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Recently, Ghysels, Hill, and Motegi (2020) proposed a test for examiningwhether a large number of coefficients in linear regression models is zero. Thetest is called the max test. The test statistic is calculated by first runningmultiple ordinary least squares (OLS) regressions, each including only one ofkey regressors, whose coefficients are supposed to be zero under the null, andthen taking the maximum value of the squared OLS coefficient estimates of thosekey regressors. They called these regressions parsimonious regressions. Thispaper answers a question raised in their Remark 2.4; whether the asymptoticcovariance matrix of the OLS estimators in the parsimonious regressions isgenerally positive definite. The paper shows that it is generally positivedefinite, and the result may be utilized to facilitate the calculation of thesimulated p value necessary for implementing the max test.

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