A Durbin–Levinson regularized estimator of high-dimensional autocovariance matrices

Abstract

The autocovariance matrix of a stationary random process plays a central role in prediction theory and time series analysis. When the dimension of the matrix is of the same order of magnitude as the number of observations, the sample autocovariance matrix gives an inconsistent estimator. In the nonparametric framework, recent proposals have concentrated on banding and tapering the sample autocovariance matrix. We introduce an alternative approach via a modified Durbin–Levinson algorithm that receives as input the banded and tapered sample partial autocorrelations and returns a consistent and positive-definite estimator of the autocovariance matrix. We establish the convergence rate of our estimator and characterize the properties of the optimal linear predictor obtained from it. The computational complexity of the latter is of the order of the square of the banding parameter, which renders our method scalable for high-dimensional time series.