Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions

Abstract

We propose an estimation method for models of conditional moment restrictions, which contain finite dimensional unknown parameters (θ) and infinite dimensional unknown functions (h). Our proposal is to approximate h with a sieve and to estimate θ and the sieve parameters jointly by applying the method of minimum distance. We show that: (i) the sieve estimator of h is consistent with a rate faster than n −1/4 under certain metric; (ii) the estimator of θ is √ n consistent and asymptotically normally distributed; (iii) the estimator for the asymptotic covariance of the θ estimator is consistent and easy to compute; and (iv) the optimally weighted minimum distance estimator of θ attains the semiparametric efficiency bound. We illustrate our results with two examples: a partially linear regression with an endogenous nonparametric part, and a partially additive IV regression with a link function.

Identification and estimation of nonclassical nonlinear errors-in-variables models with continuous distributions using instruments

“…We also use techniques described in Ai and Chen (2003) to state more primitive regularity conditions. In their paper, there are two sieve approximations: One is used to directly estimate the conditional mean as a function of the unknown parameter, the other is the sieve approximation of the infinite-dimensional parameter estimated through the maximization procedure.…”

“…In their paper, there are two sieve approximations: One is used to directly estimate the conditional mean as a function of the unknown parameter, the other is the sieve approximation of the infinite-dimensional parameter estimated through the maximization procedure. Our setup is, in some ways, simpler than in Ai and Chen (2003), because all the unknown parameters in α are estimated through a single-step semiparametric sieve MLE (Maximum Likelihood Estimator). Since our estimator takes the form of a semiparametric sieve estimator, the very general treatment of Shen (1997) and Chen and Shen (1998) can be used to establish asymptotic normality and root n consistency under a very wide variety of conditions, including dependent and nonidentically distributed data.…”

“…The random variables x, y and z can have unbounded support R. Following Ai and Chen (2003), we first show consistency under a strong norm k·k s as a stepping stone to establishing a convergence rate under a suitably constructed weaker norm. Let…”

“…In order to proceed towards our main semiparametric asymptotic normality and root n consistency result, we then need to establish convergence at the rate o p ¡ n −1/4 ¢ in a suitable norm. In order to achieve this convergence rate under relatively weak assumptions, we employ a device introduced by Ai and Chen (2003) and employ a weaker norm k·k, under which o p ¡ n −1/4 ¢ convergence is easier to establish.…”

“…Ai and Chen (2003), Shen (1997), Chen and Shen (1998)). However, once the type of sieve and the particular form of f y|x * (y|x * ; θ) are specified, more primitive assumptions can be formulated, using some of the techniques found in Blundell, Chen, and Kristensen (2003), for instance.…”