Parsimonious Time Series Modelling of High-dimensional Data with Linear and Non-Linear Models

Time series models perform statistical analysis and predictions based on sequential data, with two trends emerging in their development: high dimensionality and non-linearity. The first trend results from advances in data collection and storage capabilities, and the second arises from the dynamic evolution of sequential data that challenges traditional static linear models. In this thesis, we study these two trends in Bayesian Vector Autoregression (VAR), chosen for its flexibility and widespread applications across fields. The Bayesian framework enables the incorporation of well-designed priors for unknown parameters while providing insight into parameter uncertainty. The first part of the thesis focuses on addressing the proliferation of parameters, a challenge inherent to high dimensionality, through tensor VAR, which treats the coefficient matrix as a third-order tensor and conducts tensor decomposition to achieve parsimony. We apply different Bayesian techniques to induce shrinkage and achieve convergent Markov chains. In the second part, we introduce time-varying parameterization to the tensor VAR to capture the evolving interrelation among time series, enabling the model to accommodate both trends. The final part explores factor augmented VAR (FAVAR), an alternative VAR framework that reduces dimensionality by extracting factors from high-dimensional data. We develop a novel approach which combines time-varying parameterization in the VAR component with a proposed Grouped Sparse autoencoder, alleviating identifiability and interpretability issues of the standard autoencoder. To demonstrate the merits of the models proposed in these three parts, we apply them to macroeconomic data and functional magnetic resonance imaging data.