Difference-in-differences with Economic Factors and the Case of Housing Returns

Abstract

This paper studies how to incorporate observable factors in difference-in-differences and document their empirical relevance. We show that even under random assignment directly adding factors with unit-specific loadings into the difference-in-differences estimation results in biased estimates. This bias, which we term the “bad time control problem” arises when the treatment effect covaries with the factor variation. Researchers often control for factor structures by using: (i) unit time trends, (ii) pre-treatment covariates interacted with a time trend and (iii) group-time dummies. We show that all these methods suffer from the bad time control problem and/or omitted factor bias. We propose two solutions to the bad time control problem. To evaluate the relevance of the factor structure we study US housing returns. Adding macroeconomic factors shows that factors have additional explanatory power and estimated factor loadings differ systematically across geographic areas. This results in substantially altered treatment effects.