Multivariate Nonstationary Time Series: Spectrum Analysis and Dimension Reduction

Abstract

This dissertation tackles two important problems in modern multivariate nonstationary time series analysis: spectrum analysis and dimension reduction. The first part of the dissertation introduces a nonparametric approach to multivariate time-varying power spectrum analysis. The procedure adaptively partitions a time series into an unknown number of approximately stationary segments, where some spectral components may remain unchanged across segments, allowing components to evolve differently over time. Local spectra within segments are fit through Whittle likelihood based penalized spline models of modified Cholesky components, which provide flexible nonparametric estimates that preserve positive definite structures of spectral matrices. The approach is formulated in a Bayesian framework, in which the number and location of partitions are random, and relies on reversible jump Markov chain and Hamiltonian Monte Carlo methods that can adapt to the unknown number of segments and parameters. The second part of the dissertation aims to shed lights on the usefulness of contemporaneous aggregation for high--dimensional time series analysis, especially in the forecasting point of view. Compared to other computationally intensive methods, contemporaneous aggregation has several advantages: it is simple and easy to use, it is much more computationally efficient, and its forecasting properties are well-known. We propose a statistical measure to quantify the advantages of using contemporaneous aggregation, and provide general guidelines to researchers on when to use contemporaneous aggregation.