No abstract

Nonstationary time series data exist in various scientific disciplines, including environmental science, biology, signal processing, econometrics, among others. Many Bayesian models have been developed to handle nonstationary time series. The time-varying vector autoregressive (TV-VAR) model is a well-established model for multivariate nonstationary time series. Nevertheless, in most cases, the large number of parameters presented by the model results in a high computational burden, ultimately limiting its usage. This paper proposes a computationally efficient multivariate Bayesian Circular Lattice Filter to extend the usage of the TV-VAR model to a broader class of high-dimensional problems. Our fully Bayesian framework allows both the autoregressive (AR) coefficients and innovation covariance to vary over time. Our estimation method is based on the Bayesian lattice filter (BLF), which is extremely computationally efficient and stable in univariate cases. To illustrate the effectiveness of our approach, we conduct a comprehensive comparison with other competing methods through simulation studies and find that, in most cases, our approach performs superior in terms of average squared error between the estimated and true time-varying spectral density. Finally, we demonstrate our methodology through applications to quarterly Gross Domestic Product (GDP) data and Northern California wind data.

Nonstationary time series data exist in various scientic disciplines, including environmental science, biology, signal processing, econometrics, among others. Many Bayesian models have been developed to handle nonstationary time series. Some of the models characterize nonstationarity through the mean, e.g., regime-switching time series models and time-varying coefficients models, whereas other models characterize nonstationarity through modeling the variance/covariance. Models of nonstationary behavior can be viewed both in the time domain (e.g., stochastic volatility) or in the time-frequency domain. This dissertation proposes a multi-regime smooth transition stochastic volatility model based on ordered categorical variables and is illustrated using additional information in the form of covariates (e.g., trading volume). This model can handle the non-Gaussian behavior often found in return data, address the asymmetric effects of financial returns, and provide regime-specific inference. The time-varying autoregressive (TV-VAR) model is a well-established model for multivariate nonstationary time series. Nevertheless, in most cases, the large number of parameters presented by the model results in a high computational burden, ultimately limiting its usage. This dissertation proposes a computationally efficient multivariate Bayesian Circular Lattice Filter to extend the usage of the TV-VAR model to a broader class of highdimensional problems. Finally, modeling and forecasting nonstationary count time series presents many challenges. To address these challenges, this dissertation introduces a computationally efficient Poisson model with a latent Gaussian time-varying autoregressive process. The model is shown to provide effective multi-step ahead forecasts for nonstationary count time series, e.g., COVID-19 count data. The utility of each proposed method is illustrated through simulation and through case-studies.