Dimension Reduction for High-Dimensional Multivariate Heteroskedastic Time Series via Envelope Methods

Author

Toktam Valizadeh kozalabad, Southern Illinois University CarbondaleFollow

Date of Award

8-1-2025

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Samadi, Yaser

Abstract

Multivariate time series data, prevalent in fields such as finance, economics, and neuroscience, often display complex features including heteroskedasticity and dynamic structural changes. Ignoring these factors can lead to inefficient and biased estimation as well as inaccurate forecasting. Although traditional models, such as vector autoregressive (VAR) and matrix autoregressive (MAR) frameworks, are widely used, they often assume constant volatility or rely on oversimplified covariance structures, limiting their applicability and interpretability in high-dimensional, time-varying contexts.This dissertation addresses these limitations by developing a unified set of models that integrate envelope methods and reduce subspace techniques to efficiently handle heteroskedasticity, dimensionality reduction, and time-varying dynamics in high-dimensional multivariate time series. We introduce the unconditional heteroskedastic Envelope VAR (HEVAR) models, which link the mean function to time-varying unconditional covariances through a minimal reducing subspace, thereby improving estimation stability and forecasting accuracy. Extending this concept to matrix-valued time series, we propose the unconditional heteroskedastic Envelope MAR (HEMAR) models, which preserve the inherent row-column structures while modeling both high dimensionality and heteroskedasticity. Furthermore, we extend this framework to dynamic brain connectivity analysis by creating a reducing subspace framework for multivariate GARCH (MGARCH) models, enabling improved functional connectivity estimation and facilitating brain health classification from fMRI data. Finally, we propose a time-varying MAR (TV-MAR) model that accommodates smoothly evolving structures in both the coefficient matrices and innovation covariance components, further enhancing model flexibility for structured high-dimensional time series and capturing rich temporal dependencies often overlooked by static models. Across all proposed models, we establish theoretical and asymptotic properties, develop (quasi-)maximum likelihood estimation procedures, and demonstrate the practical advantages through extensive simulations and real-world applications. Together, this work contributes new tools for modeling, inference, forecasting, classification, and uncovering complex structures in high-dimensional, heteroskedastic, and dynamically evolving multivariate time series.