In Defense of the Pre-Test: Valid Inference when Testing Violations of Parallel Trends for Difference-in-Differences

The difference-in-differences (DID) research design is a key identification strategy which allows researchers to estimate causal effects under the parallel trends assumption. While the parallel trends assumption is counterfactual and cannot be tested directly, researchers often examine pre-treatment periods to check whether the time trends are parallel before treatment is administered. Recently, researchers have been cautioned against using preliminary tests which aim to detect violations of parallel trends in the pre-treatment period. In this paper, we argue that preliminary testing can -- and should -- play an important role within the DID research design. We propose a new and more substantively appropriate conditional extrapolation assumption, which requires an analyst to conduct a preliminary test to determine whether the severity of pre-treatment parallel trend violations falls below an acceptable level before extrapolation to the post-treatment period is justified. This stands in contrast to prior work which can be interpreted as either setting the acceptable level to be exactly zero (in which case preliminary tests lack power) or assuming that extrapolation is always justified (in which case preliminary tests are not required). Under mild assumptions on how close the actual violation is to the acceptable level, we provide a consistent preliminary test as well confidence intervals which are valid when conditioned on the result of the test. The conditional coverage of these intervals overcomes a common critique made against the use of preliminary testing within the DID research design. We use real data as well as numerical simulations to illustrate the performance of the proposed methods.