Rejoinder from Howell Tong to the discussions on 'threshold models in time series analysis - 30 years on'

This paper is a selective review of the development of the threshold model in time series analysis over the past 30 years or so. First, the review re-visits the motivation of the model. Next, it describes the various expressions of the model, highlighting the underlying principle and the main probabilistic and statistical properties. Finally, after listing some of the recent offsprings of the threshold model, the review finishes with some on-going research in the context of threshold volatility.

[1]  H. Tong,et al.  Score Based Goodness-of-fit Tests for Time Series , 2011 .

[2]  Yingcun Xia,et al.  Feature Matching in Time Series Modeling , 2011, 1104.3073.

[3]  H. Tong,et al.  A note on the invertibility of nonlinear ARMA models , 2010 .

[4]  Kung-Sik Chan,et al.  Time Series Analysis: With Applications in R , 2010 .

[5]  N. Stenseth.  The Importance of TAR-Modelling for Understanding the Structure of Ecological Dynamics: The Hare-Lynx Population Cycles as an Example , 2009 .

[6]  A. Jones,et al.  Foreword , 1967, British Journal of Cancer.

[7]  John Geweke.  The SETAR model of Tong and Lim and advances in computation , 2009 .

[8]  W. Li.  The threshold approach in volatility modelling , 2009 .

[9]  N. Shephard,et al.  The ACR Model: A Multivariate Dynamic Mixture Autoregression , 2008 .

[10]  H. Tong.  Exploring volatility from a dynamical system perspective , 2007 .

[11]  N. Stenseth,et al.  A generalized threshold mixed model for analyzing nonnormal nonlinear time series, with application to plague in Kazakhstan , 2007 .

[12]  Howell Tong,et al.  Ergodicity and invertibility of threshold moving-average models , 2007 .

[13]  Yingcun Xia,et al.  THRESHOLD VARIABLE SELECTION USING NONPARAMETRIC METHODS , 2007 .

[14]  Senlin Wu,et al.  THRESHOLD VARIABLE DETERMINATION AND THRESHOLD VARIABLE DRIVEN SWITCHING AUTOREGRESSIVE MODELS , 2007 .

[15]  H. Tong.  Birth of the threshold time series model , 2007 .

[16]  C. Robert,et al.  STOCHASTIC UNIT ROOT MODELS , 2006, Econometric Theory.

[17]  S. B. Pole,et al.  Plague dynamics are driven by climate variation , 2006, Proceedings of the National Academy of Sciences.

[18]  A note on time-reversibility of multivariate linear processes , 2006 .

[19]  H. Tong,et al.  TESTING FOR A LINEAR MA MODEL AGAINST THRESHOLD MA MODELS , 2005, math/0603040.

[20]  Bruce D. McCullough,et al.  Diagnostic Checks in Time Series , 2005, Technometrics.

[21]  N. Wermuth,et al.  Nonlinear Time Series : Nonparametric and Parametric Methods , 2005 .

[22]  Howell Tong,et al.  Some Nonlinear Threshold Autoregressive Time Series Models for Actuarial Use , 2004 .

[23]  Howell Tong,et al.  A note on testing for multi-modality with dependent data , 2004 .

[24]  Stan Lipovetsky,et al.  Chaos: A Statistical Perspective , 2003, Technometrics.

[25]  Eric R. Ziegel,et al.  Analysis of Financial Time Series , 2002, Technometrics.

[26]  B. Hansen,et al.  Testing for two-regime threshold cointegration in vector error-correction models , 2002 .

[27]  O. Stramer,et al.  On inference for threshold autoregressive models , 2002 .

[28]  Bruce E. Hansen,et al.  THRESHOLD AUTOREGRESSION WITH A UNIT ROOT , 2001 .

[29]  Andreas Galka,et al.  Topics in Nonlinear Time Series Analysis, with Implications for Eeg Analysis , 2000 .

[30]  H. Tong,et al.  Common dynamic structure of canada lynx populations within three climatic regions , 1999, Science.

[31]  Clive W. J. Granger,et al.  Unit Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates , 1998 .

