Estimation of Autocovariance Matrices for Infinite Dimensional Vector Linear Process

Abstract

Consider an infinite dimensional vector linear process. Under suitable assumptions on the parameter space, we provide consistent estimators of the autocovariance matrices. In particular, under causality, this includes the infinite‐dimensional vector autoregressive (IVAR) process. In that case, we obtain consistent estimators for the parameter matrices. An explicit expression for the estimators is obtained for IVAR(1), under a fairly realistic parameter space. We also show that under some mild restrictions, the consistent estimator of the marginal large dimensional variance–covariance matrix has the same convergence rate as that in case of i.i.d. samples.

Keywords: High‐dimensional data, IVAR, spatial variable, cross‐sectional variables, variance–covariance matrix, marginal variance–covariance matrix, coefficient matrix, parameter matrix, k‐th order autocovariance matrix, banding, consistency, convergence rate, operator norm