The Unbiasedness Approach to Linear Regression Models

TL;DR

This paper introduces an unbiasedness approach to linear regression, deriving explicit estimators without error assumptions, and shows they coincide with least-squares estimators in fixed design models, with applications to AR(p).

Contribution

It presents a novel unbiased estimator for linear regression parameters that does not rely on traditional error assumptions, expanding the theoretical framework.

Findings

Unbiased estimator equals least-squares estimator for fixed design.

Explicit expression for regression parameters without error assumptions.

Application to AR(p) models demonstrated effectiveness.

Abstract

The linear regression models are widely used statistical techniques in numerous practical applications. The standard regression model requires several assumptions about the regres- sors and the error term. The regression parameters are estimated using the least-squares method. In this paper, we consider the regression model with arbitrary regressors and with- out the error term. An explicit expression for the regression parameters vector is obtained. The unbiasedness approach is used to estimate the regression parameters and its various properties are investigated. It is shown that the resulting unbiased estimator equals the least-squares estimator for the fixed design model. The analysis of residuals and the regres- sion sum of squares can be carried out in a natural way. The unbiased estimator of the dispersion matrix of the unbiased estimator is also obtained. Applications to AR(p) model and numerical examples are also discussed.