[32]  B. Hansen,et al.  Inference in TAR Models , 1997 .

[33]  Ruth J. Williams,et al.  ON THE EXISTENCE AND APPLICATION OF CONTINUOUS TIME THRESHOLD AUTOREGRESSIONS OF ORDER TWO , 1997 .

[34]  Cathy W. S. Chen,et al.  BAYESIAN INFERENCE OF THRESHOLD AUTOREGRESSIVE MODELS , 1995 .

[35]  Qiwei Yao,et al.  Quantifying the influence of initial values on nonlinear prediction , 1994 .

[36]  K. Chan,et al.  Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model , 1993 .

[37]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[38]  An Hong-Zhi,et al.  A Kolmogorov-Smirnov type statistic with application to test for nonlinearity in time series , 1991 .

[39]  H. Tong,et al.  Strong consistency of least-squares estimator for a non-ergodic threshold autoregressive model , 1991 .

[40]  K. Chan,et al.  Percentage Points of Likelihood Ratio Tests for Threshold Autoregression , 1991 .

[41]  Howell Tong,et al.  Threshold autoregressive modelling in continuous time , 1991 .

[42]  K. Chan,et al.  Testing for threshold autoregression , 1990 .

[43]  Kung-Sik Chan,et al.  On Likelihood Ratio Tests for Threshold Autoregression , 1990 .

[44]  H. Tong.  Non-linear time series. A dynamical system approach , 1990 .

[45]  James D. Hamilton.  A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle , 1989 .

[46]  Dag Tjøstheim,et al.  An autoregressive model with suddenly changing parameters and an application to stock market prices , 1988 .

[47]  K. S. Chan.  ON THE EXISTENCE OF THE STATIONARY AND ERGODIC NEAR(p) MODEL , 1988 .

[48]  H. Tong,et al.  ON ESTIMATING THRESHOLDS IN AUTOREGRESSIVE MODELS , 1986 .

[49]  A note on certain integral equations associated with non-linear time series analysis , 1986 .

[50]  H. Tong,et al.  On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations , 1985, Advances in Applied Probability.

[51]  H. Tong,et al.  Threshold time series modelling of two Icelandic riverflow systems , 1985 .

[52]  A. Azzalini.  A class of distributions which includes the normal ones , 1985 .

[53]  J. Petruccelli,et al.  A threshold AR(1) model , 1984, Journal of Applied Probability.

[54]  E. Nummelin.  General irreducible Markov chains and non-negative operators: Embedded renewal processes , 1984 .

[55]  Ivan Netuka,et al.  On threshold autoregressive processes , 1984, Kybernetika.

[56]  Hung Man Tong,et al.  Threshold models in non-linear time series analysis. Lecture notes in statistics, No.21 , 1983 .

[57]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[58]  Howell Tong.  Discontinuous decision processes and threshold autoregressive time series modelling , 1982 .

[59]  Howell Tong,et al.  Threshold autoregression, limit cycles and cyclical data- with discussion , 1980 .

[60]  E. Nummelin,et al.  A splitting technique for Harris recurrent Markov chains , 1978 .

[61]  A. O'Hagan,et al.  Bayes estimation subject to uncertainty about parameter constraints , 1976 .

[62]  R. Tweedie.  Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space , 1975 .

[63]  P. Young,et al.  Time series analysis, forecasting and control , 1972, IEEE Transactions on Automatic Control.

[64]  P. Whittle,et al.  Prediction and Regulation. , 1965 .

[65]  M. Rosenblatt.  Some nonlinear problems arising in the study of random processes , 1964 .

[66]  J. Tukey.  Curves As Parameters, and Touch Estimation , 1961 .

[67]  F. G. Foster.  On the Stochastic Matrices Associated with Certain Queuing Processes , 1953 .

[68]  P. A. P. Moran,et al.  The statistical analysis of the Canadian Lynx cycle. , 1953 .

[69]  G. Yule.  On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers , 1927 .

[70]  ROBT. B. HAYWARD,et al.  On the Variation of Latitude , 1892, Nature.