Of course, again, we cannot observe both potential outcomes of the same unit i.

Selection Bias

When dealing with causal inference, we have to find ways to approximate what the hidden potential outcome of the treated units is.

That is, the challenge in identifying causal effects is that the untreated potential outcomes, \(Y_{i,0}\), are never observed for the treated group (\(D_i= 1\)). The “second” term in the following equation:

We need an empirical design to “observe” what we do not really observe (i.e., the counterfactual).

Selection Bias

Many options:

Matching/Balancing

Difference-in-differences (DiD)

Instrumental variables

Regression discontinuity design (RDD)

Synthetic control (Synth)

Selection Bias

The process of finding units that are comparable is called matching.

Before we continue…

We will match on observables. We cannot be on unobservables.

Thus, you may want to write in your article “selection bias due to observables”.

Cunningham:

Propensity score matching has not seen as wide adoption among economists as in other nonexperimental methods like regression discontinuity or difference-in-differences. The most common reason given for this is that economists are oftentimes skeptical that CIA can be achieved in any dataset almost as an article of faith. This is because for many applications, economists as a group are usually more concerned about selection on unobservables than they are selection on observables, and as such, they reach for matching methods less often.

CIA = CMI

Matching

Matching

Matching aims to compare the outcomes between observations that have the same values of all control variables, except that one unit is treated and the other is not.

In this literature, the control variables used to matched are often called covariates.

That is, for each treated unit, the researcher finds an untreated unit that is similar in all covariates.

The implication is that the researcher can argue that “units are comparable after matching”.

Matching

The easiest to see is exact matching: it matches observations that have the exact same values.

It might be doable if you have only one covariate.

Naturally, if you have only one covariate, you might still be left with some selection bias.

In the previous example, health history is one important covariate that makes John and Mary different.

But what about life style? Nutrition? Etc.

As the number of covariates grow, you cannot pursue exact matching. That is the job of PSM.

Matching

In exact matching, the causal effect estimator (ATET) is:

Where \(Y_{j(i)}\) is the j-th unit matched to the i-th unit based on the j-th being “closest to” the i-th unit for some covariate.

For instance, let’s say that a unit in the treatment group has a covariate with a value of 2 and we find another unit in the control group (exactly one unit) with a covariate value of 2.

Then we will impute the treatment unit’s missing counterfactual with the matched unit’s, and take a difference.

The problem with this measure of distance is that the distance measure itself depends on the scale of the variables themselves.

For this reason, researchers typically will use some modification of the Euclidean distance, such as the normalized Euclidean distance, or they’ll use a wholly different alternative distance.

The normalized Euclidean distance is a commonly used distance, and what makes it different is that the distance of each variable is scaled by the variable’s variance.

Where \(\hat{\sum_x}\) is the sample covariance matrix of X.

Distance Matching

Distance matching only goes so far…

… the larger the dimensionality, the harder is to use distance matching.

As sample size increases, for a given N of covariates, the matching discrepancies tend to zero.

But, the more covariates, the longer it takes.

At the end of the day, it is preferable to have many covariates, but it is makes distance matching harder.

Coarsened Exact Matching (CER)

Coarsened Exact Matching (CER)

In coarsened exact matching, something only counts as a match if it exactly matches on each matching variable.

The “coarsened” part comes in because, if you have any continuous variables to match on, you need to “coarsen” them first by putting them into bins, rather than matching on exact values.

Coarsening means creating bins. Fewer bins makes exact matches more likely.

CER is not used much in empirical research in finance. It is used more in the big data realm when you have many variables to match.

Propensity-score matching (PSM)

Propensity-score matching (PSM)

PSM is one way to matching using many covariates.

PSM aggregates all covariates into one score (propensity-score), which is the likelihood of receiving the treatment.

The idea is to match units that, based on observables, have the same probability (called propensity-score) of being treated.

The idea is to estimate a probit (default in stata) or logit model (fist stage):

\[P(D=1|X)\]

The propensity-score is the predicted probability of a unit being treated given all covariates X. The p-score is just a single number.

Propensity-score matching (PSM)

Considerations in PSM.

How many neighbors to match?

Nearest neighbor, radius or kernel?

With or without replacement?

With or without common support?

Common support: imposes a common support by dropping treatment observations whose pscore is higher than the maximum or less than the minimum pscore of the controls.

