Difference in Generalized-Differences with Panel Data: Effects of Moving from Private to Public School on Test Scores

Difference in differences (DD) relies on the key identifying condition that the untreated response variable would have grown equally across the control and treatment groups; i.e., the ¡®time effects¡¯ across the groups are the same. This condition can be rewritten as the ¡®group effects¡¯ across the time points being the same, with which this paper generalizes DD to difference in generalized-differences (DG). DG is indexed by a parameter ¥ç, and includes DD as a special case when ¥ç = 1. This makes it possible to use DG as a sensitivity analysis for DD by trying values of ¥ç other than one. Going further from sensitivity analysis, one may desire to fix ¥ç. For this, we provide a way to get a benchmark value of ¥ç using a dynamic panel data model for the control group. An empirical analysis is provided for the effects of moving from private to public school on test scores. In the empirical analysis, (i) DD magnitude is fairly sensitive to changes in ¥ç around one, but its statistical significance is not, (ii) ¥ç is significantly smaller than one in math score (and possibly in science score), and (iii) DG yields a significant negative effect of about 3-5% for reading score, but the effects are ambiguous or insignificant for the other scores. ¥ç being less than 1 means that, had the movers to public school stayed, the score gap between the movers and stayers would have narrowed. That is, the move to public school is likely to have been involuntary.