ABSTRACT

We investigate the performance of a class of semiparametric estimators of the treatment effect via asymptotic expansions. We derive approximations to the first two moments of the estimator that are valid to “second order.” We use these approximations to define a method of bandwidth selection. We also propose a degrees of freedom–like bias correction that improves the secondorder properties of the estimator but without requiring estimation of higher-order derivatives of the unknown propensity score. We provide some numerical calibrations of the results.

INTRODUCTION

In a series of classic papers Tom (Rothenberg 1984a,b,c, 1988) introduced Edgeworth expansions to a broad audience. His treatment of the generalized least-squares estimator (1984b) in particular was immensely influential because it dealt with an estimator of central importance and the analysis was both deep and precise, but comprehensible. This is in contrast with some of the more frenzied publications about Edgeworth expansions that had hitherto appeared in econometrics journals. The use of Basu's theorem in that paper to establish the independence of the correction terms from the leading term is a well-known example of his elegant work. The reviewpaper (1984a) was also very influential and highly cited.

It is our purpose here to present asymptotic expansions for a class of semiparametric estimators used in the program evaluation literature. We have argued elsewhere (Linton 1991, 1995; Heckman et al. 1998) that the first-order asymptotics of semiparametric procedures can be misleading and unhelpful. The limiting variance matrix of the semiparametric procedure Σ does not depend on the specific details of how the nonparametric function estimator ĝ is constructed, and thus sheds no light on how to implement this important part of the procedure.