Positive definiteness of the asymptotic covariance matrix of OLS estimators in parsimonious regressions

TL;DR

This paper proves that the asymptotic covariance matrix of OLS estimators in parsimonious regressions is generally positive definite, aiding the implementation of the max test for large coefficient null hypotheses.

Contribution

It provides a theoretical proof of positive definiteness of the covariance matrix in parsimonious regressions, addressing a question raised in prior research.

Findings

The covariance matrix is generally positive definite.

This result facilitates the calculation of simulated p-values for the max test.

Supports the validity of the max test in large coefficient testing.

Abstract

Recently, Ghysels, Hill, and Motegi (2020) proposed a test for examining whether a large number of coefficients in linear regression models is zero. The test is called the max test. The test statistic is calculated by first running multiple ordinary least squares (OLS) regressions, each including only one of key regressors, whose coefficients are supposed to be zero under the null, and then taking the maximum value of the squared OLS coefficient estimates of those key regressors. They called these regressions parsimonious regressions. This paper answers a question raised in their Remark 2.4; whether the asymptotic covariance matrix of the OLS estimators in the parsimonious regressions is generally positive definite. The paper shows that it is generally positive definite, and the result may be utilized to facilitate the calculation of the simulated p value necessary for implementing the max test.