Dynamic Orthogonal Subseries for High-Dimensional and Nonstationary Time Series

Abstract

A multivariate time series could be partitioned either horizontally (over time) to induce local stationarity or vertically (over the variables) to reduce dimension and the high computational cost. Dimension reduction for a high-dimensional time series by linearly transforming it into several lower-dimensional subseries (vertical partition) where any two subseries are uncorrelated both temporally and cross-sectionally is of central importance in the modern age of big data. It reduces the challenging multivariate estimation problem with many parameters to that of a number of disjoint lower-dimensional problems with much fewer parameters. A notable example in the previous studies is the dynamic orthogonal components (DOC) utilizing nonlinear optimization which works well for stationary and low-dimensional time series data. First we reduce the computational burden of DOC by connecting it to the time series principal components analysis (TS-PCA) method in recent studies based on eigenanalysis of a positive-definite matrix. Next, we extend DOC to nonstationary processes which can be divided into several nearly homogeneous segments. Consistency and joint asymptotic normality of the estimates of the Givens angles parameterizing orthogonal matrices in each segment are established under some regularity conditions. Applications to multivariate volatility modeling in finance are illustrated using simulated and real datasets.