Sieve Empirical Likelihood and Extensions of the Generalized Least Squares

Abstract

The empirical likelihood cannot be used directly sometimes when an infinite dimensional parameter of interest is involved. To overcome this difficulty, the sieve empirical likelihoods are introduced in this paper. Based on the sieve empirical likelihoods, a unified procedure is developed for estimation of constrained parametric or non-parametric regression models with unspecified error distributions. It shows some interesting connections with certain extensions of the generalized least squares approach. A general asymptotic theory is provided. In the parametric regression setting it is shown that under certain regularity conditions the proposed estimators are asymptotically efficient even if the restriction functions are discontinuous. In the non-parametric regression setting the convergence rate of the maximum estimator based on the sieve empirical likelihood is given. In both settings, it is shown that the estimator is adaptive for the inhomogeneity of conditional error distributions with respect to predictor, especially for heteroscedasticity.

Empirical Likelihood-Based Inference in Conditional Moment Restriction Models

“…The impact of having high-dimensional moment conditions on the finite sample performance of their estimator remains to be seen. Zhang and Gijbels (2001) independently develop a methodology close to ours. They consider parametric and nonparametric regression models, whereas we consider parametric conditional moment models that nest regression as a special case.…”

Conditional empirical likelihood estimation and inference for quantile regression models

“…We conjecture that analogous results of Donald et al (2003) will hold in our setup. y 2 Y, CEL by Kitamura et al (2004) and Zhang and Gijbels (2003) is defined as max fp ji g n i;j¼1 X n i¼1 I in X n j¼1 w ji log p ji ; s:t: p ji X0; X n j¼1 p ji ¼ 1; X n j¼1 p ji gðz j ; yÞ ¼ 0; i; j ¼ 1; . .…”

“…1 First, we apply the method of conditional empirical likelihood (CEL) by Kitamura et al (2004) and Zhang and Gijbels (2003) to quantile regression. Second, to avoid practical problems of the CEL method induced by the discontinuity in the parameters of CEL, we propose its smoothed counterpart, called smoothed conditional empirical likelihood (SCEL).…”

“…CEL is called ''smoothed'' and ''sieve'' empirical likelihood inKitamura et al (2004) andZhang and Gijbels (2003), respectively. Since we introduce additional smoothing on moment restrictions, we follow the terminology ofKitamura (2003) and describe their method as ''conditional'' empirical likelihood in order to avoid confusion.3 Zhao(2001)considered two methods for estimating nonparametric optimal weights, the kernel and naive methods (e.g., the k-nearest neighborhood mean of absolute residuals).…”

Penalized empirical likelihood estimation of semiparametric models

“…The purpose of this paper is to extend the empirical likelihood approach [23] to the semiparametric setup in (1) and propose an asymptotically efficient estimation method for 0 . Our method is based on the method of conditional empirical likelihood (CEL) by Zhang and Gijbels [37] and Kitamura et al [14], 1 and the method of penalization (see, e.g., [34,27,28] and references therein). Thus, the proposed estimator is called the penalized empirical likelihood (PEL) estimator.…”