Simple Approaches to Inference with Difference-in-Differences Estimators with Small Cross-Sectional Sample Sizes

We propose simple methods for valid inference in panel data difference-in-differences(DiD) settings with a small number of treated units or a small number of control units (including a small number of both). The approach uses a suitable transformation to collapse the panel data set into a cross-sectional data set. If the classical linear model assumptions hold in the cross-sectional regression, exact inference is available – even in cases with only two control units and one treated unit. The approach likely works best with a large number of time periods before and after the intervention so that the central limit theorem across time makes normality a better approximation. Nevertheless, under joint normality and homoskedasticity of the time-varying errors, the exact approach can be applied with few time periods – both before and after the intervention – as well as few cross-sectional units. We also show how randomization inference can be implemented. In addition to standard DiD estimation, the approach permits removal of unit-specific trends. With large enough sample sizes, control variables may be included. If the cross-sectional sample size is not too small, one can use a particular heteroskedasticity-robust standard error. We illustrate the approach using increased smoking restrictions in California, where there is only a single treated unit, as an alternative to the synthetic control approach. In the staggered intervention case, we reexamine the expansion of so-called “castle” laws in the United States.