It is expected that, after PSM, you show the overlap of propensity-scores.

Kernel matching: Each treated observation i is matched with several control observations, with weights inversely proportional to the distance between treated and control observations.

use files/cps1re74.dta, clearqui estpost tabstat age black educ , by(treat) c(s) s(me v sk n) nototalesttab . ,varwidth(20) cells("mean(fmt(3)) variance(fmt(3)) skewness(fmt(3)) count(fmt(0))") noobs nonumber compress

(DW Subset of LaLonde Data)
------------------------------------------------------------
mean variance skewness count
------------------------------------------------------------
0
age 33.225 121.997 0.348 15992
black 0.074 0.068 3.268 15992
educ 12.028 8.242 -0.423 15992
------------------------------------------------------------
1
age 25.816 51.194 1.115 185
black 0.843 0.133 -1.888 185
educ 10.346 4.043 -0.721 185
------------------------------------------------------------

Example

Clearly, the treated group is younger, mainly black, and less educated.

Also note that the variance and skewness of the two subsamples are different.

If we were to use these two subsamples in any econometric analysis without preprocessing to make them comparable, we would likely have coefficients biased by selection bias.

Therefore, it is important to perform some matching method.

Let’s start with Propensity Score Matching (PSM). We will use the simplest matching, that is, without using any additional functions.

# install.packages("MatchIt")library(haven)library(psych)library(MatchIt)data <-read_dta("files/cps1re74.dta")model <-matchit(treat ~ age + black + educ, data = data, method ="nearest")summary(model)

Call:
matchit(formula = treat ~ age + black + educ, data = data, method = "nearest")
Summary of Balance for All Data:
Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean
distance 0.1445 0.0099 1.3615 7.5662 0.4987
age 25.8162 33.2252 -1.0355 0.4196 0.1863
black 0.8432 0.0735 2.1171 . 0.7697
educ 10.3459 12.0275 -0.8363 0.4905 0.0908
eCDF Max
distance 0.7741
age 0.3427
black 0.7697
educ 0.4123
Summary of Balance for Matched Data:
Means Treated Means Control Std. Mean Diff. Var. Ratio eCDF Mean
distance 0.1445 0.1443 0.0020 1.0039 0.0002
age 25.8162 25.7081 0.0151 0.9244 0.0073
black 0.8432 0.8432 0.0000 . 0.0000
educ 10.3459 10.4054 -0.0296 0.7190 0.0117
eCDF Max Std. Pair Dist.
distance 0.0162 0.0029
age 0.0270 0.1481
black 0.0000 0.0000
educ 0.0432 0.2554
Sample Sizes:
Control Treated
All 15992 185
Matched 185 185
Unmatched 15807 0
Discarded 0 0

Stata

use files/cps1re74.dta, clearpsmatch2 treat age black educ , n(1) noreplacementsum _weight , d

# install.packages("MatchIt")library(haven)library(MatchIt)data <-read_dta("files/cps1re74.dta")model <-matchit(treat ~ age + black + educ, data = data, method ="exact")summary(model$weights)

Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000 0.0000 0.0000 0.1457 0.0000 52.7952

Stata

use files/cps1re74.dta, clearqui psmatch2 treat age black educ , kernelsum _weight , d

(DW Subset of LaLonde Data)
psmatch2: weight of matched controls
-------------------------------------------------------------
Percentiles Smallest
1% .0024355 .0007862
5% .0024375 .0024348
10% .00244 .0024348 Obs 16,177
25% .0024517 .0024348 Sum of wgt. 16,177
50% .0024919 Mean .022872
Largest Std. dev. .1130791
75% .0026476 1
90% .0038379 1 Variance .0127869
95% .0876547 1 Skewness 7.604874
99% 1 1 Kurtosis 64.26276

use files/cps1re74.dta, clearqui psmatch2 treat age black educ , kernelqui estpost tabstat age black educ [aweight = _weight], by(treat) c(s) s(me v sk n) nototalesttab . ,varwidth(20) cells("mean(fmt(3)) variance(fmt(3)) skewness(fmt(3)) count(fmt(0))") noobs nonumber compress

(DW Subset of LaLonde Data)
------------------------------------------------------------
mean variance skewness count
------------------------------------------------------------
0
age 27.033 85.548 1.077 15992
black 0.791 0.165 -1.434 15992
educ 10.710 8.146 -0.883 15992
------------------------------------------------------------
1
age 25.816 51.194 1.115 185
black 0.843 0.133 -1.888 185
educ 10.346 4.043 -0.721 185
------------------------------------------------------------

Entropy Balancing

Entropy Balancing

Here, instead of matching units, we reweight the observations such that the moments of the distributions (mean, variance, skewness) are similar.

