Independence test for high dimensional data based on regularized canonical correlation coefficients

This paper proposes a new statistic to test independence between two high dimensional random vectors X:p₁ × 1 and Y:p₂ × 1. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of X and Y. The asymptotic distribution of the statistic under the null hypothesis is established as a corollary of general central limit theorems (CLT) for the linear statistics of classical and regularized sample canonical correlation coefficients when p₁ and p₂ are both comparable to the sample size n. As applications of the developed independence test, various types of dependent structures, such as factor models, ARCH models and a general uncorrelated but dependent case, etc., are investigated by simulations. As an empirical application, cross-sectional dependence of daily stock returns of companies between different sections in the New York Stock Exchange (NYSE) is detected by the proposed test.