A multiple testing approach to the regularisation of large sample correlation matrices

This paper proposes a regularisation method for the estimation of large covariance matrices that uses insights from the multiple testing (MT) literature. The approach tests the statistical significance of individual pair-wise correlations and sets to zero those elements that are not statistically significant, taking account of the multiple testing nature of the problem. By using the inverse of the normal distribution at a predetermined significance level, it circumvents the challenge of estimating the theoretical constant arising in the rate of convergence of existing thresholding estimators, and hence it is easy to implement and does not require cross-validation. The MT estimator of the sample correlation matrix is shown to be consistent in the spectral and Frobenius norms, and in terms of support recovery, so long as the true covariance matrix is sparse. The performance of the proposed MT estimator is compared to a number of other estimators in the literature using Monte Carlo experiments. It is shown that the MT estimator performs well and tends to outperform the other estimators, particularly when the cross section dimension, N, is larger than the time series dimension, T.