Which <scp>GRS</scp> Statistic Is Appropriate for Cross‐Sectional Tests of Linear Multi‐Factor Pricing Models?

ABSTRACT Fama and French (2015, 2017) introduce the five‐factor asset pricing model in the former paper and test their model on data from international financial markets in the latter paper. Each paper tests whether the five‐factor model represents returns by way of the Gibbons, Ross and Shanken (1989) (hereafter GRS) statistic. That statistic's null hypothesis jointly sets all cross‐section intercepts (alpha) to zero. The GRS statistic developed and presented in equation (4) on page 1124 of GRS (1989) is a cross‐section test of the one‐factor capital asset pricing model. Using the same data as Fama and French (2015, 2017), we show that the latter authors did not use the GRS (1989) statistic given in equation (4) on page 1124. In fact, they used a version of that statistic appropriate for the five‐factor model. To provide clarity on this issue, this paper provides a detailed mathematical derivation of the cross‐sectional variance of the OLS estimators of the intercepts when N versions of the K ‐factor model are estimated. This variance is then used to construct the enhanced version of the GRS statistic. Its finite sample distribution is then rigorously established. To obtain that distribution, restrictions are made on cross‐sectional variances and covariances of the errors of pricing models that are inconsistent with times series data. We derive the variance–covariance of the estimated intercepts of the K ‐factor model without making these restrictions. An almost sure approximation to that estimator is constructed here which is then used to obtain the asymptotic distribution of the GRS statistic. We call it the robust GRS statistic. Using data of Fama and French (2015, 2017), we use the robust GRS statistic to reconstruct their tables 5 and 4, respectively. As the distribution of the robust GRS does not change with the number of factors, in contrast to the finite sample version of this statistic, it allows for a more nuanced comparison of three‐, four‐ and five‐factor models. The power functions of the GRS (1989) statistic are compared with the enhanced version of the GRS appropriate for K factors.