A Practical Asymptotic Variance Estimator for Two-Step Semiparametric Estimators

Abstract

The goal of this paper is to develop techniques to simplify semiparametric inference. We do this by deriving a number of numerical equivalence results. These illustrate that in many cases, one can obtain estimates of semiparametric variances using standard formulas derived in the well-known parametric literature. This means that for computational purposes, an empirical researcher can ignore the semiparametric nature of the problem and do all calculations as if it were a parametric situation. We hope that this simplicity will promote the use of semiparametric procedures.

Design-free estimation of variance matrices

“…We start by addressing the question of how quickly k can be allowed to grow as n increases, a case where one could use a sieve version of our estimator subject to a restriction on the speed of growth of k. For example, the mean vector can be estimated by a sieve as in Antoniadis (1988, Theorem 3.1) and Ackerberg, Chen, and Hahn (2012), then the variance matrix obtained consistently as in Lemma 2 of the latter reference. Using the element-wise definition of convergence employed so far, consistency requires k = o(n) for both estimators Σ and Σ m,q .…”

Generalized Jackknife Estimators of Weighted Average Derivatives

“…Newey (1994a, Example 3 and Theorem 7.2) considered a linear sieve (series) estimation of average derivative parameter E ∂ ∂x g(x) . Moreover, , Ai and Chen (2007), and others have shown how to consistently estimate the variance of a sieve semiparametric two-step estimator easily, while and Ackerberg, Chen, and Hahn (2012) provided a numerically equivalent way to compute standard errors of a large class of semiparametric two-step estimator when the first step nuisance functions are estimated via linear sieves. One additional benefit of using sieve estimation in the first step is that a cross-validated choice of sieve number of terms to get optimal mean squared error rate in the first step would typically lead to root-n asymptotic normality of the second step plug-in estimate of θ .…”

“…There are alternative consistent variance estimators that might have better finite sample performance: (a) a jackknife variance estimator (e.g., Shao and Wu (1989) and the references therein); (b) instead of computing a standard error based on the asymptotic variance expression, one could use a finite sample (or "fixing smoothing parameter") version such as in Newey (1994a,b), Ai and Chen (2007), Ackerberg, Chen, and Hahn (2012).…”

Penalized sieve estimation and inference of semi-nonparametric dynamic models: a selective review

“…For iid data, Ackerberg, Chen and Hahn (2010) show that in a large class of semiparametric models, one can greatly simplify the estimation of Avar( b n ), provided that the …rst stage unknown function h is estimated by a sieve (or series) method. They show, by extending earlier work of Newey (1994), that the consistent estimate of the semiparametric Avar( b n ) using the method of Ai and Chen (2007) is numerically identical to the estimate of the parametric asymptotic variance using the standard parametric two-step framework of Murphy and Topel (1985).…”