Efficient regularised estimation methods for sparse time series models

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Regularised regression methods such as LASSO and its variants have been widely used in many areas of science. These methods successfully overcome the challenge of having more parameters (p) than observations (n) by exploiting the underlying sparsity in the data. In the case of time series, similarly we are often interested in fitting models with a large number of parameters, for relatively short time series. The dependency between the elements of the time series means that in order to fit such models, we need to find ways to exploit sparsity by appropriate regularisation. An example of such a situation is the vector autoregression (VAR) process, where the parameters form a series of matrices. In high dimensional settings, the total number of parameters can exceed the number of data points available. There have been several methods proposed and some theoretical results obtained in the literature, but the field is still very active and no method has yet been widely applied by practitioners. The goal of this project is to develop regularized regression methods that are both theoretically well founded and computationally efficient in high dimensions. In order to achieve this goal, we are going to use state-of-the-art sparse optimization and Monte Carlo methods.