Empirical Likelihood Estimation for Linear Regression Models with AR(p) Error Terms

TL;DR

This paper introduces a distribution-free empirical likelihood method for estimating parameters in linear regression models with AR(p) error terms, outperforming traditional CML methods in simulations and real data.

Contribution

It proposes a novel empirical likelihood estimation approach for linear regression with AR(p) errors, avoiding distributional assumptions and demonstrating superior performance.

Findings

EL estimators have lower MSE than CML in simulations.

EL estimators exhibit less bias across configurations.

Numerical and real data examples confirm the effectiveness of EL method.

Abstract

Linear regression models are useful statistical tools to analyze data sets in several different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors with zero mean and constant variance. However, for some data sets error terms may not satisfy these or some of these assumptions. If error terms are correlated, such as the regression models with autoregressive (AR(p)) error terms, the Conditional Maximum Likelihood (CML) under normality assumption or the Least Square (LS) methods are often used to estimate the parameters of interest. For CML estimation a distributional assumption on error terms is needed to carry on estimation, but, in practice, such distributional assumptions on error terms may not be plausible. Therefore, in such cases some alternative distribution free methods are needed to conduct the parameter estimation. In this paper, we propose to estimate the parameters of a linear regression model with AR(p) error term using the Empirical Likelihood (EL) method, which is one of the distribution free estimation methods. A small simulation study and a numerical example are provided to evaluate the performance of the proposed estimation method over the CML method. The results of simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from the CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.