Most propensity score matching (PSM) examples typically use cross-sectional data rather than panel data. However, in accounting research, panel data (observations with two subscripts i and t, e.g., firm-years) are often used in a difference-in-differences (DID) research design. This involves two dummy variables, TREATMENT and POST , in the following regression:
Outcome = TREATMENT + POST + TREATMENT * POST
where TREATMENT indicates a treatment event and POST indicates before or after that event. In this context, it is common to perform one-to-one matching using selected pre-event and firm-level variables (Xs). These pre-event variables can be measured either at the most recent date before the event (e.g., total assets at the most recent quarter end before the event) or as an average over the pre-event period (e.g., average total assets in the four quarters preceding the event).
To conduct PSM, a probit or logit regression is needed:
TREATMENT = X1 + X2 + …
The single nearest neighbour based on propensity score is selected as the matched control observation. The treatment observations and their respective matched control observations then form the sample for subsequent DID regressions.
In Stata, the third-party module psmatch2 is commonly used to find matched control observations using PSM. To install the module, the following command can be used:
ssc install psmatch2
Once installed, the following command is typically used:
psmatch2 TREATMENT X1 X2 ..., [noreplacement logit descending]
There are three options in the above command:
noreplacement – Perform one-to-one matching without replacement. I would add this option if I want to find more unique matches.logit – Uses logit instead of the default probit regression to estimate the propensity score. I would be indifference between using logit and probit.descending – More details about this option can be found in Lunt (2014). The author concludes that “in the absence of a caliper (another option that I would omit to maximize the number of matches), the descending method provides the best matches, particularly when there is a large separation between exposed (treated) and unexposed (untreated) subjects.” Therefore, I would add this option.
psmatch2 creates several variables, with _id and _n1 being the most useful for subsequent DID regressions:
_id is a new identifier created for all observations in the case of one-to-one and nearest-neighbors matching._n1 stores the new identifier (_id) of the matched control observation for every treatment observation.
There is one limitation with psmatch2. Sometimes, we may want the treatment and its matched control to have the same value on a variable X. For example, we may want the treatment and its matched control to be drawn from the same industry, or both to be male or female. psmatch2 lacks a direct solution for this requirement. Some imperfect workarounds, such as adding i.industry or i.gender in Xs, are discussed in this post. In contrast, the PSMATCH procedure in SAS provides a perfect solution by offering the EXACT= statement. I am not sure if SAS achieves this by implementing a stratification method, but if it does, it is possible that psmatch2 in Stata could achieve similar results by tweaking its options. More details on the PSMATCH procedure in SAS can be found in this manual.
It is worth noting that that psmatch2 is preferable to Stata’s built-in command teffects because the variables generated by psmatch2 (particularly _id and _n1) are necessary for subsequent DID regressions, whereas teffects does not return such variables.
This article aims to provide a quick how-to and may omit some necessary steps for PSM, such as assessing covariate balance. A more rigorous discussion on PSM in accounting research can be found in Shipman, Swanquist, and Whited (2017).
I would like to express my gratitude to the authors of the following articles that have been beneficial in preparing this post:
