                                                                 (0.187)          
                                                                                                  
factor(nr)6942                                                                   0.549***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)6954                                                                   0.565***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)6955                                                                   0.678***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)6964                                                                   0.473***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)6987                                                                   1.456***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)7025                                                                   0.906***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)7043                                                                   0.498***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)7060                                                                   0.460***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)7087                                                                    -0.207          
                                                                                 (0.176)          
                                                                                                  
factor(nr)7238                                                                   0.680***         
                                                                                 (0.193)          
                                                                                                  
factor(nr)7279                                                                    0.270           
                                                                                 (0.178)          
                                                                                                  
factor(nr)7342                                                                   0.432**          
                                                                                 (0.179)          
                                                                                                  
factor(nr)7343                                                                   0.350**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)7411                                                                    -0.038          
                                                                                 (0.176)          
                                                                                                  
factor(nr)7424                                                                   0.651***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)7429                                                                   0.515***         
                                                                                 (0.182)          
                                                                                                  
factor(nr)7454                                                                   0.730***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)7472                                                                   0.402**          
                                                                                 (0.177)          
                                                                                                  
factor(nr)7474                                                                    0.303*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)7509                                                                   0.689***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)7539                                                                    -0.012          
                                                                                 (0.176)          
                                                                                                  
factor(nr)7769                                                                   0.655***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)7783                                                                   0.449**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)7784                                                                   1.929***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)7801                                                                    -0.048          
                                                                                 (0.176)          
                                                                                                  
factor(nr)7824                                                                   0.546***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)7874                                                                   1.107***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)7887                                                                   0.639***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)7923                                                                   1.238***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)7926                                                                   0.708***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)8021                                                                   0.824***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8087                                                                   0.460***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8089                                                                    0.222           
                                                                                 (0.177)          
                                                                                                  
factor(nr)8090                                                                   1.226***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8096                                                                   1.299***         
                                                                                 (0.221)          
                                                                                                  
factor(nr)8106                                                                   0.948***         
                                                                                 (0.200)          
                                                                                                  
factor(nr)8107                                                                   1.186***         
                                                                                 (0.221)          
                                                                                                  
factor(nr)8142                                                                    0.182           
                                                                                 (0.176)          
                                                                                                  
factor(nr)8168                                                                   0.551***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)8173                                                                   0.909***         
                                                                                 (0.202)          
                                                                                                  
factor(nr)8203                                                                   1.072***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)8211                                                                    0.294           
                                                                                 (0.183)          
                                                                                                  
factor(nr)8224                                                                   0.807***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)8272                                                                   0.959***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)8300                                                                   0.386**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)8304                                                                   0.470***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8364                                                                   0.348**          
                                                                                 (0.177)          
                                                                                                  
factor(nr)8370                                                                   0.845***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)8381                                                                    0.218           
                                                                                 (0.176)          
                                                                                                  
factor(nr)8388                                                                   0.743***         
                                                                                 (0.194)          
                                                                                                  
factor(nr)8406                                                                   0.626***         
                                                                                 (0.184)          
                                                                                                  
factor(nr)8415                                                                    0.054           
                                                                                 (0.176)          
                                                                                                  
factor(nr)8496                                                                    -0.112          
                                                                                 (0.176)          
                                                                                                  
factor(nr)8501                                                                    -0.070          
                                                                                 (0.179)          
                                                                                                  
factor(nr)8518                                                                   1.243***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)8520                                                                   -0.353**         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8524                                                                    -0.118          
                                                                                 (0.177)          
                                                                                                  
factor(nr)8548                                                                    0.115           
                                                                                 (0.176)          
                                                                                                  
factor(nr)8556                                                                   0.625***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8564                                                                    0.106           
                                                                                 (0.176)          
                                                                                                  
factor(nr)8581                                                                    -0.249          
                                                                                 (0.176)          
                                                                                                  
factor(nr)8586                                                                    0.258           
                                                                                 (0.176)          
                                                                                                  
factor(nr)8587                                                                   -0.370**         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8597                                                                   0.771***         
                                                                                 (0.183)          
                                                                                                  
factor(nr)8656                                                                    0.153           
                                                                                 (0.176)          
                                                                                                  
factor(nr)8722                                                                   0.637***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)8743                                                                   0.515***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)8749                                                                   0.940***         
                                                                                 (0.202)          
                                                                                                  
factor(nr)8758                                                                   0.535***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)8796                                                                   0.780***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8838                                                                   0.406**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)8842                                                                    0.145           
                                                                                 (0.176)          
                                                                                                  
factor(nr)8846                                                                    0.112           
                                                                                 (0.177)          
                                                                                                  
factor(nr)8860                                                                    0.124           
                                                                                 (0.178)          
                                                                                                  
factor(nr)8862                                                                   0.399**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)8880                                                                   1.219***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)8886                                                                   0.631***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)8903                                                                    -0.239          
                                                                                 (0.176)          
                                                                                                  
factor(nr)8908                                                                    -0.023          
                                                                                 (0.177)          
                                                                                                  
factor(nr)8911                                                                    -0.034          
                                                                                 (0.177)          
                                                                                                  
factor(nr)8917                                                                   0.796***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)8991                                                                   0.913***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)8997                                                                   0.898***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9014                                                                   0.540***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)9015                                                                   0.697***         
                                                                                 (0.187)          
                                                                                                  
factor(nr)9027                                                                   1.002***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9066                                                                    -0.153          
                                                                                 (0.176)          
                                                                                                  
factor(nr)9082                                                                    0.274           
                                                                                 (0.178)          
                                                                                                  
factor(nr)9131                                                                   0.758***         
                                                                                 (0.201)          
                                                                                                  
factor(nr)9132                                                                   0.709***         
                                                                                 (0.188)          
                                                                                                  
factor(nr)9154                                                                   1.113***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9158                                                                    0.086           
                                                                                 (0.179)          
                                                                                                  
factor(nr)9184                                                                    0.164           
                                                                                 (0.177)          
                                                                                                  
factor(nr)9230                                                                    0.158           
                                                                                 (0.176)          
                                                                                                  
factor(nr)9265                                                                   0.437**          
                                                                                 (0.175)          
                                                                                                  
factor(nr)9367                                                                    0.207           
                                                                                 (0.176)          
                                                                                                  
factor(nr)9390                                                                   0.745***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)9391                                                                   0.370**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)9418                                                                   1.133***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9424                                                                   0.806***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)9447                                                                   0.503***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9449                                                                    0.318*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)9453                                                                   0.745***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9468                                                                   0.493***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9502                                                                   0.895***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)9505                                                                    0.046           
                                                                                 (0.179)          
                                                                                                  
factor(nr)9603                                                                    0.310*          
                                                                                 (0.188)          
                                                                                                  
factor(nr)9643                                                                   0.433**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)9667                                                                   1.067***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9683                                                                    0.128           
                                                                                 (0.180)          
                                                                                                  
factor(nr)9694                                                                   0.847***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9710                                                                    0.280           
                                                                                 (0.177)          
                                                                                                  
factor(nr)9718                                                                   0.938***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9725                                                                    0.322*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)9744                                                                   0.689***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9752                                                                   1.390***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9776                                                                   0.953***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)9786                                                                   0.624***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9791                                                                    0.328*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)9794                                                                   0.473***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)9810                                                                    -0.144          
                                                                                 (0.176)          
                                                                                                  
factor(nr)9846                                                                   0.825***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)9859                                                                    0.148           
                                                                                 (0.178)          
                                                                                                  
factor(nr)9868                                                                    0.029           
                                                                                 (0.176)          
                                                                                                  
factor(nr)9876                                                                   0.853***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)9883                                                                   0.714***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)9889                                                                   0.988***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)9901                                                                   0.689***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)9936                                                                   0.895***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)9964                                                                   0.412**          
                                                                                 (0.177)          
                                                                                                  
factor(nr)10043                                                                   -0.172          
                                                                                 (0.178)          
                                                                                                  
factor(nr)10067                                                                  0.510***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)10091                                                                  0.697***         
                                                                                 (0.181)          
                                                                                                  
factor(nr)10120                                                                   -0.118          
                                                                                 (0.176)          
                                                                                                  
factor(nr)10121                                                                   0.325*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)10167                                                                  0.567***         
                                                                                 (0.178)          
                                                                                                  
factor(nr)10209                                                                   0.112           
                                                                                 (0.176)          
                                                                                                  
factor(nr)10230                                                                  0.724***         
                                                                                 (0.180)          
                                                                                                  
