Generalized Difference in Differences With Panel Data and Least Squares Estimator

With one treated and one untreated periods, difference in differences (DD) requires the untreated response changes to be the same across the treatment and control groups, if the treatment were withheld contrary to the fact. A natural way to check the condition is to backtrack one period and examine the response changes in two pretreatment periods. If the condition does not hold in the pretreatment periods, then a modified DD takes the form of “generalized difference in differences (GDD),” which is a triple difference (TD) with one more time-wise difference than DD. GDD generalizes DD with a weaker identification condition in the sense that a time-constant, but not necessarily zero, time/selection effect is allowed. One more time-wise differencing (quadruple difference [QD]) than GDD allows for the time/selection effect even to change over time, which makes it possible to test for the GDD identification condition. Simple panel least squares estimators (LSEs)/tests for DD and GDD are proposed and an empirical illustration is presented.