On singular values of large dimensional lag-tau sample autocorrelation matrices
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We study the limiting behavior of singular values of a lag-τ sample auto-correlation matrix R_τ^ϵ of error term ϵ in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) which 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. Our results take the first step to identify the number of factors in factor analysis using lag-τ sample auto-correlation matrices. Our theoretical results are fully supported by numerical experiments as well.
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