Abstract
Optimal subset selection among a general family of threshold autoregressive moving-average (TARMA) models is considered. The usual complexity of model/order selection is increased by capturing the uncertainty of unknown threshold levels and an unknown delay lag. The Monte Carlo method of Bayesian model averaging provides a possible way to overcome such model uncertainty. Incorporating with the idea of Bayesian model averaging, a modified stochastic search variable selection method is adapted to consider subset selection in TARMA models, by adding latent indicator variables for all potential model lags as part of the proposed Markov chain Monte Carlo sampling scheme. Metropolis–Hastings methods are employed to deal with the well-known difficulty of including moving-average terms in the model and a novel proposal mechanism is designed for this purpose. Bayesian comparison of two hyper-parameter settings is carried out via a simulation study. The results demonstrate that the modified method has favourable performance under reasonable sample size and appropriate settings of the necessary hyper-parameters. Finally, the application to four real datasets illustrates that the proposed method can provide promising and parsimonious models from more than 16 million possible subsets.
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References
Amendola A, Niglio M, Vitale C (2006) The moments of SETARMA models. Stat Probab Lett 76: 625–633
Baragona R, Battaglia F, Cucina D (2004) Estimating threshold subset autoregressive moving-average models by genetic algorithms. Int J Stat 62: 39–61
Brockwell P, Liu J, Tweedie RL (1992) On the existence of stationary threshold autoregressive moving-average processes. J Time Ser Anal 13: 95–107
Chen CWS (1999) Subset selection of autoregressive time series models. J Forecast 18: 505–516
Chen CWS, Gerlach RH, Lin AMH (2010) Falling and explosive, dormant, and rising markets via multiple-regime financial time series models. Appl Stoch Model Bus Ind 26: 28–49
Chen CWS, Lin EMH, Liu FC, Gerlach R (2008) Bayesian estimation for parsimonious threshold autoregressive models in R. Newsl R proj 8: 26–33
Chen CWS, So MKP, Gerlach R (2005) Assessing and testing for threshold nonlinearity in stock returns. Aust New Zealand Appl Stat 47: 473–488
De Gooijer G (1998) On threshold moving-average models. J Time Ser Anal 19: 1–18
Fernandez C, Ley E, Steel MFJ (2001) Benchmark priors for Bayesian model averaging. J Econ 100: 381–427
Franses PH, van Dijk D (2000) Nonlinear time series models in empirical finance. Cambridge University Press, Cambridge
George EI, McCulloch RE (1993) Variable selection via Gibbs sampling. J Am Stat Assoc 88: 881–889
Gerlach R, Carter CK, Kohn R (1999) Diagnostics for time series analysis. J Time Ser Anal 20: 309–330
Geweke J (1995) Bayesian comparison of econometric models. Working paper 532. Research Department, Federal Reserve Bank of Minneapolis
Ghaddar DK, Tong H (1981) Data transformation and self-exciting threshold autoregression. Appl Stat 30: 238–248
Green PJ (1995) Reversible jump MCMC computation and Bayesian model determination. Biometrika 82: 711–732
Hoeting JA, Madigan D, Raftery AE, Volinsky CT (1999) Bayesian model averaging: a tutorial. Stat Sci 14: 382–417
Li CW, Li WK (1996) On a double-threshold autoregressive heteroscedastic time series model. J Appl Econ 11: 253–274
Ling S (1999) On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model. J Appl Probab 36: 688–705
Ling S, Tong H, Li D (2007) Ergodicity and invertibility of threshold moving-average models. Bernoulli 13: 161–168
Liu FC (2009) Volatility forecasting and model selection for nonlinear time series, Doctoral Thesis, Department of Statistics, Feng Chia University, December 2009
Nakatsuma T (2000) Bayesian analysis of ARMA-GARCH models: a Markov chain sampling approach. J Econ 95: 57–69
Petruccelli J, Woolford SW (1984) A threshold AR(1) model. J Appl Probab 21: 270–286
Philippe A (2006) Bayesian analysis of autoregressive moving average processes with unknown orders. Comput Stat Data Anal 51: 1904–1923
Potter SM (1995) A nonlinear approach to US GNP. J Appl Econ 10: 109–125
Raftery AE, Madigan D, Hoeting JA (1997) Bayesian model averaging for linear regression models. J Am Stat Assoc 92: 179–191
Smith M, Kohn R (1996) Nonparametric regression using Bayesian variable selection. J Econ 75: 317–343
So MKP, Chen CWS (2003) Subset threshold autoregression. J Forecast 22: 49–66
So MKP, Chen CWS, Chen MT (2005) A Bayesian threshold nonlinearity test for financial time series. J Forecast 24: 61–75
So MKP, Chen CWS, Liu FC (2006) Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors. J Roy Stat Soc Ser C 55: 201–224
Tanner MA, Wong WH (1987) The calculation of posterior distributions by data augmentation. J Am Stat Assoc 82: 528–540
Tong H (1978) On a threshold model. In: Chen CH (ed) Pattern recognition and signal processing. Sijthoff & Noordhoff, Amsterdam
Tong H, Lim KS (1980) Threshold autoregression, limit cycles and cyclical data (with discussion). J Roy Stat Soc Ser B 42: 245–292
Tong H (1983) Threshold models in nonlinear time series analysis. Lecture notes in statistics. Springer, New York
Tong H (1990) Non-linear time series. A dynamical system approach. Clarendon Press, Oxford
Tsay RS (1989) Testing and modeling threshold autoregressive processes. J Am Stat Assoc 84: 231–240
Tsay RS (1998) Testing and modeling multivariate threshold models. J Am Stat Assoc 93: 1188–1202
Tsay RS (2005) Analysis of financial time series, 2nd edn. John Wiley & Sons, New Jersey
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Chen, C.W.S., Liu, F.C. & Gerlach, R. Bayesian subset selection for threshold autoregressive moving-average models. Comput Stat 26, 1–30 (2011). https://doi.org/10.1007/s00180-010-0198-0
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DOI: https://doi.org/10.1007/s00180-010-0198-0