Recent Developments of Threshold Estimation for Nonlinear Time Series

In this article, several important problems of threshold estimation in a Bayesian framework for nonlinear time series models are discussed. The paper starts with the issue of calculating the maximum likelihood and the Bayesian estimators for threshold autoregressive models. It turns out that the asymptotic efficiency of the Bayesian estimators in this type of singular estimation problems is superior than the maximum likelihood estimators. To illustrate the properties of these estimators and to explain the proposed method, the paper begins with the study of a linear threshold autoregressive model with i.i.d. Gaussian noise. The paper then extends the idea to other nonlinear and non-Gaussian models and illustrates the paradigm of limiting likelihood ratio, which is applicable to a much wider class of nonlinear models. The article also investigates the robustness issue and the possibility of restricting the observation window by narrow bands, which allows one to obtain asymptotically efficient estimators. Finally, the paper indicates how these results can be generalized from a TAR(1) model to a higher-order TAR(p) model with multiple thresholds. The paper concludes with a discussion of other related problems and illustrates the methodology by numerical simulations.