🤖 AI Summary
This paper addresses bias and efficiency concerns in Difference-in-Differences (DiD) estimation of the Average Treatment Effect on the Treated (ATT) under compositional change—i.e., time-varying population structure—in repeated cross-sectional data. Methodologically, it introduces: (i) the first nonparametric Hausman-type test to detect compositional change; (ii) a rate-form doubly robust estimation framework; and (iii) a consistent stochastic expansion for a local polynomial multinomial logit estimator. Theoretically, it achieves the semiparametric efficiency bound with optimal convergence rates within a semiparametric efficiency framework and characterizes the bias–efficiency trade-off arising from ignoring compositional change. Monte Carlo simulations and empirical applications demonstrate that the proposed method substantially outperforms existing approaches in bias control, statistical efficiency, and test power.
📝 Abstract
This paper studies Difference-in-Differences (DiD) setups with repeated cross-sectional data and potential compositional changes across time periods. We begin our analysis by deriving the efficient influence function and the semiparametric efficiency bound for the average treatment effect on the treated (ATT). We introduce nonparametric estimators that attain the semiparametric efficiency bound under mild rate conditions on the estimators of the nuisance functions, exhibiting a type of rate doubly robust (DR) property. Additionally, we document a trade-off related to compositional changes: We derive the asymptotic bias of DR DiD estimators that erroneously exclude compositional changes and the efficiency loss when one fails to correctly rule out compositional changes. We propose a nonparametric Hausman-type test for compositional changes based on these trade-offs. The finite sample performance of the proposed DiD tools is evaluated through Monte Carlo experiments and an empirical application. We consider extensions of our framework that accommodate double machine learning procedures with cross-fitting, and setups when some units are observed in both pre- and post-treatment periods. As a by-product of our analysis, we present a new uniform stochastic expansion of the local polynomial multinomial logit estimator, which may be of independent interest.