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

TL;DR

This paper introduces a new statistical test for independence between high-dimensional vectors using regularized canonical correlation coefficients, with theoretical distribution results and applications to financial data.

Contribution

It develops a novel independence test based on regularized canonical correlations and derives its asymptotic distribution for high-dimensional data.

Findings

The test accurately detects dependence structures in simulations.

It effectively identifies cross-sectional dependence in stock returns.

The asymptotic distribution under the null hypothesis is established.

Abstract

This paper proposes a new statistic to test independence between two high dimensional random vectors ${\mathbf{X}}:p_1\times1$ and ${\mathbf{Y}}:p_2\times1$. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of ${\mathbf{X}}$ and ${\mathbf{Y}}$. 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 $p_1$ and $p_2$ 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.