factor(nr)10265                                                                  0.993***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)10274                                                                   0.237           
                                                                                 (0.176)          
                                                                                                  
factor(nr)10311                                                                   0.301*          
                                                                                 (0.176)          
                                                                                                  
factor(nr)10392                                                                   0.108           
                                                                                 (0.177)          
                                                                                                  
factor(nr)10425                                                                  0.945***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)10441                                                                   0.176           
                                                                                 (0.176)          
                                                                                                  
factor(nr)10457                                                                  0.446**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)10469                                                                  0.447**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)10524                                                                   0.088           
                                                                                 (0.177)          
                                                                                                  
factor(nr)10552                                                                   -0.185          
                                                                                 (0.176)          
                                                                                                  
factor(nr)10553                                                                  0.706***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)10570                                                                 -0.747***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)10593                                                                   0.284           
                                                                                 (0.184)          
                                                                                                  
factor(nr)10666                                                                  -0.324*          
                                                                                 (0.177)          
                                                                                                  
factor(nr)11275                                                                   0.146           
                                                                                 (0.176)          
                                                                                                  
factor(nr)11328                                                                  0.495***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)11750                                                                  0.384**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)11821                                                                  0.808***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)11857                                                                  0.934***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)11887                                                                  0.396**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)11890                                                                  1.093***         
                                                                                 (0.211)          
                                                                                                  
factor(nr)11892                                                                  1.132***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)11924                                                                  0.354**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)11925                                                                  0.408**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)11957                                                                   0.004           
                                                                                 (0.177)          
                                                                                                  
factor(nr)11973                                                                  0.578***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)11990                                                                  1.010***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)12012                                                                   0.224           
                                                                                 (0.176)          
                                                                                                  
factor(nr)12013                                                                  0.568***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)12045                                                                   0.200           
                                                                                 (0.176)          
                                                                                                  
factor(nr)12055                                                                  0.674***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)12084                                                                  0.450**          
                                                                                 (0.179)          
                                                                                                  
factor(nr)12088                                                                   0.117           
                                                                                 (0.176)          
                                                                                                  
factor(nr)12122                                                                  0.586***         
                                                                                 (0.202)          
                                                                                                  
factor(nr)12179                                                                   0.301*          
                                                                                 (0.177)          
                                                                                                  
factor(nr)12182                                                                  0.515***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)12220                                                                  0.673***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)12221                                                                   -0.121          
                                                                                 (0.183)          
                                                                                                  
factor(nr)12245                                                                   -0.110          
                                                                                 (0.176)          
                                                                                                  
factor(nr)12276                                                                  0.655***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)12385                                                                  0.819***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)12410                                                                   0.040           
                                                                                 (0.178)          
                                                                                                  
factor(nr)12420                                                                  0.840***         
                                                                                 (0.177)          
                                                                                                  
factor(nr)12433                                                                  0.393**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)12451                                                                  0.431**          
                                                                                 (0.176)          
                                                                                                  
factor(nr)12477                                                                  1.004***         
                                                                                 (0.179)          
                                                                                                  
factor(nr)12500                                                                   0.222           
                                                                                 (0.178)          
                                                                                                  
factor(nr)12534                                                                  0.942***         
                                                                                 (0.176)          
                                                                                                  
factor(nr)12548                                                                   0.347*          
                                                                                 (0.180)          
                                                                                                  
educ                                                                                              
                                                                                                  
                                                                                                  
black                                                                                             
                                                                                                  
                                                                                                  
hisp                                                                                              
                                                                                                  
                                                                                                  
exper                                                                                             
                                                                                                  
                                                                                                  
Constant                    -0.445***                                            0.933***         
                             (0.030)                                             (0.125)          
                                                                                                  
--------------------------------------------------------------------------------------------------
Observations                  4,360                     4,360                     4,360           
R2                            0.181                     0.181                     0.621           
Adjusted R2                   0.179                     0.061                     0.566           
Residual Std. Error     0.328 (df = 4349)                                   0.351 (df = 3805)     
F Statistic         95.840*** (df = 10; 4349) 83.851*** (df = 10; 3805) 11.250*** (df = 554; 3805)
==================================================================================================
Note:                                                                  *p<0.1; **p<0.05; ***p<0.01
Stata
use "files/WAGEPAN.dta" , clear

bys nr:  egen lwage_mean = mean(lwage) 
bys nr:  egen expersq_mean = mean(expersq) 
bys nr:  egen married_mean = mean(married) 
bys nr:  egen union_mean = mean(union)

gen lwage_demean = lwage - lwage_mean
gen expersq_demean = expersq - expersq_mean
gen married_demean = married - married_mean
gen union_demean = union - union_mean

xtset nr year 
eststo: qui reg lwage_demean expersq_demean married_demean union_demean i.year
eststo: qui xtreg lwage expersq married union i.year  educ black hisp exper , fe
eststo: qui reg lwage expersq married union i.year i.nr  educ black hisp exper 
esttab , mtitles("Demean" "FE" "LSDV") compress
Panel variable: nr (strongly balanced)
 Time variable: year, 1980 to 1987
         Delta: 1 unit

(est1 stored)

(est2 stored)

(est3 stored)


-------------------------------------------------
                 (1)          (2)          (3)   
              Demean           FE         LSDV   
-------------------------------------------------
expe~emean  -0.00519***                          
             (-7.87)                             

marr~emean    0.0467**                           
              (2.73)                             

union_de~n    0.0800***                          
              (4.43)                             

1980.year          0            0            0   
                 (.)          (.)          (.)   

1981.year      0.151***     0.151***     0.151***
              (7.36)       (6.89)       (6.89)   

1982.year      0.253***     0.253***     0.253***
             (11.08)      (10.36)      (10.36)   

1983.year      0.354***     0.354***     0.354***
             (12.96)      (12.12)      (12.12)   

1984.year      0.490***     0.490***     0.490***
             (14.46)      (13.53)      (13.53)   

1985.year      0.617***     0.617***     0.617***
             (14.59)      (13.65)      (13.65)   

1986.year      0.765***     0.765***     0.765***
             (14.58)      (13.64)      (13.64)   

1987.year      0.925***     0.925***     0.925***
             (14.38)      (13.45)      (13.45)   

expersq                  -0.00519***  -0.00519***
                          (-7.36)      (-7.36)   

married                    0.0467*      0.0467*  
                           (2.55)       (2.55)   

union                      0.0800***    0.0800***
                           (4.14)       (4.14)   

educ                            0            0   
                              (.)          (.)   

black                           0            0   
                              (.)          (.)   

hisp                            0            0   
                              (.)          (.)   

exper                           0            0   
                              (.)          (.)   

13.nr                                        0   
                                           (.)   

17.nr                                    0.579** 
                                        (3.26)   

18.nr                                    0.929***
                                        (5.20)   

45.nr                                    0.554** 
                                        (3.15)   

110.nr                                   1.046***
                                        (5.82)   

120.nr                                   0.239   
                                        (1.36)   

126.nr                                   0.783***
                                        (4.45)   

150.nr                                  -0.140   
                                       (-0.80)   

162.nr                                   0.269   
                                        (1.53)   

166.nr                                   0.138   
                                        (0.78)   

189.nr                                   0.191   
                                        (1.09)   

193.nr                                   0.981***
                                        (5.58)   

209.nr                                   0.707***
                                        (4.01)   

212.nr                                   0.833***
                                        (4.74)   

218.nr                                   1.231***
                                        (6.99)   

243.nr                                   0.586***
                                        (3.29)   

259.nr                                   0.478** 
                                        (2.69)   

260.nr                                   0.789***
                                        (4.48)   

309.nr                                   1.052***
                                        (5.95)   

351.nr                                   0.358*  
                                        (2.02)   

353.nr                                   0.583***
                                        (3.31)   

383.nr                                   0.564** 
                                        (3.21)   

408.nr                                   0.801***
                                        (4.54)   

424.nr                                   1.345***
                                        (7.45)   

464.nr                                  0.0803   
                                        (0.46)   

483.nr                                   0.713***
                                        (3.98)   

556.nr                                   1.089***
                                        (6.20)   

560.nr                                   0.205   
                                        (1.17)   

569.nr                                  -0.431*  
                                       (-2.45)   

583.nr                                   0.852***
                                        (4.80)   

647.nr                                   0.360*  
                                        (2.04)   

658.nr                                   0.838***
                                        (4.73)   

684.nr                                   1.132***
                                        (6.42)   

711.nr                                 -0.0972   
                                       (-0.55)   

