Recent Developments of Threshold Estimation for Nonlinear Time Series

Abstract

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.

Estimation in threshold autoregressive models with correlated innovations

“…there exist positive constants C and r < 1, such that for a measurable function |h| ≤ 1 and m ≥ k E θ 0 h(X m , ξ m )|X k = x, ξ k = y − E θ 0 h(X k , ξ k ) ≤ Cr m−k (|x| + |y|), x, y ∈ R, (A. 4) and consequently, for an F m,∞ -measurable random variable |H| ≤ 1…”

On cusp location estimation for perturbed dynamical systems

“…The change‐point problem can be described by the following example: that is, we have a switching diffusion process with unknown threshold ϑ . Such models are called threshold diffusion processes like threshold autoregressive time series (Chan & Kutoyants, ), and statistical problems related to this model are singular (Kutoyants, ). If we have a cusp‐type singularity as where , then, for κ close to zero, we have cusp‐type switching similar to the change‐point case, but without jump.…”

“…that is, we have a switching diffusion process with unknown threshold . Such models are called threshold diffusion processes like threshold autoregressive time series (Chan & Kutoyants, 2010), and statistical problems related to this model are singular (Kutoyants, 2012). If we have a cusp-type singularity as…”

Statistical Inference for Structurally Changed Threshold Autoregressive Models

“…The threshold autoregressive (TAR) model proposed by Tong (1978) is to capture these phenomena. It has been extensively applied in many areas, including economics, finance, biological and environmental sciences, among others, see Chan and Kutoyants (2010) and Tong (2011) for nice reviews. The asymptotic theory of the least squares estimator (LSE) of two-regime TAR models was established by Chan (1993) and Chan and Tsay (1998), and was further developed by Li and Ling (2012) and Li et al (2013) for the multiple-regime TAR model and the TMA model, respectively.…”