Wilks' theorem for semiparametric regressions with weakly dependent data

TL;DR

This paper extends empirical likelihood inference to semiparametric models with weakly dependent data, incorporating a partially linear single-index model and a variance model, and establishes Wilks' theorem in this context.

Contribution

It develops a Wilks' theorem for empirical likelihood in semiparametric regressions with dependent data, including models for mean and variance.

Findings

Empirical likelihood ratio converges to a chi-squared distribution.

The method handles weak dependence and nonparametric components.

The approach includes models for both mean and variance functions.

Abstract

The empirical likelihood inference is extended to a class of semiparametric models for stationary, weakly dependent series. A partially linear single-index regression is used for the conditional mean of the series given its past, and the present and past values of a vector of covariates. A parametric model for the conditional variance of the series is added to capture further nonlinear effects. We propose a fixed number of suitable moment equations which characterize the mean and variance model. We derive an empirical log-likelihood ratio which includes nonparametric estimators of several functions, and we show that this ratio has the same limit as in the case where these functions are known.