729.nr                                   0.837***
                                        (4.74)   

731.nr                                   0.574** 
                                        (3.23)   

732.nr                                   0.531** 
                                        (3.02)   

793.nr                                   0.498** 
                                        (2.84)   

797.nr                                   0.841***
                                        (4.75)   

800.nr                                   0.380*  
                                        (2.15)   

813.nr                                  -0.657***
                                       (-3.74)   

823.nr                                  -0.393*  
                                       (-2.23)   

827.nr                                   0.507** 
                                        (2.88)   

847.nr                                   0.489** 
                                        (2.79)   

851.nr                                   0.735***
                                        (4.17)   

863.nr                                   0.550** 
                                        (3.13)   

873.nr                                  -0.127   
                                       (-0.72)   

891.nr                                   0.735***
                                        (4.16)   

908.nr                                  -0.405*  
                                       (-2.30)   

910.nr                                   0.679***
                                        (3.87)   

916.nr                                   0.564** 
                                        (3.21)   

919.nr                                   0.238   
                                        (1.34)   

922.nr                                   1.011***
                                        (5.76)   

924.nr                                 -0.0709   
                                       (-0.40)   

925.nr                                   0.809***
                                        (4.60)   

944.nr                                   0.106   
                                        (0.60)   

945.nr                                   0.106   
                                        (0.60)   

947.nr                                   1.154***
                                        (6.52)   

955.nr                                   1.006***
                                        (5.64)   

965.nr                                   0.434*  
                                        (2.42)   

996.nr                                 0.00740   
                                        (0.04)   

1007.nr                                  0.996***
                                        (5.61)   

1054.nr                                  0.133   
                                        (0.75)   

1064.nr                                  0.858***
                                        (4.81)   

1081.nr                                  0.179   
                                        (1.02)   

1085.nr                                  0.114   
                                        (0.65)   

1091.nr                                  0.226   
                                        (1.23)   

1094.nr                                  0.600***
                                        (3.38)   

1096.nr                                  0.251   
                                        (1.37)   

1098.nr                                  0.837***
                                        (4.77)   

1102.nr                                  0.948***
                                        (5.38)   

1107.nr                                  0.762***
                                        (4.23)   

1142.nr                                  0.715***
                                        (4.07)   

1156.nr                                  0.569** 
                                        (3.17)   

1180.nr                                  0.636***
                                        (3.61)   

1190.nr                                  0.119   
                                        (0.68)   

1204.nr                                  0.888***
                                        (4.93)   

1249.nr                                  0.468** 
                                        (2.60)   

1272.nr                                  0.612***
                                        (3.47)   

1311.nr                                  0.481** 
                                        (2.74)   

1316.nr                                  0.921***
                                        (5.17)   

1318.nr                                  0.625** 
                                        (3.25)   

1345.nr                                  0.549** 
                                        (3.08)   

1397.nr                                  0.653***
                                        (3.62)   

1434.nr                                -0.0818   
                                       (-0.46)   

1492.nr                                  0.408*  
                                        (2.30)   

1496.nr                                 -0.219   
                                       (-1.24)   

1506.nr                                 0.0822   
                                        (0.47)   

1515.nr                                  0.281   
                                        (1.60)   

1520.nr                                  0.375*  
                                        (2.07)   

1528.nr                                  0.136   
                                        (0.77)   

1554.nr                                  0.312   
                                        (1.77)   

1575.nr                                  0.577** 
                                        (3.26)   

1576.nr                                  0.504** 
                                        (2.86)   

1628.nr                                  0.440*  
                                        (2.50)   

1641.nr                                  0.974***
                                        (5.46)   

1644.nr                                  0.263   
                                        (1.50)   

1653.nr                                  0.629***
                                        (3.58)   

1654.nr                                  0.663***
                                        (3.73)   

1721.nr                                  1.160***
                                        (6.47)   

1742.nr                                  0.861***
                                        (4.90)   

1744.nr                                  0.557** 
                                        (3.16)   

1763.nr                                 -0.194   
                                       (-1.11)   

1777.nr                                  0.115   
                                        (0.65)   

1843.nr                                  1.310***
                                        (7.41)   

1891.nr                                -0.0896   
                                       (-0.51)   

1895.nr                                  0.604***
                                        (3.41)   

1899.nr                                  0.171   
                                        (0.97)   

1925.nr                                  0.488** 
                                        (2.78)   

1930.nr                                  0.251   
                                        (1.43)   

1961.nr                                  0.745***
                                        (4.18)   

1963.nr                                  0.470** 
                                        (2.66)   

1979.nr                                  0.828***
                                        (4.63)   

1988.nr                                -0.0621   
                                       (-0.35)   

2000.nr                                  0.637***
                                        (3.60)   

2014.nr                                  0.310   
                                        (1.77)   

2025.nr                                  0.432*  
                                        (2.45)   

2038.nr                                  0.180   
                                        (1.02)   

2075.nr                                  0.928***
                                        (5.14)   

2101.nr                                  1.158***
                                        (6.57)   

2106.nr                                  0.595***
                                        (3.33)   

2107.nr                                  0.460** 
                                        (2.62)   

2108.nr                                  0.816***
                                        (4.61)   

2147.nr                                 -0.860***
                                       (-4.90)   

2157.nr                                 0.0546   
                                        (0.30)   

2163.nr                                  1.092***
                                        (6.09)   

2173.nr                                  0.148   
                                        (0.84)   

2180.nr                                  0.411*  
                                        (2.34)   

2183.nr                                  0.392*  
                                        (2.23)   

2216.nr                                  0.540** 
                                        (3.07)   

2220.nr                                  0.991***
                                        (5.63)   

2227.nr                                  0.255   
                                        (1.44)   

2264.nr                                -0.0930   
                                       (-0.53)   

2306.nr                                -0.0521   
                                       (-0.29)   

2312.nr                                  0.816***
                                        (4.64)   

2314.nr                                  0.969***
                                        (5.39)   

2329.nr                                  0.545** 
                                        (3.09)   

2335.nr                                  0.506** 
                                        (2.82)   

2341.nr                                  0.471** 
                                        (2.68)   

2351.nr                                 0.0458   
                                        (0.26)   

2386.nr                                 -0.154   
                                       (-0.87)   

2401.nr                                  0.644***
                                        (3.57)   

2413.nr                                  0.999***
                                        (5.59)   

2421.nr                                 0.0360   
                                        (0.20)   

2445.nr                                  0.463** 
                                        (2.60)   

2451.nr                                  0.592***
                                        (3.35)   

2494.nr                                  0.168   
                                        (0.94)   

2508.nr                                  0.848***
                                        (4.79)   

2535.nr                                  0.486** 
                                        (2.72)   

2540.nr                                -0.0591   
                                       (-0.34)   

2685.nr                                  1.031***
                                        (5.87)   

2711.nr                                 0.0516   
                                        (0.29)   

2718.nr                                  1.059***
                                        (5.98)   

2721.nr                                  0.157   
                                        (0.86)   

2733.nr                                 -0.146   
                                       (-0.82)   

2741.nr                                  0.419*  
                                        (2.32)   

2745.nr                                  0.556** 
                                        (3.15)   

2751.nr                                  0.456*  
                                        (2.55)   

2774.nr                                  0.903***
                                        (5.09)   

2801.nr                                 0.0497   
                                        (0.28)   

2813.nr                                  0.222   
                                        (1.26)   

2833.nr                                  0.621***
                                        (3.52)   

2839.nr                                  0.646***
                                        (3.66)   

2842.nr                                  0.942***
                                        (5.31)   

2866.nr                                  0.707***
                                        (3.99)   

2868.nr                                  0.978***
                                        (5.54)   

2874.nr                                0.00693   
                                        (0.04)   

2916.nr                                  0.345*  
                                        (1.96)   

2951.nr                                  0.313   
                                        (1.71)   

2980.nr                                  0.327   
                                        (1.85)   

2994.nr                                  0.506** 
                                        (2.86)   

2997.nr                                  0.601***
                                        (3.41)   

3017.nr                                  0.936***
                                        (5.31)   

3037.nr                                  1.040***
                                        (5.81)   

3059.nr                                  0.694***
                                        (3.84)   

3062.nr                                  1.166***
                                        (6.64)   

3100.nr                                  0.637***
                                        (3.57)   

3102.nr                                 -0.259   
                                       (-1.48)   

3127.nr                                 -0.407*  
                                       (-2.31)   

3136.nr                                  0.698***
                                        (3.94)   

3137.nr                                  0.892***
                                        (5.08)   

