Central Mean Subspace in Time Series

Abstract

We propose a notion of central mean dimension reduction subspace for time series {xt} which does not require specification of a model but seeks to find a p×d matrix Φd, d≤p, so that the d×1 vector ΦdTXt−1, where Xt−1=(xt−1, …, xt−p)T for some p≥1, includes all the information about xt that is available from E(xt|Xt−1). For known p and d, we estimate the mean central subspace through the Nadaraya–Watson kernel smoother and establish the strong consistency of our estimator. In addition, we propose estimation of d and p using a modified Schwarz Bayesian criterion, if either of d and p is unknown. Finally, we examine the performance of all the estimators extensively through a variety of simulations and provide a new analysis of the well-known Canadian lynx data. Supplemental materials for this article are available online.