Statistics and Its Interface Volume 4 (2011) 167–181 A review of threshold time series models in finance Cathy W. S. Chen * , Mike K. P. So and Feng-Chi Liu Since the pioneering work by Tong (1978, 1983), thresh- old time series modelling and its applications have become increasingly important for research in economics and fi- nance. A number of books and a vast number of research papers published in this area have been motivated by Tong’s threshold models. The goal of this paper is to give a through review on the development of the family of threshold time series model in finance and to provide a streamlined ap- proach to financial time series analysis. It covers threshold modeling, nonlinearity tests, statistical inference, diagnos- tic checking, and model selection, as well as applications of the threshold autoregressive model and its generalizations in finance. Keywords and phrases: Asymmetry, Heteroskedasticity, MCMC, Markov switching, Smooth transition, Nonlinearity, Volatility models. 1. INTRODUCTION Nonlinear time series modelling drew much attention in the 1970’s, during the time when many classes of models were proposed. As compared to the linear models, the non- linear time series models provide a much wider spectrum of possible dynamics for economic and financial time series data. The threshold autoregressive (TAR) model is proposed by Tong (1978, 1983) and Tong and Lim (1980) for describ- ing periodic time series. This model captures the dynamic behavior of time series by switching the regimes. The fea- tures of this class of models include limit cycles, amplitude dependent frequencies and jump phenomena which linear models fail to capture. The TAR model plays an important role in nonlinear time series modelling. The new era of non- linear time series analysis, following the proposal of the TAR model, offers us very exciting possibilities. A general form of Tong’s TAR model is given as follows: y t = φ (Jt) 0 + p  i=1 φ (Jt) i y t-i + θ (Jt) a t , (1) where a t s are i.i.d. D(0,σ 2 ) and {J t } are indicator random variables, taking integer values in {1,...,J }. Typical thresh- old modelling assumes that the value of J t is determined by * Corresponding author. a threshold variable z t . {J t } can be a Markov chain driven TAR model. The Markov chain can be either observable or hidden. (see Tong and Lim 1980, p. 285, Tong 1983, p. 62, and Tyssedal and Tjøstheim 1988). When {J t } is hid- den, (1) is also called Markov switching models (Hamilton 1989). The indicator time series acts as a switching mecha- nism. Note that the TAR model can be easily extended to a threshold autoregressive moving average model (TARMA model for short) by replacing θ (Jt) a t with ∑ q j=0 θ (Jt) j a t-j . Further extension to include some exogenous time series is straightforward and called TARMAX. It is well known that financial market volatility changes over time and often exhibits the volatility clustering prop- erty: large changes in prices tend to cluster together, result- ing in the persistence of the amplitudes of price changes. A common way to capture this phenomenon is that proposed by Engle (1982) who introduces the autoregressive condi- tional heteroscedastic (ARCH) model for modelling time- varying conditional variance of financial returns. Boller- slev (1986) proposes an extension to a generalized ARCH (GARCH) model which is a widely accepted model to de- scribe the time series properties of financial market returns. However, symmetric ARCH and GARCH formulations are not well suited for capturing an asymmetric response of volatility. This phenomenon is discovered by Black (1976) and subsequently confirmed by Nelson (1991) and Glosten, Jagannathan and Runkle (1993), among others. Numerous improvements have been made to the GARCH model. For example, Chan (2009) provides a useful tool for researchers and students interested in the theory and practice of nonlin- ear time series analysis, while Li (2009) examines threshold approaches in volatility modelling. The idea of nonlinearity is also extended to the threshold unit-root test and the dis- continuous adjustment to a long-run equilibrium (threshold co-integration), e.g. Kapetanios, Shin, and Snell (2003). A number of books and a vast number of research pa- pers have been published in diverse areas, such as ecology, econometrics, economics, finance, actuarial science and hy- drology, motivated by Tong’s threshold models. The goal of this paper is to give a thorough review of the vast and important development of the threshold time series model in financial applications and to provide a streamlined ap- proach to financial time series analysis. This paper is orga- nized as follows. We discuss representations of the threshold time series models in Section 2. Testing for the presence of