Near Optimal Estimation of Average Regression Functionals; Chen Qiu (Cornell University)

Abstract

This paper considers estimation of a continuous linear regression functional that can be written as a weighted population mean of observed outcome. A leading example is the population average treatment effect under unconfoundedness and overlap conditions, when the weight function is the product of binary treatment and inverse propensity score. We propose a new plug-in estimator of the functional when the weight function is approximated in series space. This estimator is near optimal in the sense that its mean square remainder error is controlled as small as possible in finite sample. We characterize its asymptotic distribution, allowing the number of basis functions k to grow proportionally to sample size n. We compare both finite sample and asymptotic performance of our estimator with doubly robust (DR) estimators. We find DR estimators often do not have materially smaller mean square remainder error in finite sample. DR estimators also do not improve the asymptotic performance when the ratio of k to n is smaller than 1. When the ratio is larger than 1, we propose a modified DR estimator to improve the asymptotic performance of our plug-in estimator. We apply our method to the work of Ferraz and Finan (2011) and conduct simulations to support theoretic findings.

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