3138.nr                                  0.665***
                                        (3.75)   

3140.nr                                  0.843***
                                        (4.77)   

3193.nr                                  0.686***
                                        (3.90)   

3196.nr                                  0.681***
                                        (3.85)   

3200.nr                                  0.771***
                                        (4.12)   

3202.nr                                  0.593***
                                        (3.34)   

3207.nr                                  0.471** 
                                        (2.64)   

3208.nr                                  0.855***
                                        (4.82)   

3210.nr                                  0.708***
                                        (4.02)   

3215.nr                                  0.324   
                                        (1.82)   

3219.nr                                  0.237   
                                        (1.35)   

3226.nr                                  0.642***
                                        (3.64)   

3235.nr                                  0.144   
                                        (0.81)   

3239.nr                                 -0.375*  
                                       (-2.14)   

3271.nr                                  0.756***
                                        (4.15)   

3275.nr                                  0.542** 
                                        (3.09)   

3282.nr                                  0.660***
                                        (3.75)   

3289.nr                                  0.454** 
                                        (2.58)   

3290.nr                                  0.456** 
                                        (2.60)   

3307.nr                                  1.321***
                                        (7.47)   

3333.nr                                -0.0781   
                                       (-0.44)   

3353.nr                                 0.0825   
                                        (0.47)   

3380.nr                                  0.522** 
                                        (2.96)   

3381.nr                                  0.587***
                                        (3.34)   

3389.nr                                  0.672***
                                        (3.81)   

3394.nr                                  0.656***
                                        (3.73)   

3401.nr                                  0.397*  
                                        (2.25)   

3414.nr                                  0.701***
                                        (3.99)   

3420.nr                                  0.301   
                                        (1.71)   

3440.nr                                 -0.211   
                                       (-1.20)   

3461.nr                                  0.711***
                                        (4.05)   

3468.nr                                  0.195   
                                        (1.11)   

3482.nr                                  1.056***
                                        (5.99)   

3495.nr                                  0.538** 
                                        (2.99)   

3503.nr                                  0.965***
                                        (5.42)   

3525.nr                                  1.040***
                                        (5.93)   

3526.nr                                  0.594***
                                        (3.33)   

3538.nr                                  0.962***
                                        (5.33)   

3563.nr                                  0.636***
                                        (3.60)   

3575.nr                                  0.997***
                                        (5.59)   

3580.nr                                  0.361*  
                                        (2.06)   

3581.nr                                  0.671***
                                        (3.82)   

3589.nr                                  0.287   
                                        (1.62)   

3591.nr                                  0.674***
                                        (3.82)   

3598.nr                                  0.864***
                                        (4.90)   

3602.nr                                  1.082***
                                        (6.00)   

3607.nr                                -0.0189   
                                       (-0.10)   

3621.nr                                  0.528** 
                                        (2.97)   

3628.nr                                  0.984***
                                        (5.55)   

3653.nr                                  0.723***
                                        (4.10)   

3706.nr                                  0.650***
                                        (3.67)   

3707.nr                                  0.556** 
                                        (3.15)   

3708.nr                                  0.617***
                                        (3.51)   

3743.nr                                  0.607***
                                        (3.45)   

3777.nr                                  0.732***
                                        (4.13)   

3831.nr                                  0.227   
                                        (1.29)   

3844.nr                                  0.510** 
                                        (2.89)   

3847.nr                                  1.104***
                                        (6.22)   

3848.nr                                  0.651***
                                        (3.71)   

3882.nr                                 0.0468   
                                        (0.27)   

3937.nr                                  0.520** 
                                        (2.94)   

4000.nr                                  1.072***
                                        (5.34)   

4004.nr                                  0.976***
                                        (5.50)   

4025.nr                                  0.135   
                                        (0.77)   

4032.nr                                  0.157   
                                        (0.89)   

4046.nr                                  0.564** 
                                        (3.07)   

4088.nr                                  1.447***
                                        (8.16)   

4091.nr                                  1.271***
                                        (7.14)   

4122.nr                                  0.713***
                                        (4.04)   

4127.nr                                 0.0934   
                                        (0.53)   

4128.nr                                  0.297   
                                        (1.67)   

4159.nr                                  0.605***
                                        (3.43)   

4204.nr                                  0.324   
                                        (1.82)   

4229.nr                                  0.314   
                                        (1.79)   

4258.nr                                  0.718***
                                        (4.07)   

4261.nr                                  1.342***
                                        (6.70)   

4264.nr                                 0.0729   
                                        (0.41)   

4278.nr                                  0.594***
                                        (3.35)   

4297.nr                                  0.538** 
                                        (3.05)   

4302.nr                                 -0.129   
                                       (-0.74)   

4321.nr                                  0.473** 
                                        (2.68)   

4328.nr                                  0.239   
                                        (1.35)   

4332.nr                                 -0.371*  
                                       (-2.11)   

4335.nr                                  0.342   
                                        (1.93)   

4357.nr                                  0.527** 
                                        (2.98)   

4365.nr                                -0.0388   
                                       (-0.22)   

4380.nr                                  0.880***
                                        (4.94)   

4394.nr                                  0.724***
                                        (4.12)   

4510.nr                                  0.643***
                                        (3.66)   

4559.nr                                  0.374*  
                                        (2.08)   

4563.nr                                 0.0190   
                                        (0.11)   

4569.nr                                  0.791***
                                        (4.48)   

4603.nr                                  0.734***
                                        (4.16)   

4607.nr                                  0.166   
                                        (0.94)   

4633.nr                                  0.505** 
                                        (2.86)   

4676.nr                                  0.125   
                                        (0.71)   

4701.nr                                  0.952***
                                        (5.42)   

4716.nr                                  0.122   
                                        (0.69)   

4720.nr                                  1.137***
                                        (6.28)   

4759.nr                                  0.596***
                                        (3.33)   

4791.nr                                  0.678***
                                        (3.74)   

4811.nr                                 0.0962   
                                        (0.55)   

4828.nr                                  0.108   
                                        (0.61)   

4857.nr                                -0.0984   
                                       (-0.56)   

4858.nr                                  0.976***
                                        (5.54)   

4859.nr                                 0.0313   
                                        (0.18)   

4866.nr                                  0.818***
                                        (4.62)   

4881.nr                                  0.358*  
                                        (2.02)   

4884.nr                                  0.522** 
                                        (2.93)   

4888.nr                                 0.0963   
                                        (0.55)   

4901.nr                                  0.218   
                                        (1.22)   

4917.nr                                 0.0595   
                                        (0.34)   

4926.nr                                  0.440*  
                                        (2.47)   

4982.nr                                  0.102   
                                        (0.57)   

5017.nr                                  1.088***
                                        (6.19)   

5033.nr                                  1.061***
                                        (6.03)   

5048.nr                                  0.315   
                                        (1.79)   

5122.nr                                  0.657***
                                        (3.70)   

5141.nr                                  0.711***
                                        (4.03)   

5147.nr                                  1.045***
                                        (5.84)   

5158.nr                                  0.700***
                                        (3.97)   

5221.nr                                  0.538** 
                                        (3.04)   

5223.nr                                  0.486** 
                                        (2.76)   

5227.nr                                  0.652***
                                        (3.63)   

5248.nr                                  0.179   
                                        (1.02)   

5252.nr                                 0.0879   
                                        (0.50)   

5263.nr                                  0.494** 
                                        (2.79)   

5274.nr                                  0.756***
                                        (4.28)   

5335.nr                                  0.318   
                                        (1.80)   

5345.nr                                  0.233   
                                        (1.33)   

5359.nr                                -0.0821   
                                       (-0.46)   

5368.nr                                 0.0598   
                                        (0.34)   

5377.nr                                  0.382*  
                                        (2.18)   

5390.nr                                  0.288   
                                        (1.63)   

5419.nr                                  0.125   
                                        (0.71)   

5435.nr                                 0.0333   
                                        (0.19)   

5437.nr                                  0.265   
                                        (1.50)   

5497.nr                                  0.195   
                                        (1.10)   

5525.nr                                  0.315   
                                        (1.75)   

5529.nr                                  0.573** 
                                        (3.21)   

5531.nr                                  0.927***
                                        (5.16)   

5579.nr                                  0.229   
                                        (1.30)   

5588.nr                                  0.936***
                                        (4.83)   

5599.nr                                  0.569** 
                                        (3.23)   

5650.nr                                  0.330   
                                        (1.87)   

5660.nr                                  0.724***
                                        (4.11)   