The ebalance function implements a reweighting scheme. The user starts by choosing the covariates that should be included in the reweighting.

For each covariate, the user then specifies a set of balance constraints (in Equation 5) to equate the moments of the covariate distribution between the treatment and the reweighted control group.

The moment constraints may include the mean (first moment), the variance (second moment), and the skewness (third moment).

The outcome is a vector containing the weights to weight the observations, such that the weighted average, weighted variance, and weighted skewness of the covariates in control group are similar to those in the treatment group

# means in treatment group dataapply(vars[treatment==1,],2,mean)

[1] 25.8162162 10.3459459 0.8432432

R

# means in reweighted control group dataapply(vars[treatment==0,],2,weighted.mean,w=eb$w)

[1] 25.8163688 10.3460391 0.8431526

R

# means in raw data control group dataapply(vars[treatment==0,],2,mean)

[1] 33.22523762 12.02751376 0.07353677

Stata

use files/cps1re74.dta, clearebalance treat age black educ, targets(3)

(DW Subset of LaLonde Data)
Data Setup
Treatment variable: treat
Covariate adjustment: age black educ (1st order). age black educ (2nd order). a
> ge black educ (3rd order).
Optimizing...
Iteration 1: Max Difference = 580799.347
Iteration 2: Max Difference = 213665.688
Iteration 3: Max Difference = 78604.7628
Iteration 4: Max Difference = 28918.6249
Iteration 5: Max Difference = 10640.1108
Iteration 6: Max Difference = 3915.82197
Iteration 7: Max Difference = 1442.09731
Iteration 8: Max Difference = 532.07826
Iteration 9: Max Difference = 197.376777
Iteration 10: Max Difference = 74.6380533
Iteration 11: Max Difference = 29.9524313
Iteration 12: Max Difference = 11.4337344
Iteration 13: Max Difference = 4.43722698
Iteration 14: Max Difference = 1.76899046
Iteration 15: Max Difference = .420548538
Iteration 16: Max Difference = .037814194
Iteration 17: Max Difference = .001164231
maximum difference smaller than the tolerance level; convergence achieved
Treated units: 185 total of weights: 185
Control units: 15992 total of weights: 185
Before: without weighting
| Treat | Control
| mean variance skewness | mean variance
-------------+---------------------------------+----------------------
age | 25.82 51.19 1.115 | 33.23 122
black | .8432 .1329 -1.888 | .07354 .06813
educ | 10.35 4.043 -.7212 | 12.03 8.242
| Control
| skewness
-------------+-----------
age | .3478
black | 3.268
educ | -.4233
After: _webal as the weighting variable
| Treat | Control
| mean variance skewness | mean variance
-------------+---------------------------------+----------------------
age | 25.82 51.19 1.115 | 25.8 51.17
black | .8432 .1329 -1.888 | .8421 .133
educ | 10.35 4.043 -.7212 | 10.34 4.04
| Control
| skewness
-------------+-----------
age | 1.122
black | -1.877
educ | -.7121

use files/cps1re74.dta, clearqui ebalance treat age black educ, targets(3)qui estpost tabstat age black educ [aweight = _webal], by(treat) c(s) s(me v sk n) nototalesttab . ,varwidth(20) cells("mean(fmt(3)) variance(fmt(3)) skewness(fmt(3)) count(fmt(0))") noobs nonumber compress

(DW Subset of LaLonde Data)
------------------------------------------------------------
mean variance skewness count
------------------------------------------------------------
0
age 25.801 51.167 1.122 15992
black 0.842 0.133 -1.877 15992
educ 10.340 4.040 -0.712 15992
------------------------------------------------------------
1
age 25.816 51.194 1.115 185
black 0.843 0.133 -1.888 185
educ 10.346 4.043 -0.721 185
------------------------------------------------------------