Banded Regularization of Autocovariance Matrices in Application to Parameter Estimation and Forecasting of Time Series

Abstract

SummaryThe paper addresses a ‘large p–small n’ problem in a time series framework and considers properties of banded regularization of an empirical autocovariance matrix of a time series process. Utilizing the banded autocovariance matrix enables us to fit a much longer auto-regressive AR(p) model to the observed data than typically suggested by the Akaike information criterion, while controlling how many parameters are to be estimated precisely and the level of accuracy. We present results on asymptotic consistency of banded autocovariance matrices under the Frobenius norm and provide a theoretical justification on optimal band selection by using cross-validation. Remarkably, the cross-validation loss function for banded prediction is related to the conditional mean-square prediction error and, thus, may be viewed as an alternative model selection criterion. The procedure proposed is illustrated by simulations and application to predicting the sea surface temperature index in the Niño 3.4 region.