Identification of Threshold Autoregressive Moving Average Models

  • Qiang Xia
  • Heung WongEmail author
Part of the Fields Institute Communications book series (FIC, volume 78)


Due to the lack of a suitable modeling procedure and the difficulty to identify the threshold variable and estimate the threshold values, the threshold autoregressive moving average (TARMA) model with multi-regime has not attracted much attention in application. Therefore, the chief goal of our paper is to propose a simple and yet widely applicable modeling procedure for multi-regime TARMA models. Under no threshold case, we utilize extended least squares estimate (ELSE) and linear arranged regression to obtain a test statistic \(\hat{F}\), which is proved to follow an approximate F distribution. And then, based on the statistic \(\hat{F}\), we employ some scatter plots to identify the number and locations of the potential thresholds. Finally, the procedures are considered to build a TARMA model by these statistics and the Akaike information criterion (AIC). Simulation experiments and the application to a real data example demonstrate that both the power of the test statistic and the model-building can work very well in the case of TARMA models.


Arranged regression Nonlinearity test TMA Model 



We thank the two Referees for their criticisms and suggestions which have led to improvements of the paper. The research of Qiang Xia was supported by National Social Science Foundation of China (No:12CTJ019) and Ministry of Education in China Project of Humanities and Social Sciences (Project No.11YJCZH195).The research of Heung Wong was supported by a grant of the Research Committee of The Hong Kong Polytechnic University (Code: G-YBCV).


  1. 1.
    Akaike, H. (1974). A new look at statistical model identification. IEEE Transactions on Automatic Control, 19, 716–722.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Billingsley, P. (1961). The Lindeberg–Levy theorem for martingales. Proceedings of the American Mathematical Society, 12, 788–792.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brockwell, P., Liu, J., & Tweedie, R. L. (1992). On the existence of stationary threshold autoregressive moving-average processes. Journal of Time Series Analysis, 13, 95–107.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Christopeit, N., & Helmes, K. (1980). Strong consistency of least squares estimators in linear regression models. The Annals of Statistics, 4, 778–788.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan, K. S. (1990). Testing for threshold autoregression. Annals of Statistics, 18, 1886–1894.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chan, K. S., Petruccelli, J. D., Tong, H., & Woolford, S. W. (1985). A multiple-threshold AR(1) model. Journal of Applied Probability, 22, 267–279.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chan, K. S., & Tong, H. (1986). On estimating thresholds in autoregressive models. Journal of Time Series Analysis, 7, 179–190.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, W. S. C., So, K. P. M., & Liu, F. C. (2011). A review of threshold time series models in finance. Statistics and Its Interface, 4, 167–181.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cryer, J. D., & Chan, K. S. (2008). Time series analysis with applications in R. Springer texts in statistics (2nd ed.). Berlin: Springer.Google Scholar
  10. 10.
    de Gooijer, J. G. (1998). On threshold moving-average models. Journal of Time Series Analysis, 19, 1–18.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ertel, J. E., & Fowlkes, E. B. (1976). Some algorithms for linear spline and piecewise multiple linear regression. Journal of the American Statistical Association, 71, 640–648.CrossRefzbMATHGoogle Scholar
  12. 12.
    Goodwin, G. C., & Payne, R. L. (1977). Dynamic system identification: Experiment design and data analysis. New York, NY: Academic Press.zbMATHGoogle Scholar
  13. 13.
    Haggan, V., Heravi, S. M., & Priestley, M. B. (1984). A study of the application of state-dependent models in nonlinear time series analysis. Journal of Time Series Analysis, 5, 69–102.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hannan, E. J., & Rissanen, J. (1982). Recursive estimation of mixed autoregressive-moving average order. Biometrika, 69, 81–94.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hansen, B. E. (2000). Sample splitting and threshold estimation. Econometrica, 68, 575–603.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Keenan, D. M. (1985). A Tukey nonadditivity-type test for time series nonlinearity. Biometrika, 72, 39–44.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lai, T. L., & Wei, C. Z. (1982). Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. The Annals of Statistics, 10, 154–166.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, G., & Li, W. K. (2011). Testing a linear time series model against its threshold extension. Biometrika, 98, 243–250.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, D., Li, W. K., & Ling, S. (2011). On the least squares estimation of threshold autoregressive and moving-average models. Statistics and Its Interface, 4, 183–196.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, D., & Ling, S. (2012). On the least squares estimation of multiple-regime threshold autoregressive models. Journal of Econometrics, 167, 240–253.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liang, R., Niu, C., Xia, Q., & Zhang, Z. (2015). Nonlinearity testing and modeling for threshold moving average models. Journal of Applied Statistics, 42, 2614–2630.Google Scholar
  22. 22.
    Ling, S. (1999). On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model. Journal of Applied Probability, 36, 688–705.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ling, S., & Tong, H. (2005). Testing a linear moving-average model against threshold moving-average models. The Annals of Statistics, 33, 2529–2552.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ling, S., Tong, H., & Li, D. (2007). The ergodicity and invertibility of threshold moving-average models. Bernoulli, 13, 161–168.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Liu, J., & Susko, E. (1992). On strict stationarity and ergodicity of a nonlinear ARMA model. Journal of Applied Probability, 29, 363–373.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ljung, L., & Soderstrom, T. (1983). Theory and practice of recursive identification. Cambridge: MIT Press.zbMATHGoogle Scholar
  27. 27.
    Priestley, M. B. (1980). State-dependent models: A general approach to nonlinear time series analysis. Journal of Time Series Analysis, 1, 47–71.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Qian, L. (1998). On maximum likelihood estimators for a threshold autoregression. Journal of Statistical Planning and Inference, 75, 21–46.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tong, H. (1978). On a threshold model. In C. H. Chen (Ed.), Pattern recognition and signal processing (pp. 101–141). Amsterdam: Sijthoff and Noordhoff.Google Scholar
  30. 30.
    Tong, H., & Lim, K. S. (1980). Threshold autoregressions, limit cycles, and data. Journal of the Royal Statistical Society B, 42, 245–292.zbMATHGoogle Scholar
  31. 31.
    Tong, H. (1990). Non-linear time series: A dynamical system approach. Oxford: Oxford University Press.zbMATHGoogle Scholar
  32. 32.
    Tsay, R. S. (1986). Nonlinearity tests for time series. Biometrika, 73, 461–466.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tsay, R. S. (1987). Conditional heteroscedastic time series models. Journal of the American Statistical Association, 82, 590–604.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Tsay, R. S. (1989). Testing and modeling threshold autoregressive process. Journal of the American Statistical Association, 84, 231–240.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tsay, R. S. (2005). Analysis of financial time series (2nd ed.). London: Wiley.CrossRefzbMATHGoogle Scholar
  36. 36.
    Wong, C. S., & Li, W. K. (1997). Testing for threshold autoregression with conditional heteroscedasticity. Biometrika, 84, 407–418.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wong, C. S., & Li, W. K. (2000). Testing for double threshold autoregressive conditional heteroscedastic model. Statistica Sinica, 10, 173–189.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.College of Mathematics and InformaticsSouth China Agricultural UniversityGuangzhouChina
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong, China

Personalised recommendations