On singular values of large dimensional lag-tau sample autocorrelation matrices

TL;DR

This paper analyzes the spectral properties of lag-$\tau$ sample auto-correlation matrices in high-dimensional factor models, establishing their limiting spectral distribution and the behavior of their largest singular value.

Contribution

It provides the first theoretical characterization of the spectral distribution and largest singular value of lag-$\tau$ auto-correlation matrices in high-dimensional settings, aiding factor number identification.

Findings

The LSD of the auto-correlation matrix matches that of the auto-covariance matrix.

The largest singular value converges to the edge of the LSD support.

Numerical experiments support the theoretical results.

Abstract

We study the limiting behavior of singular values of a lag-$\tau$ sample auto-correlation matrix $\bf{R}_{\tau}^{\epsilon}$ of error term $\epsilon$ in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) which characterizes the global spectrum of $\bf{R}_{\tau}^{\epsilon}$, and derive the limit of its largest singular value. All the asymptotic results are derived under the high-dimensional asymptotic regime where the data dimension and sample size go to infinity proportionally. Under mild assumptions, we show that the LSD of $\bf{R}_{\tau}^{\epsilon}$ is the same as that of the lag-$\tau$ sample auto-covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of $\bf{R}_{\tau}^{\epsilon}$ converges almost surely to the right end point of the support of its LSD. Our results take the first step to identify the number of factors in factor analysis using lag-$\tau$ sample auto-correlation matrices. Our theoretical results are fully supported by numerical experiments as well.