On singular values of large dimensional lag-[formula omitted] sample auto-correlation matrices.

We study the limiting behavior of singular values of a lag- τ sample auto-correlation matrix R τ ϵ of large dimensional vector white noise process, the error term ϵ in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) that characterizes the global spectrum of R τ ϵ , 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 R τ ϵ is the same as that of the lag- τ sample auto-covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of R τ ϵ converges almost surely to the right end point of the support of its LSD. Based on these results, we further propose two estimators of total number of factors with lag- τ sample auto-correlation matrices in a factor model. Our theoretical results are fully supported by numerical experiments as well. [ABSTRACT FROM AUTHOR]