Asymptotic Properties of Empirical Quantile-Based Estimators

Abstract

We consider inference for parameters of the form $\hat{I}{¸}_0=E[F_Y^{-1}\circ F_Z(X)]$ for some variables $X$, $Y$ and $Z$. Such parameters appear, in particular, in the ``changes-in-changes'' model of \cite{AtheyImbens2006}. We first establish that $\hat{\hat{I}}¸$, a plug-in estimator of $\hat{I}{¸}_0$, is root-$n$ consistent and asymptotically normal under weaker conditions than those previously available, allowing in particular for unbounded variables. Next, we propose a new estimator of the asymptotic variance of $\hat{\hat{I}}¸$ and show its consistency, also allowing for unbounded variables. Monte Carlo simulations suggest that the conditions for root-$n$ consistency and asymptotic normality are, in some sense, minimal. These simulations highlight that our variance estimator also leads to more accurate inference than some alternative approaches.