Testing for Threshold Effects in the TARMA Framework

We present supremum Lagrange Multiplier tests to compare a linear ARMA specification against its threshold ARMA extension. We derive the asymptotic distribution of the test statistics both under the null hypothesis and contiguous local alternatives. Moreover, we prove the consistency of the tests. The Monte Carlo study shows that the tests enjoy good finite-sample properties, are robust against model mis-specification and their performance is not affected if the order of the model is unknown. The tests present a low computational burden and do not suffer from some of the drawbacks that affect the quasi-likelihood ratio setting. Lastly, we apply our tests to a time series of standardized tree-ring growth indexes and this can lead to new research in climate studies.

[1]  H. Tong,et al.  Threshold Autoregression, Limit Cycles and Cyclical Data , 1980 .

[2]  E. Hannan,et al.  Recursive estimation of mixed autoregressive-moving average order , 1982 .

[3]  Neville Davies,et al.  A portmanteau test for self-exciting threshold autoregressive-type nonlinearity in time series , 1986 .

[4]  Timo Teräsvirta,et al.  Testing linearity against smooth transition autoregressive models , 1988 .

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

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

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

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

[9]  ByoungSeon Choi,et al.  Arma Model Identification , 1992 .

[10]  W. K. Li,et al.  Testing for threshold autoregression with conditional heteroscedasticity , 1997 .

[11]  R. Tsay.  Testing and modeling multivariate threshold models , 1998 .

[12]  Shiqing Ling.  On the Probabilistic Properties of a Double Threshold ARMA Conditional Heteroskedastic Model , 1999 .

[13]  Wai Keung Li,et al.  TESTING FOR DOUBLE THRESHOLD AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTIC MODEL , 2000 .

[14]  Julian C. Fox,et al.  Stochastic structure and individual-tree growth models , 2001 .

[15]  D. Andrews.  Tests for Parameter Instability and Structural Change with Unknown Change Point , 1993 .

[16]  Howell Tong,et al.  TESTING FOR A LINEAR MA MODEL AGAINST THRESHOLD MA MODELS , 2005 .

[17]  Guodong Li,et al.  TESTING FOR THRESHOLD MOVING AVERAGE WITH CONDITIONAL HETEROSCEDASTICITY , 2008 .

[18]  Kung-Sik Chan.  Exploration of a nonlinear world : an appreciation of Howell Tong's contributions to statistics , 2009 .

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

[20]  Dong Li,et al.  On the least squares estimation of threshold autoregressive and moving-average models ∗ , 2011 .

[21]  Guodong Li,et al.  Testing a linear time series model against its threshold extension , 2011 .

[22]  Howell Tong,et al.  Threshold Models in Time Series Analysis-30 Years On , 2011 .

[23]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[24]  B. Hansen.  Threshold autoregression in economics , 2011 .

[25]  H. Tong,et al.  ON CONDITIONALLY HETEROSCEDASTIC AR MODELS WITH THRESHOLDS , 2014 .

[26]  E. Maasoumi,et al.  Entropy testing for nonlinear serial dependence in time series , 2015 .

[27]  Kung-Sik Chan,et al.  Guest Editors’ Introduction: Regime Switching and Threshold Models , 2017 .

[28]  Kung-Sik Chan,et al.  Testing for Threshold Diffusion , 2017 .

[29]  Greta Goracci,et al.  On the Ergodicity of First‐Order Threshold Autoregressive Moving‐Average Processes , 2018, Journal of Time Series Analysis.

[30]  Greta Goracci.  Revisiting the Canadian Lynx Time Series Analysis Through TARMA Models , 2020 .

[31]  H. Tong,et al.  Unit-root test within a threshold ARMA framework , 2020 .

[32]  Greta Goracci.  An empirical study on the parsimony and descriptive power of TARMA models , 2020 .