Department of Statistics

In this article, we develop a sufficient dimension reduction theory for time series. This does not require specification of a model but seeks to find a p x d matrix Phi(d) with the smallest possible number d (<= p) such that the conditional distribution of x(t)vertical bar Xt-1 is the same as that of x(t)vertical bar Phi X-T(d)t-1, where Xt-1 = (x(t-1), ... , x(t-p))(T). resulting in no loss of information about the conditional distribution of the series given its past p values. We define the subspace spanned by the columns of Phi(d) as the time series central subspace and estimate it by maximizing Kullback-Leibler distance. We show that the estimator is consistent when p and d are known. In addition, for unknown d and p, we propose a consistent estimator of d and a graphical method to determine p. Finally, we present examples and a data analysis to illustrate a theory that may open new research avenues in time series analysis.