5665.nr                                  0.155   
                                        (0.88)   

5666.nr                                  0.495** 
                                        (2.81)   

5698.nr                                  0.686***
                                        (3.84)   

5699.nr                                  0.679***
                                        (3.85)   

5731.nr                                  0.318   
                                        (1.80)   

5750.nr                                  0.661***
                                        (3.66)   

5755.nr                                  0.546** 
                                        (3.06)   

5772.nr                                  0.126   
                                        (0.71)   

5816.nr                                  0.211   
                                        (1.19)   

5823.nr                                  0.107   
                                        (0.61)   

5851.nr                                 0.0606   
                                        (0.35)   

5857.nr                                  0.438*  
                                        (2.49)   

5859.nr                                  0.427*  
                                        (2.42)   

6016.nr                                  0.794***
                                        (4.52)   

6020.nr                                 -0.282   
                                       (-1.59)   

6025.nr                                 -0.425*  
                                       (-2.41)   

6056.nr                                 -0.104   
                                       (-0.59)   

6094.nr                                  0.484** 
                                        (2.72)   

6186.nr                                  0.487** 
                                        (2.76)   

6395.nr                                  0.979***
                                        (5.21)   

6430.nr                                  0.403*  
                                        (2.24)   

6446.nr                                  0.555** 
                                        (3.14)   

6463.nr                                  0.352   
                                        (1.88)   

6558.nr                                  0.807***
                                        (4.58)   

6559.nr                                  0.581** 
                                        (3.29)   

6561.nr                                  0.953***
                                        (5.42)   

6574.nr                                 0.0977   
                                        (0.55)   

6648.nr                                  0.774***
                                        (4.40)   

6813.nr                                  0.274   
                                        (1.55)   

6824.nr                                  0.419*  
                                        (2.34)   

6888.nr                                  0.494** 
                                        (2.64)   

6942.nr                                  0.549** 
                                        (3.12)   

6954.nr                                  0.565** 
                                        (3.20)   

6955.nr                                  0.678***
                                        (3.85)   

6964.nr                                  0.473** 
                                        (2.64)   

6987.nr                                  1.456***
                                        (8.26)   

7025.nr                                  0.906***
                                        (5.07)   

7043.nr                                  0.498** 
                                        (2.82)   

7060.nr                                  0.460** 
                                        (2.61)   

7087.nr                                 -0.207   
                                       (-1.18)   

7238.nr                                  0.680***
                                        (3.52)   

7279.nr                                  0.270   
                                        (1.51)   

7342.nr                                  0.432*  
                                        (2.41)   

7343.nr                                  0.350*  
                                        (1.99)   

7411.nr                                -0.0378   
                                       (-0.21)   

7424.nr                                  0.651***
                                        (3.70)   

7429.nr                                  0.515** 
                                        (2.83)   

7454.nr                                  0.730***
                                        (4.16)   

7472.nr                                  0.402*  
                                        (2.27)   

7474.nr                                  0.303   
                                        (1.72)   

7509.nr                                  0.689***
                                        (3.80)   

7539.nr                                -0.0121   
                                       (-0.07)   

7769.nr                                  0.655***
                                        (3.71)   

7783.nr                                  0.449*  
                                        (2.55)   

7784.nr                                  1.929***
                                       (10.99)   

7801.nr                                -0.0478   
                                       (-0.27)   

7824.nr                                  0.546** 
                                        (3.11)   

7874.nr                                  1.107***
                                        (6.30)   

7887.nr                                  0.639***
                                        (3.59)   

7923.nr                                  1.238***
                                        (6.88)   

7926.nr                                  0.708***
                                        (3.91)   

8021.nr                                  0.824***
                                        (4.69)   

8087.nr                                  0.460** 
                                        (2.61)   

8089.nr                                  0.222   
                                        (1.26)   

8090.nr                                  1.226***
                                        (6.97)   

8096.nr                                  1.299***
                                        (5.89)   

8106.nr                                  0.948***
                                        (4.74)   

8107.nr                                  1.186***
                                        (5.38)   

8142.nr                                  0.182   
                                        (1.04)   

8168.nr                                  0.551** 
                                        (3.12)   

8173.nr                                  0.909***
                                        (4.51)   

8203.nr                                  1.072***
                                        (6.07)   

8211.nr                                  0.294   
                                        (1.61)   

8224.nr                                  0.807***
                                        (4.50)   

8272.nr                                  0.959***
                                        (5.40)   

8300.nr                                  0.386*  
                                        (2.20)   

8304.nr                                  0.470** 
                                        (2.67)   

8364.nr                                  0.348*  
                                        (1.96)   

8370.nr                                  0.845***
                                        (4.78)   

8381.nr                                  0.218   
                                        (1.24)   

8388.nr                                  0.743***
                                        (3.83)   

8406.nr                                  0.626***
                                        (3.41)   

8415.nr                                 0.0542   
                                        (0.31)   

8496.nr                                 -0.112   
                                       (-0.64)   

8501.nr                                -0.0699   
                                       (-0.39)   

8518.nr                                  1.243***
                                        (6.89)   

8520.nr                                 -0.353*  
                                       (-2.00)   

8524.nr                                 -0.118   
                                       (-0.67)   

8548.nr                                  0.115   
                                        (0.65)   

8556.nr                                  0.625***
                                        (3.55)   

8564.nr                                  0.106   
                                        (0.60)   

8581.nr                                 -0.249   
                                       (-1.41)   

8586.nr                                  0.258   
                                        (1.47)   

8587.nr                                 -0.370*  
                                       (-2.10)   

8597.nr                                  0.771***
                                        (4.21)   

8656.nr                                  0.153   
                                        (0.87)   

8722.nr                                  0.637***
                                        (3.59)   

8743.nr                                  0.515** 
                                        (2.90)   

8749.nr                                  0.940***
                                        (4.67)   

8758.nr                                  0.535** 
                                        (3.02)   

8796.nr                                  0.780***
                                        (4.42)   

8838.nr                                  0.406*  
                                        (2.30)   

8842.nr                                  0.145   
                                        (0.82)   

8846.nr                                  0.112   
                                        (0.63)   

8860.nr                                  0.124   
                                        (0.70)   

8862.nr                                  0.399*  
                                        (2.27)   

8880.nr                                  1.219***
                                        (6.85)   

8886.nr                                  0.631***
                                        (3.52)   

8903.nr                                 -0.239   
                                       (-1.35)   

8908.nr                                -0.0227   
                                       (-0.13)   

8911.nr                                -0.0340   
                                       (-0.19)   

8917.nr                                  0.796***
                                        (4.48)   

8991.nr                                  0.913***
                                        (5.18)   

8997.nr                                  0.898***
                                        (5.09)   

9014.nr                                  0.540** 
                                        (3.05)   

9015.nr                                  0.697***
                                        (3.72)   

9027.nr                                  1.002***
                                        (5.68)   

9066.nr                                 -0.153   
                                       (-0.87)   

9082.nr                                  0.274   
                                        (1.54)   

9131.nr                                  0.758***
                                        (3.77)   

9132.nr                                  0.709***
                                        (3.77)   

9154.nr                                  1.113***
                                        (6.32)   

9158.nr                                 0.0857   
                                        (0.48)   

9184.nr                                  0.164   
                                        (0.92)   

9230.nr                                  0.158   
                                        (0.90)   

9265.nr                                  0.437*  
                                        (2.49)   

9367.nr                                  0.207   
                                        (1.18)   

9390.nr                                  0.745***
                                        (4.20)   

9391.nr                                  0.370*  
                                        (2.10)   

9418.nr                                  1.133***
                                        (6.44)   

9424.nr                                  0.806***
                                        (4.56)   

9447.nr                                  0.503** 
                                        (2.85)   

9449.nr                                  0.318   
                                        (1.81)   

9453.nr                                  0.745***
                                        (4.22)   

9468.nr                                  0.493** 
                                        (2.81)   

9502.nr                                  0.895***
                                        (4.95)   

9505.nr                                 0.0464   
                                        (0.26)   

9603.nr                                  0.310   
                                        (1.65)   

9643.nr                                  0.433*  
                                        (2.46)   

9667.nr                                  1.067***
                                        (6.07)   

9683.nr                                  0.128   
                                        (0.71)   

9694.nr                                  0.847***
                                        (4.82)   

9710.nr                                  0.280   
                                        (1.58)   

9718.nr                                  0.938***
                                        (5.34)   

9725.nr                                  0.322   
                                        (1.84)   

