Large‐sample approximations and change testing for high‐dimensional covariance matrices of multivariate linear time series and factor models

Various statistical problems can be formulated in terms of a bilinear form of the covariance matrix. Examples are testing whether coordinates of a high‐dimensional random vector are uncorrelated, constructing confidence intervals for the risk of optimal portfolios or testing for the stability of a covariance matrix, especially for factor models. Extending previous works to a general high‐dimensional multivariate linear process framework and factor models, we establish distributional approximations for the associated bilinear form of the sample covariance matrix. These approximations hold for increasing dimension without any constraint relative to the sample size. The results are used to construct change‐point tests for the covariance structure, especially in order to check the stability of a high‐dimensional factor model. Tests based on the cumulated sum (CUSUM), self‐standardized CUSUM and the CUSUM statistic maximized over all subsamples are considered. Size and power of the proposed testing methodology are investigated by a simulation study and illustrated by analyzing the Fama and French factors for a change due to the SARS‐CoV‐2 pandemic.