Quantile Difference in Differences with Time-Varying Qualification in Panel Data

Abstract

Abstract This paper investigates the identification and estimation of the quantile treatment effect in a difference in differences (DID) setting when treatment is provided only to qualified individuals at a certain point in time and the qualification is time-varying. The time-varying qualification may affect an outcome variable even when the actual effect of treatment is zero. We show how to account for this “movers effect” bias and propose the quantile treatment effect on “in-stayers” that are qualified both before and after the treatment. The estimate is identified under three main assumptions: (i) panel data availability, (ii) a distributional common trend assumption conditional on covariates, and (iii) a copula stability assumption. We then apply our method to estimate the effects of an increase in the benefits of the Supplemental Nutrition Assistance Program (SNAP) on recipients’ food expenditure shares. The results show significant heterogeneity and highlight the importance of accounting for time-varying qualification.