LASSO-type estimations for threshold autoregressive and heteroscedastic time series models.

In this thesis, we propose Least Absolute Shrinkage and Selection Operator (LASSO) type estimators to perform simultaneous parameter estimation and model selection for five specific univariate and multivariate time series models, and develop several algorithms to compute these estimators.

The first model is the self-exciting threshold autoregressive model (SETAR). Although the group LASSO has been applied previously for this model to estimate thresholds, we propose a slightly different definition of the group LASSO estimator for the reformulated SETAR model which does not penalize the first set of parameters. For the reformulated model, our group LASSO definition has lower prediction errors compared to the standard group LASSO definition. We develop an active-set based block coordinate descent algorithm (BCD) to optimize exactly the group LASSO. To consistently estimate relevant thresholds from the threshold set obtained by the group LASSO, the backward elimination algorithm (BEA) is utilized. We extend BCD algorithm for the multivariate SETAR model, our second model. Empirical studies using these univariate and multivariate models show that the BCD algorithms estimate less irrelevant thresholds compared to the approximation group LASSO algorithms of group least angle regression (GLAR). Furthermore, the ensemble algorithms of BCD-BEA perform better in terms of correctly estimating the number of thresholds in simulation studies, and in identifying important thresholds in case studies compared to the ensemble algorithms of GLAR-BEA.

Motivated by an existing work, we propose the adaptive LASSO estimator for simultaneous parameter estimation and terms selection for the pure autoregressive conditional heteroscedasticity (ARCH) model, our third model, with its generalized model form (GARCH). A new algorithm of coordinate gradient descent (CGD) is developed to optimize the adaptive LASSO. We extend the application of the adaptive LASSO via the CGD algorithm for the multivariate Baba-Engle-Kroner-Kraft (BEKK) ARCH/GARCH, our fourth model. From our empirical studies using both pure ARCH and pure multivariate BEKK-ARCH models, our CGD algorithms exclude irrelevant terms more often, and have more stable parameter convergence compared to the existing modified shooting algorithm. Furthermore, our CGD algorithms are also capable of estimating the pure GARCH model, unlike any similar algorithm for the same model in the literature.

As an extension to the SETAR model, the SETAR model with GARCH error or SETARGARCH model is proposed to account for volatility in time series, our fifth and final model. We propose the iteratively reweighted group LASSO estimator for estimating thresholds. This is an iterative two-stage procedure, where the weighted conditional mean model is estimated in the first stage and the heteroscedastic weights are estimated in the second stage. We introduce the weighted BCD (wBCD) and weighted BEA (wBEA) algorithms, which utilize the above two-stage procedure where the weights are updated via the CGD algorithm. Our simulation studies show that the ensemble wBCD-wBEA algorithm estimates the correct number of thresholds at a high percentage, and the biases in the empirical standard deviations for the estimated thresholds are small.