Positive Definiteness of the Asymptotic Covariance Matrix of OLS Estimators in Parsimonious Regressions; Comments on ’Testing a Large Set of Zero Restrictions in Regression Models, With an Application to Mixed Frequency Granger Causality’ by Ghysels, Hill, and Motegi (2020)

Abstract

Recently, Ghysels, Hill, and Motegi (2020) have proposed a test for a large set of zero restrictions on coefficients in regression models. They referred to the test as a ``max test''. The test statistic for the max test is calculated by first running OLS regressions, each of which includes only one of explanatory variables whose coefficients are under examination, and then taking the maximum value of the squared OLS estimates of those coefficients. They called those regressions ``parsimonious regressions''. In this paper, we answer a question raised in Remark 2.4 in Ghysels, Hill, and Motegi (2020), namely, whether the asymptotic covariance matrix of the OLS estimators in the parsimonious regressions is, in general, positive definite. We show that it is generally positive definite. The result may be utilized to facilitate the calculation of the simulated p values necessary for implementing the max test.