Sparse causal time series analysis

The increasing prevalence of multiple, long time series data fuels the hope for gaining better understanding and more precise prediction of various phenomena in many domains, hence calling for the development of new methodologies in time series analysis. Two features pertinent to the practical usefulness of any time series model are sparsity and causality. A sparse time series model means that only a small subset of model parameters are non-zero, which is usually preferable to its non-sparse counterparts in terms of interpretability, robustness, parsimony and estimation variability. Causality refers to the dependence structure for which only past and present information is used. Causality is an indispensable feature for certain purposes such as prediction, and is usually desirable when modeling real-world phenomena. This dissertation devotes to the development of two time series methodologies that incorporate sparsity and causality.

In the first part of the thesis, we introduce the sparse causal dynamic linear regression model. Two approaches are proposed for construction of such models. The first one utilizes the frequency domain solution of a classic two-sided dynamic linear model and augments it with a simple soft-thresholding operator. This approach is computationally efficient and enjoys good performance theoretically and empirically under the assumption that the true dynamic regression relationship is causal and sparse. The second approach constructs a sparse causal dynamic linear model by minimizing certain penalized mean squared prediction error. This approach is computationally intensive but is superior to the first approach as it does not require the causal assumption. It enjoys better empirical performance and opens up meaningful extensions.

In the second part of the thesis, we introduce the sparse causal dynamic principal component analysis. This procedure also has two variants. The first one computes the principal component series as a sparse and causal linear filter of the original data. The filter is obtained by minimizing a penalized reconstruction loss functional in the frequency domain. The principal component series so obtained enjoys good performance in terms of summarizing high-dimensional data, estimating factors in dynamic factor models, and prediction. However, it is computationally intensive. The second approach imposes sparsity structure on the innovation representation of the data generating process, which leads to simpler computational algorithms. In addition, this method is of interest in its own right for certain applications in finance and econometrics.