dissertationRecent economic crises have exposed a critical need for appropriate methods to measure, model, and predict financial volatility. Generalized autoregressive conditional heteroskedastic (GARCH) models have been among the most successful and widely studied tools for this task due to their ability to capture the stylized characteristics of financial data. Extending the original univariate GARCH processes to the multivariate framework is important because, in many applications, the primary quantity of interest is the interdependence, or covariance, between different univariate processes. Covariances are used for calculations of hedge ratios, betas of CAPM (Capital Asset Pricing Model), portfolio VaR (Value at Risk) estimates, asset weights in portfolios, and to investigate contagion across financial markets. In Chapter 1 of this dissertation, we briefly review concepts and terminology related to stochastic processes and time series analysis. In Chapter 2, we prove sufficient conditions for existence, uniqueness, and stochastic stability of multivariate GARCH processes. In Chapter 3, we explore the QMLE and VTE methods for estimating multivariate GARCH parameters. We prove sufficient conditions for strong consistency and asymptotic normality of the QMLE and VTE estimators, and we conduct simulation studies to compare the performance of the VTE and QMLE
