Martingale Difference Divergence Matrix and Its Application to Dimension Reduction for Stationary Multivariate Time Series

In this article, we introduce a new methodology to perform dimension reduction for a stationary multivariate time series. Our method is motivated by the consideration of optimal prediction and focuses on the reduction of the effective dimension in conditional mean of time series given the past information. In particular, we seek a contemporaneous linear transformation such that the transformed time series has two parts with one part being conditionally mean independent of the past. To achieve this goal, we first propose the so-called martingale difference divergence matrix (MDDM), which can quantify the conditional mean independence of V ∈ Rp given U ∈ Rq and also encodes the number and form of linear combinations of V that are conditional mean independent of U. Our dimension reduction procedure is based on eigen-decomposition of the cumulative martingale difference divergence matrix, which is an extension of MDDM to the time series context. Interestingly, there is a static factor model representation for our dimension reduction framework and it has subtle difference from the existing static factor model used in the time series literature. Some theory is also provided about the rate of convergence of eigenvalue and eigenvector of the sample cumulative MDDM in the fixed-dimensional setting. Favorable finite sample performance is demonstrated via simulations and real data illustrations in comparison with some existing methods. Supplementary materials for this article are available online.