Matching

I first learned matching in 2021 spring, when I studied Econometrics at University of Wisconsin-Madison. My professor was Harold D. Chiang, a patient, detail-oriented, young professor, whose research interests include machine learning and causal inference. In 2023 spring, it is my second time to meet matching. Thanks to Professor Chiang (and other great Econometrics professors at UW-Madison) and Professor Guido W. Imbens, I have some improved understanding of matching (especially propensity score matching).

In this repository, I will share my Stata coding for estimating propensity score and applying a matching estimator to estimating treatment effect. Comments on better coding and error corrections are welcomed. Contact: ianho0815@outlook.com.

"psmatch2" versus "teffects psmatch"

There are two popular commands in Stata for propensity score matching, and their basic syntaxes are

psmatch2 t x, out(y)
teffects psmatch (y) (t x, logit)

Although both of them can estimate treatment effect by propensity score matching, they are very different in some respects:

1. psmatch2 (written by Edwin Leuven and Barbara Sianesi): To use it, please install the psmatch2 package. It by default estimates the average treatment effect on the treated (ATT or ATET) and uses a probit model to estimate propensity score. It is outdated because its estimation of standard error doesn't take into account that the propensity score is estimated instead of given/known.
2. teffects psmatch: It by default estimates the average treatment effect (ATE) and uses a logit model to estimate propensity score. It was introduced in Stata 13 (no need for manual installation) for estimating treatment effects in various ways (including propensity score). Importantly, it takes into account the fact that propensity scores are estimated when estimating standard errors, due to Abadie & Imbens (2016).

Their major differences are summarized in the following table.

	psmatch2	teffects psmatch
Installation	psmatch2	no need
Default estimate	ATT (or ATET)	ATE
Model for estimating PS	probit	logit
Estimating s.e.	incorrect	correct
Default number of tied neighbors	one	all
Weighting variable	automatically generate	no
Control unit(s) indicator	no	gen(match) option

One suggestion: If a propensity score matching model can be done using both teffects psmatch and psmatch2, then

• to get the correct standard error, use teffects psmatch;
• to get a variable for weighting, use psmatch2.

teffects actually is a powerful command not only designed to do propensity score matching. It has totally six subcommands:

• teffects psmatch for propensity score matching, which is my focus here;
• teffects ra for regression adjustment;
• teffects ipw for inverse probability weighting (IPW);
• teffects aipw for augmented IPW, also called "doubly robust";
• teffects ipwra for IPW regression adjustment;
• teffects nnmatch for nearest neighbor matching.

For greater details, please read Stata documentation (which you can find by running help teffects in Stata).

Also, please take a look at my do file (here) for examples. I used the two commands (psmatch2 and teffects psmatch) with a pseudo dataset from Social Science Computing Cooperative at University of Wisconsin-Madison. Note that in the dataset,

• probability of getting treated, $Pr(t=1)$, is positively correlated with x1 and x2.
• Both x1 and x2 are positively correlated with y.

Selecting a Specification for Estimating Propensity Score

Regardless of using psmatch2 or teffects psmatch, we need to specify what model will be used and what covariates will be included in propensity score estimation. A natural question is: How do we find a "good" specification for estimating the propensity score?

In their book "Causal Inference for Statistics, Social, and Biomedical Sciences：An Introduction" (2015), Imbens and Rubin proposed a stepwise procedure for selecting the covariates and higher-order terms for estimating the propensity score. They focus on the logit regression model, and the main problem is how to select the functions of the basic covariates included in the specification. Please read Chapter 13 in their book for details. Here I summarize their selection procedure in three steps:

1. Select a set of basic covariates: The selection of basic covariates is usually motivated by substantive knowledge. After selecting the $K_B$ basic covariates from totally $K$ covariates, we run the model and save this estimation result for later comparison.
2. Select the additional linear terms: Consider one of the remaining $K-K_B$ covariates at a time. Each time, we run the new model and perform a likelihood ratio test for the null hypothesis that the included covariate's coefficient is equal to zero. If at least one of the likehood ratio test statistics is greater than a pre-set constant ($C_L$), we add the covariate with the largest likehood ratio test statistic. After adding the new term, we check whether any of the remaining $K-K_B-1$ covariates should be included by performing the likelihood ratio test, and we continue this process until none of the likelihood ratio statistics exceeds $C_L$. Let's say that finally we include $K_L$ covariates in linear form.
3. Select the additional linear terms: We do the same process as in step 2, but in this step we only consider the second-order (i.e., quadratic and interaction) forms of the selected $K_L$ covariates and their interactions. Of course, we should have a different pre-set constant ($C_Q$) to compare with each likelihood ratio statistic.

Alvaro Carril wrote a Stata package, psestimate, for using the procedure above to estimate the propensity score. The threshold values for the likelihood ratio statistics are $C_L = 1$ and $C_Q = 2.71$ by default, corresponding to z-statistics of 1 and 1.645, respectively. We can change them by the clinear(#) and cquadratic(#) options. After running the command, we can use return list, all to see what linear and second-order terms are selected. Thus, we have a "good" specification for estimating the propensity score.

An example using National Supported Work (NSW) data files (Dehejia-Wahha Sample) is here. The dataset I used consists of 185 treated units and 260 control units, exactly the same as in Imbens (2015). I did summary statistics and created a LaTeX table (here) to report them; all statistics are identical to Table 10 in Imbens (2015). Then, I used psestimate to find the specification for estimating propensity score. Finally, by inputting this specification into the teffects psmatch, I got the estimate of the ATE.

Note: To run my codes successfully, you have to install two Stata packages: listtex and estout.