“Difference in Quasi-Differences with Panel Data: Reversed Effects of Private School” by Myoung-jae LEE
School of Economics and Social Sciences Singapore Management University
Difference in differences (DD) is a popular way of finding treatment effects with observational data. DD relies on the identification condition that, without the treatment, the response variable would have grown equally across the control and treatment groups, controlling the observed covariates. This selection-on-observable assumption, however, may be violated, and this paper generalizes DD to quasi-difference in differences (QD) to allow for the violation, as well as to difference in quasi-differences (DQ). QD indexed by a parameter γ and DQ indexed by η include DD as a special case when γ = 1 and η = 1, respectively, and it is shown that there is little justification for setting γ = 1 and η = 1 a prior. In view of this, QD and DQ provide sensitivity analyses for DD by trying values of γ and η other than unity. Going further, for DQ, we explore constructive and data-dependent ways to find the right value of η. An empirical analysis is provided for the effects of being in public school (i.e., “reversed effects” of being in primary school) at grade 10 on those who were in private school at grade 8, where the response variables are various logged test scores. In the empirical analysis, (i) DD is fairly sensitive to changes in η and γ, (ii) η is significantly smaller than one in math score, and (iii) DQ yields a significant negative effect of about 3-5% for reading score, but the effects are ambiguous or insignificant for the other scores. The DQ parameter η being less than 1 means that, had the movers to public school stayed contrary to the fact, the score gap between the movers and stayers would have narrowed. That is, the move to public school is likely to have been involuntary.