9744.nr                                  0.689***
                                        (3.92)   

9752.nr                                  1.390***
                                        (7.88)   

9776.nr                                  0.953***
                                        (5.37)   

9786.nr                                  0.624***
                                        (3.55)   

9791.nr                                  0.328   
                                        (1.86)   

9794.nr                                  0.473** 
                                        (2.67)   

9810.nr                                 -0.144   
                                       (-0.82)   

9846.nr                                  0.825***
                                        (4.65)   

9859.nr                                  0.148   
                                        (0.83)   

9868.nr                                 0.0289   
                                        (0.16)   

9876.nr                                  0.853***
                                        (4.78)   

9883.nr                                  0.714***
                                        (4.05)   

9889.nr                                  0.988***
                                        (5.59)   

9901.nr                                  0.689***
                                        (3.88)   

9936.nr                                  0.895***
                                        (5.06)   

9964.nr                                  0.412*  
                                        (2.32)   

10043.nr                                -0.172   
                                       (-0.97)   

10067.nr                                 0.510** 
                                        (2.89)   

10091.nr                                 0.697***
                                        (3.86)   

10120.nr                                -0.118   
                                       (-0.67)   

10121.nr                                 0.325   
                                        (1.85)   

10167.nr                                 0.567** 
                                        (3.18)   

10209.nr                                 0.112   
                                        (0.63)   

10230.nr                                 0.724***
                                        (4.03)   

10265.nr                                 0.993***
                                        (5.63)   

10274.nr                                 0.237   
                                        (1.35)   

10311.nr                                 0.301   
                                        (1.71)   

10392.nr                                 0.108   
                                        (0.61)   

10425.nr                                 0.945***
                                        (5.34)   

10441.nr                                 0.176   
                                        (1.00)   

10457.nr                                 0.446*  
                                        (2.53)   

10469.nr                                 0.447*  
                                        (2.54)   

10524.nr                                0.0885   
                                        (0.50)   

10552.nr                                -0.185   
                                       (-1.05)   

10553.nr                                 0.706***
                                        (3.96)   

10570.nr                                -0.747***
                                       (-4.23)   

10593.nr                                 0.284   
                                        (1.54)   

10666.nr                                -0.324   
                                       (-1.83)   

11275.nr                                 0.146   
                                        (0.83)   

11328.nr                                 0.495** 
                                        (2.76)   

11750.nr                                 0.384*  
                                        (2.19)   

11821.nr                                 0.808***
                                        (4.60)   

11857.nr                                 0.934***
                                        (5.27)   

11887.nr                                 0.396*  
                                        (2.24)   

11890.nr                                 1.093***
                                        (5.18)   

11892.nr                                 1.132***
                                        (6.38)   

11924.nr                                 0.354*  
                                        (2.02)   

11925.nr                                 0.408*  
                                        (2.31)   

11957.nr                               0.00425   
                                        (0.02)   

11973.nr                                 0.578** 
                                        (3.26)   

11990.nr                                 1.010***
                                        (5.73)   

12012.nr                                 0.224   
                                        (1.27)   

12013.nr                                 0.568** 
                                        (3.24)   

12045.nr                                 0.200   
                                        (1.13)   

12055.nr                                 0.674***
                                        (3.81)   

12084.nr                                 0.450*  
                                        (2.51)   

12088.nr                                 0.117   
                                        (0.66)   

12122.nr                                 0.586** 
                                        (2.91)   

12179.nr                                 0.301   
                                        (1.70)   

12182.nr                                 0.515** 
                                        (2.93)   

12220.nr                                 0.673***
                                        (3.81)   

12221.nr                                -0.121   
                                       (-0.66)   

12245.nr                                -0.110   
                                       (-0.63)   

12276.nr                                 0.655***
                                        (3.69)   

12385.nr                                 0.819***
                                        (4.57)   

12410.nr                                0.0399   
                                        (0.22)   

12420.nr                                 0.840***
                                        (4.74)   

12433.nr                                 0.393*  
                                        (2.23)   

12451.nr                                 0.431*  
                                        (2.45)   

12477.nr                                 1.004***
                                        (5.62)   

12500.nr                                 0.222   
                                        (1.25)   

12534.nr                                 0.942***
                                        (5.35)   

12548.nr                                 0.347   
                                        (1.93)   

_cons         -0.445***     1.426***     0.933***
            (-14.96)      (77.75)       (7.44)   
-------------------------------------------------
N               4360         4360         4360   
-------------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001

Practical Tips

Notice that the parameter \(\delta\) does not have meaning.

\[y_{i,t} = \alpha + \beta_1 x_{i,t} + \delta FE + \epsilon_{i,t}\]

In fact, the previous slides have shown that you will find the same results of a FE model if you include the dummies for the units in the panel (i.e., dummies for the firms or individuals, etc.).

This is called least squares dummy variable (LSDV) model.

  • the SE are also identical to the within-transformation model.

  • But the R2 of the LSDV will be very high because you are including a lot of “explanatory variables”.

Note

At the end of the day, you will use the package for the unit’s FE (i.e., the firm), and will include the additional FE as dummies, just like a LSDV model.

Practical Tips

When you estimate a LSDV, the software will inform an \(\alpha\).

But this coefficient has no interpretation whatsoever.

  • it will be FE for the dropped unit of FE.

You can simply ignore it, you even don’t need to include in your final table.

No problem if you do, just don’t make inferences from it.

Practical Tips

A FE model helps a lot, but it only does what it can do.

That is, FE models do not capture time-variant unobserved heterogeneity.

Also, if you have constant Xs in your model, you will have to drop them.

  • More technically, if there is no within-variation in a X, you cannot include it (the software will drop them).

  • For instance, the software will drop \(year_{birth}\) below if you include CEO FE.

\[Y_{i,t} = \alpha + \beta_1 year_{birth} + CEO \;FE + ... + \epsilon_{i,t}\]

If you attempt to include the CEO FE manually, the software will drop a random CEO FE or the variable \(year_{birth}\). If you get a beta for \(year_{birth}\) it has no meaning.

Practical Tips

Adding many FE can demand a lot of computational power.

Consider the multidimensional model as follows:

\[Y_{i,t} = \alpha + \beta_1 X_{i,t} + Firm \;FE + Year\; FE + Year.Industry \;FE + CEO \;FE + ... + \epsilon_{i,t}\]

It would take a while to estimate in an average computer.

Random Effects

Random Effects

Remember that:

\[\epsilon_{i,t} = c_i + \mu_{i,t}\]

The most important thing here is whether \(x_{it}\) and \(c_i\) are correlated.

  • If they are, you should estimate Fixed Effects

  • If \(x_{it}\) and \(c_i\) are not correlated, then \(c_i\) is referred to as a random effect.

    • Endogeneity is not a concern; however, the computation of standard errors is affected.

But, if the \(x_{it}\) and \(c_i\) are not correlated, there is no endogeneity concern.

\(c_i\) can be let as part of the \(\epsilon_{i,t}\) without bias in the estimated betas.

Random Effects

Additionally, the assumption that \(x_{it}\) and \(c_i\) are not correlated is rather strong and not practical to most applications of corporate finance, economics or public policy.

RE is a model not used often. Cunningham does not even discuss it.

If the key explanatory variable is constant over time, we cannot use FE to estimate its effect on y.

Of course, we can only use RE because we are willing to assume the unobserved effect is uncorrelated with all explanatory variables.

Typically, if one uses RE, and as many time-constant controls as possible are included among the explanatory variables (with an FE analysis, it is not necessary to include such controls) RE is preferred to pooled OLS because RE is generally more efficient.

