Inference in differences-in-differences with few treated groups and heteroskedasticity

Abstract

We show that the usual inference methods used in Di fferences-in-Di fferences (DID) might not perform well with few treated groups and heteroskedastic errors. One important example is when there is variation in the number of observations per group, as this generates heteroskedasticity in the aggregate DID model. In this case, methods designed to work with few treated groups tend to (under-) over-reject when the treated groups are (large) small relative to the control groups. We provide Monte Carlo simulations and placebo regressions with real datasets showing that this problem is relevant even in datasets with many observations per group. We then derive an alternative inference method that works when there are few treated groups (oreven just one) and many control groups in the presence of heteroskedasticity. Our method assumes that wecan model the heteroskedasticity of a linear combination of the errors. We show that this assumption can be satis ed without imposing strong restrictions in common DID applications. Importantly, we do not need to specify the structure of the serial correlation of the errors. Our inference method can also be combined with feasible generalized least squares (FGLS) estimation. This way, we attain an asymptotically uniformly most powerful (UMP) test if the FGLS t-test is asymptotically UMP, while still provide correct size if the serial correlation is misspeci ed. We also provide an alternative inference method that relaxes our main assumption when the number of pre-treatment periods is large and we extend our methods to linear factor models with few treated groups.