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

Abstract

The 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 autoregressive 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.

A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices

“…The main objective is to compare the performance with the banded and tapered estimator (4) by (McMurry and Politis, 2010), referred to as the MP estimator; the overall conclusions can be extended to the class of banded estimator (3) proposed by Bickel andGel (2011) All the reported results are based on 1,000 replications. The selection of the banding parameter is based on the empirical rule proposed by Politis (2003) and used by McMurry and Politis (2010), applied to the sample autocorrelations for the MP estimator and to the sample partial autocorrelations for the RDL estimator; see section 5.…”

“…Naturally, we can apply the MP rule (19) with K n varying with the forecast lead time. Bickel and Gel (2011) propose a cross-validation method which divides the time series into two consecutive segments of length n 0 (e.g. n/3) and n 1 , respectively.…”

“…Wu and Pourahmadi (2009) proved the consistency of (3) for the class of non-linear short-range dependent processes considered by Wu (2005), and obtained an explicit upper bound for the operator norm ofΓ n, −Γ n See Appendix A for a review of the different matrix norms. Bickel and Gel (2011) obtained the consistency of (3) under the Frobenius norm. Bickel and Gel (2011), and proposed a crossvalidation method for selecting the optimal banding parameter.…”

“…Bickel and Gel (2011) obtained the consistency of (3) under the Frobenius norm. Bickel and Gel (2011), and proposed a crossvalidation method for selecting the optimal banding parameter.…”

“…The introduction of p n > L is needed for having (2015) also need to introduce p n to establish the convergence of the best linear predictor to the oracle predictor. Bickel and Gel (2011) defined their -banded estimator in terms of a submatrix ofΓ n of dimension p n < n, where p n = o(n).…”