(Wooldridge, p.496)

Random Effects

R
# Load necessary packages
library(plm)
library(jtools)
library(foreign)
data <- read.dta("files/WAGEPAN.dta")
pdata <- pdata.frame(data, index = c("nr", "year"))

po_model <- lm(lwage ~ expersq + married + union + factor(year) + educ + black + hisp + exper, data = data)
fe_model <- plm(lwage ~ expersq + married + union + factor(year) + educ + black + hisp + exper, data = pdata, model = "within")
re_model <- plm(lwage ~ expersq + married + union + factor(year) + educ + black + hisp + exper, data = pdata, model = "random")

stargazer(po_model, fe_model , re_model ,title = "Regression Results", column.labels=c("OLS","FE","RE"),  type = "text")

Regression Results
==================================================================================
                                         Dependent variable:                      
                    --------------------------------------------------------------
                                                lwage                             
                               OLS                           panel                
                                                             linear               
                               OLS                       FE                 RE    
                               (1)                       (2)               (3)    
----------------------------------------------------------------------------------
expersq                     -0.002***                 -0.005***         -0.005*** 
                             (0.001)                   (0.001)           (0.001)  
                                                                                  
married                     0.108***                   0.047**           0.064*** 
                             (0.016)                   (0.018)           (0.017)  
                                                                                  
union                       0.182***                  0.080***           0.106*** 
                             (0.017)                   (0.019)           (0.018)  
                                                                                  
factor(year)1981             0.058*                   0.151***            0.040   
                             (0.030)                   (0.022)           (0.025)  
                                                                                  
factor(year)1982             0.063*                   0.253***            0.031   
                             (0.033)                   (0.024)           (0.032)  
                                                                                  
factor(year)1983             0.062*                   0.354***            0.020   
                             (0.037)                   (0.029)           (0.042)  
                                                                                  
factor(year)1984             0.090**                  0.490***            0.043   
                             (0.040)                   (0.036)           (0.051)  
                                                                                  
factor(year)1985             0.109**                  0.617***            0.058   
                             (0.043)                   (0.045)           (0.061)  
                                                                                  
factor(year)1986            0.142***                  0.765***            0.092   
                             (0.046)                   (0.056)           (0.071)  
                                                                                  
factor(year)1987            0.174***                  0.925***            0.135*  
                             (0.049)                   (0.069)           (0.081)  
                                                                                  
educ                        0.091***                                     0.092*** 
                             (0.005)                                     (0.011)  
                                                                                  
black                       -0.139***                                   -0.139*** 
                             (0.024)                                     (0.048)  
                                                                                  
hisp                          0.016                                       0.022   
                             (0.021)                                     (0.043)  
                                                                                  
exper                       0.067***                                     0.106*** 
                             (0.014)                                     (0.015)  
                                                                                  
Constant                      0.092                                       0.024   
                             (0.078)                                     (0.151)  
                                                                                  
----------------------------------------------------------------------------------
Observations                  4,360                     4,360             4,360   
R2                            0.189                     0.181             0.181   
Adjusted R2                   0.187                     0.061             0.178   
Residual Std. Error     0.480 (df = 4345)                                         
F Statistic         72.459*** (df = 14; 4345) 83.851*** (df = 10; 3805) 957.774***
==================================================================================
Note:                                                  *p<0.1; **p<0.05; ***p<0.01
Stata
use "files/WAGEPAN.dta" , clear

xtset nr year 
eststo: qui reg   lwage expersq married union i.year  educ black hisp exper 
eststo: qui xtreg lwage expersq married union i.year  educ black hisp exper , fe
eststo: qui xtreg lwage expersq married union i.year  educ black hisp exper , re

esttab , mtitles("OLS" "FE" "RE") compress
Panel variable: nr (strongly balanced)
 Time variable: year, 1980 to 1987
         Delta: 1 unit

(est1 stored)

(est2 stored)

(est3 stored)


-------------------------------------------------
                 (1)          (2)          (3)   
                 OLS           FE           RE   
-------------------------------------------------
expersq     -0.00241**   -0.00519***  -0.00472***
             (-2.94)      (-7.36)      (-6.85)   

married        0.108***    0.0467*      0.0640***
              (6.90)       (2.55)       (3.81)   

union          0.182***    0.0800***     0.106***
             (10.63)       (4.14)       (5.94)   

1980.year          0            0            0   
                 (.)          (.)          (.)   

1981.year     0.0583        0.151***    0.0405   
              (1.92)       (6.89)       (1.64)   

1982.year     0.0628        0.253***    0.0309   
              (1.89)      (10.36)       (0.96)   

1983.year     0.0620        0.354***    0.0203   
              (1.69)      (12.12)       (0.49)   

1984.year     0.0905*       0.490***    0.0431   
              (2.26)      (13.53)       (0.84)   

1985.year      0.109*       0.617***    0.0578   
              (2.52)      (13.65)       (0.94)   

1986.year      0.142**      0.765***    0.0919   
              (3.06)      (13.64)       (1.29)   

1987.year      0.174***     0.925***     0.135   
              (3.52)      (13.45)       (1.66)   

educ          0.0913***         0       0.0919***
             (17.44)          (.)       (8.62)   

black         -0.139***         0       -0.139** 
             (-5.90)          (.)      (-2.92)   

hisp          0.0160            0       0.0217   
              (0.77)          (.)       (0.51)   

exper         0.0672***         0        0.106***
              (4.91)          (.)       (6.88)   

_cons         0.0921        1.426***    0.0236   
              (1.18)      (77.75)       (0.16)   
-------------------------------------------------
N               4360         4360         4360   
-------------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001

FE vs. RE

FE vs. RE

The idea is that one uses the random effects estimates unless the Hausman test rejects.

In practice, a failure to reject means either that the RE and FE estimates are sufficiently close so that it does not matter which is used, or the sampling variation is so large in the FE estimates that one cannot conclude practically significant differences are statistically significant. (Wooldridge)

If the p-value of the Hausman test is significant then use FE, if not use RE.

FE vs. RE

Stata
use "files/WAGEPAN.dta", clear
xtset nr year
qui xtreg lwage expersq married union i.year educ black hisp exper, fe
estimates store fe_model
qui xtreg lwage expersq married union i.year educ black hisp exper, re
estimates store re_model
hausman fe_model re_model
Panel variable: nr (strongly balanced)
 Time variable: year, 1980 to 1987
         Delta: 1 unit

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |    fe_model     re_model      Difference       Std. err.
-------------+----------------------------------------------------------------
     expersq |   -.0051855    -.0047239       -.0004616        .0001443
     married |    .0466804      .063986       -.0173057        .0073414
       union |    .0800019     .1061344       -.0261326        .0073572
        year |
       1981  |    .1511912      .040462        .1107292               .
       1982  |    .2529709     .0309212        .2220497               .
       1983  |    .3544437     .0202806        .3341631               .
       1984  |    .4901148     .0431187        .4469961               .
       1985  |    .6174822     .0578154        .5596668               .
       1986  |    .7654965     .0919475         .673549               .
       1987  |    .9250249     .1349289         .790096               .
------------------------------------------------------------------------------
                          b = Consistent under H0 and Ha; obtained from xtreg.
           B = Inconsistent under Ha, efficient under H0; obtained from xtreg.

Test of H0: Difference in coefficients not systematic

   chi2(10) = (b-B)'[(V_b-V_B)^(-1)](b-B)
            =  26.36
Prob > chi2 = 0.0033
(V_b-V_B is not positive definite)

First differences

First differences

In most applications, the main reason for collecting panel data is to allow for the unobserved effect, \(c_i\), to be correlated with the explanatory variables.

For example, in the crime equation, we want to allow the unmeasured city factors in \(c_i\) that affect the crime rate also to be correlated with the unemployment rate.

It turns out that this is simple to allow: because \(c_i\) is constant over time, we can difference the data across the two years.

More precisely, for a cross-sectional observation \(i\), write the two years as:

\[y_{i,1} = \beta_0 + \beta_1 x_{i,1} + c_i + \mu_{i,1}, t=1\]

\[y_{i,2} = (\beta_0 + \delta_0) + \beta_1 x_{i,2} + c_i + \mu_{i,2}, t=2\]

If we subtract the second equation from the first, we obtain

\[(y_{i,2} - y_{i,1}) = \delta_0 + \beta_1 (x_{i,2} - x_{i,1}) + (\mu_{i,2}-\mu_{i,1})\]

\[\Delta y_{i} = \delta_0 + \beta_1 \Delta x_{i} + \Delta \mu_{i}\]

First differences

So, rather than subtracting the group mean of each variable, you subtract the lagged observation.

Not hard to see that, when t=2, FE and FD will give identical solutions

  • FE is more efficient if disturbances \(\mu_{i,t}\) have low serial correlation

  • FD is more efficient if disturbance \(\mu_{i,t}\) follow a random walk

At the end of the day, you can estimate both.

Empirical research usually estimate FD only in specific circumstances, when they are interested in how changes of X affect changes of Y.

