A Test of the Efficiency of a Given Portfolio in High Dimensions

We extend the Gibbons–Ross–Shanken test to high-dimensional cases, when the num-ber of test assets far exceeds the sample size and the return covariance matrix is ill-conditioned or singular, as inevitably occurs with large, richly specified test port-folios. In such cases, one must use a regularized (and therefore biased) estimator of the covariance matrix, which distorts the original GRS test statistic. We use Random Matrix Theory to correct for this bias and characterize the asymptotic power of the resulting test. Power increases with the number of test assets and reaches its maximum across a broad range of local alternatives. These findings are supported by extensive simulations. We empirically implement the test on state-of-the-art candidate factor portfolios and test assets to evaluate conditional asset pricing performance.