Simple Transformation Approach to Difference-in-Differences Estimation for Panel Data

Abstract

In the case of panel data, we propose a simple time-series transformation that can be combined with various treatment effect estimators, including regression adjustment, matching methods, and doubly robust estimators. The approach is motivated by the fact that, in the common timing case, our transformation, when applied with linear regression adjustment, numerically reproduces the pooled OLS estimator in Wooldridge (2021). In the general staggered case, the transformation is at the unit level, and simply requires computing the average outcome prior to an intervention, subtracting it from a post-treatment outcome, and then carefully selecting the control units in each time period. We show formally that, allowing for staggered entry under no anticipation and parallel trends assumptions, the cohort treatment indicators satisfy the key unconfoundedness assumption with respect to the transformed potential outcome. Given identification, any number of treatment effect estimators can be applied for each treated cohort and calendar time pair where the average treatment effects on the treated are identified. In effect, we establish the consistency of intuitively appealing rolling methods. The doubly robust method of combining inverse probability weighting with linear regression works particularly well in terms of bias and efficiency. Long differencing methods, such as those proposed by Callaway and Sant'Anna (2021), can be considerably less efficient. We also show how to modify the transformation to account for unit-specific trends.