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Inference in Differences-in-Differences with Few Treated Groups and Heteroskedasticity

Ferman, Bruno and Pinto, Cristine (2015): Inference in Differences-in-Differences with Few Treated Groups and Heteroskedasticity.

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Abstract

Differences-in-Differences (DID) is one of the most widely used identification strategies in applied economics. However, how to draw inferences in DID models when there are few treated groups remains an open question. We show that the usual inference methods used in DID models might not perform well when there are few treated groups and errors are heteroskedastic. In particular, when there is variation in the number of observations per group, inference methods designed to work when there are few treated groups tend to (under-) over-reject the null hypothesis when the treated groups are (large) small relative to the control groups. This happens because larger groups tend to have lower variance, generating heteroskedasticity in the group x time aggregate DID model. We provide evidence from Monte Carlo simulations and from placebo DID regressions with the American Community Survey (ACS) and the Current Population Survey (CPS) datasets to show that this problem is relevant even in datasets with large numbers of observations per group. We then derive an alternative inference method that provides accurate hypothesis testing in situations where there are few treated groups (or even just one) and many control groups in the presence of heteroskedasticity. Our method assumes that we can model the heteroskedasticity of a linear combination of the errors. We show that this assumption can be satisfied without imposing strong assumptions on the errors 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 square (FGLS) estimation. This way, it is possible to attain an asymptotically uniformly most powerful (UMP) test if the FGLS t-test is asymptotically UMP, while still provide a test with correct size if the serial correlation is misspecified. With many pre-treatment periods, we provide an alternative inference method that relies on strict stationarity and ergodicity of the time series instead of relying on the correct specification of the heteroskedasticity. Finally, we extend our inference methods to linear factor models when there are few treated groups.

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