Things like stationarity or trends are often not concerns in panel data

  • where N is 10 to 20

First differences

R
# Load necessary packages
# Load necessary libraries
library(plm)
library(lmtest)
library(stargazer)

data <- read.dta("files/WAGEPAN.dta")
pdata <- pdata.frame(data, index = c("nr", "year"))

ols_model <- lm(lwage ~ expersq + married + union + factor(year) + educ + black + hisp + exper, data = pdata)
fe_model <- plm(lwage ~ expersq + married + union + educ + black + hisp + exper, data = pdata, model = "within")
re_model <- plm(lwage ~ expersq + married + union + educ + black + hisp + exper, data = pdata, model = "random")
fd_model <- plm(lwage ~ expersq + married + union + educ + black + hisp + exper, data = pdata, model = "fd")

stargazer(ols_model, fe_model ,re_model, fd_model,title = "Regression Results",   type = "text")

Regression Results
==========================================================================================================
                                                     Dependent variable:                                  
                    --------------------------------------------------------------------------------------
                                                            lwage                                         
                               OLS                                       panel                            
                                                                         linear                           
                               (1)                       (2)               (3)               (4)          
----------------------------------------------------------------------------------------------------------
expersq                     -0.002***                 -0.004***         -0.004***         -0.004***       
                             (0.001)                   (0.001)           (0.001)           (0.001)        
                                                                                                          
married                     0.108***                   0.045**           0.063***          0.038*         
                             (0.016)                   (0.018)           (0.017)           (0.023)        
                                                                                                          
union                       0.182***                  0.082***           0.107***          0.043**        
                             (0.017)                   (0.019)           (0.018)           (0.020)        
                                                                                                          
factor(year)1981             0.058*                                                                       
                             (0.030)                                                                      
                                                                                                          
factor(year)1982             0.063*                                                                       
                             (0.033)                                                                      
                                                                                                          
factor(year)1983             0.062*                                                                       
                             (0.037)                                                                      
                                                                                                          
factor(year)1984             0.090**                                                                      
                             (0.040)                                                                      
                                                                                                          
factor(year)1985             0.109**                                                                      
                             (0.043)                                                                      
                                                                                                          
factor(year)1986            0.142***                                                                      
                             (0.046)                                                                      
                                                                                                          
factor(year)1987            0.174***                                                                      
                             (0.049)                                                                      
                                                                                                          
educ                        0.091***                                     0.101***                         
                             (0.005)                                     (0.009)                          
                                                                                                          
black                       -0.139***                                   -0.144***                         
                             (0.024)                                     (0.048)                          
                                                                                                          
hisp                          0.016                                       0.020                           
                             (0.021)                                     (0.043)                          
                                                                                                          
exper                       0.067***                  0.117***           0.112***                         
                             (0.014)                   (0.008)           (0.008)                          
                                                                                                          
Constant                      0.092                                       -0.107          0.116***        
                             (0.078)                                     (0.111)           (0.020)        
                                                                                                          
----------------------------------------------------------------------------------------------------------
Observations                  4,360                     4,360             4,360             3,815         
R2                            0.189                     0.178             0.178             0.004         
Adjusted R2                   0.187                     0.060             0.177             0.003         
Residual Std. Error     0.480 (df = 4345)                                                                 
F Statistic         72.459*** (df = 14; 4345) 206.375*** (df = 4; 3811) 943.951*** 5.362*** (df = 3; 3811)
==========================================================================================================
Note:                                                                          *p<0.1; **p<0.05; ***p<0.01
Stata
use "files/WAGEPAN.dta" , clear

xtset nr year 
eststo: qui reg   lwage expersq married union i.year  educ black hisp exper 
eststo: qui xtreg lwage expersq married union i.year  educ black hisp exper , fe
eststo: qui xtreg lwage expersq married union i.year  educ black hisp exper , re
eststo: qui reg D.lwage D.expersq D.married D.union i.year  D.educ D.black D.hisp D.exper 

esttab , mtitles("OLS" "FE" "RE" "FD") compress
Panel variable: nr (strongly balanced)
 Time variable: year, 1980 to 1987
         Delta: 1 unit

(est1 stored)

(est2 stored)

(est3 stored)

(est4 stored)


--------------------------------------------------------------
                 (1)          (2)          (3)          (4)   
                 OLS           FE           RE           FD   
--------------------------------------------------------------
expersq     -0.00241**   -0.00519***  -0.00472***             
             (-2.94)      (-7.36)      (-6.85)                

married        0.108***    0.0467*      0.0640***             
              (6.90)       (2.55)       (3.81)                

union          0.182***    0.0800***     0.106***             
             (10.63)       (4.14)       (5.94)                

1980.year          0            0            0                
                 (.)          (.)          (.)                

1981.year     0.0583        0.151***    0.0405            0   
              (1.92)       (6.89)       (1.64)          (.)   

1982.year     0.0628        0.253***    0.0309      -0.0482   
              (1.89)      (10.36)       (0.96)      (-1.77)   

1983.year     0.0620        0.354***    0.0203      -0.0479   
              (1.69)      (12.12)       (0.49)      (-1.70)   

1984.year     0.0905*       0.490***    0.0431      -0.0122   
              (2.26)      (13.53)       (0.84)      (-0.41)   

1985.year      0.109*       0.617***    0.0578      -0.0208   
              (2.52)      (13.65)       (0.94)      (-0.65)   

1986.year      0.142**      0.765***    0.0919      0.00151   
              (3.06)      (13.64)       (1.29)       (0.04)   

1987.year      0.174***     0.925***     0.135       0.0167   
              (3.52)      (13.45)       (1.66)       (0.45)   

educ          0.0913***         0       0.0919***             
             (17.44)          (.)       (8.62)                

black         -0.139***         0       -0.139**              
             (-5.90)          (.)      (-2.92)                

hisp          0.0160            0       0.0217                
              (0.77)          (.)       (0.51)                

exper         0.0672***         0        0.106***             
              (4.91)          (.)       (6.88)                

D.expersq                                          -0.00575** 
                                                    (-2.65)   

D.married                                            0.0381   
                                                     (1.66)   

D.union                                              0.0411*  
                                                     (2.09)   

oD.educ                                                   0   
                                                        (.)   

oD.black                                                  0   
                                                        (.)   

oD.hisp                                                   0   
                                                        (.)   

oD.exper                                                  0   
                                                        (.)   

_cons         0.0921        1.426***    0.0236        0.156***
              (1.18)      (77.75)       (0.16)       (6.36)   
--------------------------------------------------------------
N               4360         4360         4360         3815   
--------------------------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001

Lagged independent variables

Lagged independent variables

When you have a panel data and are concerned with simultaneity between Y and X, you can endeavor in lagging the Xs.

\[y_{i,t} = \beta_0 + \beta_1 x_{i,t-1} + c_i + \mu_{i,t}\]

As a matter of fact, this is often expected in finance research.

There is a limitation, however.

The usual proxy of corporate finance research is highly autocorrelated.

  • e.g., total assets do not vary much throughout time.

Thus, lagging the X often does not make much of a difference.

Tip

Always do it. Otherwise, you will have to explain why you didn’t do it.

Lagged dependent variables

Lagged dependent variables

Sometimes you may have something like

\[y_{i,t} = \beta_0 + \beta_1 y_{i,t-1}+ \beta_2 x_{i,t} + c_i + \mu_{i,t}\]

This is called a Dynamic Panel Model. It includes \(y_{i,t-1}\) as X.

Consider a FE model.

\[y_{i,t} - \bar{y_i} = \beta_0 + \gamma_1 (y_{i,t-1} - \bar{y}_{i,t-1}) + \omega_2 (x_{i,t-1} - \bar{x_i} ) + (FE_i - \bar{FE}_i) + (\mu_{i,t} - \bar{\mu}_i )\]

The within transformation removes the time-invariant unobserved heterogeneity from the model.

However, it introduces a correlation between the transformed lag \((y_{i,t−1}−\bar{y}_{i,t-1})\) and the transformed error \((\mu_{i,t−1}−\bar{\mu}_{i,t-1})\) because the average error (\(\bar{\mu} = \sum_{i=1}^{T} \mu_{i,t}\)) includes \(\mu_{i,t-1}\), which is also “included” in \(y_{i,t−1}\)

  • \(y_{i,t-1} = \beta_0 + \beta_1 y_{i,t-2}+ \beta_2 x_{i,t-1} + c_i + \mu_{i,t-1}\)

Lagged dependent variables

The bias declines with panel length because \(\epsilon_{i,t−1}\) becomes a smaller component of the average error term as T increases.

In other words, with higher T the correlation between the lagged dependent variable and the regression errors becomes smaller.

Flannery and Hankins (2013) have a good review with applications in corporate finance.

They conclude that FE is biased when estimating these models.

They suggest to estimate Sys-GMM or Least Squares Dummy Variable Correction. We do not discuss these models in the course.

THANK YOU!

QUESTIONS?