Independence Test for High Dimensional Random Vectors

TL;DR

This paper introduces a novel statistical test for assessing independence among high-dimensional random vectors, leveraging spectral distribution characteristics, with applications to financial and time series data.

Contribution

The paper develops a new mutual independence test based on spectral distribution, with proven asymptotic properties and broad applicability to various dependent structures.

Findings

Test effectively detects nonlinear and uncorrelated dependencies.

Application to stock data reveals prevalent cross-sectional dependence.

Simulation confirms the test's ability to identify complex dependence structures.

Abstract

This paper proposes a new mutual independence test for a large number of high dimensional random vectors. The test statistic is based on the characteristic function of the empirical spectral distribution of the sample covariance matrix. The asymptotic distributions of the test statistic under the null and local alternative hypotheses are established as dimensionality and the sample size of the data are comparable. We apply this test to examine multiple MA(1) and AR(1) models, panel data models with some spatial cross-sectional structures. In addition, in a flexible applied fashion, the proposed test can capture some dependent but uncorrelated structures, for example, nonlinear MA(1) models, multiple ARCH(1) models and vandermonde matrices. Simulation results are provided for detecting these dependent structures. An empirical study of dependence between closed stock prices of several companies from New York Stock Exchange (NYSE) demonstrates that the feature of cross--sectional dependence is popular in stock markets.