Inference in Difference in Difference Models

Difference in differences methods have become very popular in applied work. These models are typically quite easy to implement and to interpret. However, performing inference with these models is not. This paper addresses one particular aspect that is likely to be very important in most implementations of these estimators. Identification of the key parameter often arises when a unit "changes" some particular policy. The asymptotics that is typically employed assumes that the number of observationsXtime periods is large. However, even when the number of units or time periods is large, the number of actual policy changes observed in the data is often small. In this case these estimates are not even consistent and the standard methods that researchers use for inference in these models is not appropriate. We develop a different approaches which allow researchers to perform inference in these cases. We perform asymptotics assuming that there are a finite number of policy changes in the data, but use asymptotic approximations as the number of unitsXtime periods gets large. Although one can not obtain consistent estimates, one can perform hypothesis testing and obtain confidence intervals. We demonstrate the approach using data on merit aid programs