Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 9, 2015

Information criteria for nonlinear time series models

  • Saskia Rinke and Philipp Sibbertsen EMAIL logo


In this paper the performance of different information criteria for simultaneous model class and lag order selection is evaluated using simulation studies. We focus on the ability of the criteria to distinguish linear and nonlinear models. In the simulation studies, we consider three different versions of the commonly known criteria AIC, SIC and AICc. In addition, we also assess the performance of WIC and evaluate the impact of the error term variance estimator. Our results confirm the findings of different authors that AIC and AICc favor nonlinear over linear models, whereas weighted versions of WIC and all versions of SIC are able to successfully distinguish linear and nonlinear models. However, the discrimination between different nonlinear model classes is more difficult. Nevertheless, the lag order selection is reliable. In general, information criteria involving the unbiased error term variance estimator overfit less and should be preferred to using the usual ML estimator of the error term variance.

JEL: C15; C22

Corresponding author: Philipp Sibbertsen, Institute of Statistics, Leibniz University Hannover, School of Economics and Management, Königsworther Platz 1, D-30167 Hannover, Germany, Tel.: +49-511-762-3783, Fax: +49-511-762-3923, e-mail:


The authors are grateful to two anonymous referees, the associate editor and the participants of the Statistische Woche in Hannover for helpful comments and suggestions which improved the paper. Financial support by the Deutsche Forschungsgemeinschaft ( under grant SI 745/9–1 is gratefully acknowledged.


Akaike, H. 1974. “A New Look at The Statistical Model Identification.” IEEE Transactions on Automatic Control 19: 716–723.10.1007/978-1-4612-1694-0_16Search in Google Scholar

Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31: 307–327.10.1016/0304-4076(86)90063-1Search in Google Scholar

Clements, M. P., and H.-M. Krolzig. 1998. “A Comparison of The Forecast Performance of Markov-Switching and Threshold Autoregressive Models of US GNP.” The Econometrics Journal 1: 47–75.10.1111/1368-423X.11004Search in Google Scholar

Emiliano, P. C., M. J. Vivanco, and F. S. De Menezes. 2014. “Information Criteria: How Do They Behave in different Models?” Computational Statistics & Data Analysis 69: 141–153.10.1016/j.csda.2013.07.032Search in Google Scholar

Engle, R. F. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50 (4): 987–1007.10.2307/1912773Search in Google Scholar

Gonzalo, J., and J.-Y. Pitarakis. 2002. “Estimation and Model Selection Based Inference in Single and Multiple Threshold Models.” Journal of Econometrics 110: 319–352.10.1016/S0304-4076(02)00098-2Search in Google Scholar

Hamaker, E. 2009. Using Information Criteria to Determine The Number of Regimes in Threshold Autoregressive Models.” Journal of Mathematical Psychology 53: 518–529.10.1016/ in Google Scholar

Hamilton, J. D. 1989. “A New Approach To The Economic Analysis of Nonstationary Time Series and The Business Cycle.” Econometrica 57: 357–384.10.2307/1912559Search in Google Scholar

Hamilton, J. D. 1994. Time Series Analysis. Princeton University Press.10.1515/9780691218632Search in Google Scholar

Hansen, B. E. 1997. “Inference in TAR Models.” Studies in Nonlinear Dynamics & Econometrics 2.10.2202/1558-3708.1024Search in Google Scholar

Hughes, A. W., and M. L. King. 2003. “Model Selection Using AIC in the Presence of One-Sided Information.” Journal of Statistical Planning and Inference 115: 397–411.10.1016/S0378-3758(02)00159-3Search in Google Scholar

Hughes, A. W., M. L. King, and K. T. Kwek. 2004. “Selecting the Order of An ARCH model.” Economics Letters 83: 269–275.10.1016/j.econlet.2003.05.003Search in Google Scholar

Hurvich, C. M., and C.-L. Tsai. 1989. “Regression and Time Series Model Selection in Small Samples.” Biometrika 76: 297–307.10.1093/biomet/76.2.297Search in Google Scholar

Kapetanios, G. 2001. “Model Selection in Threshold Models.” Journal of Time Series Analysis 22: 733–754.10.1111/1467-9892.00251Search in Google Scholar

Li, W. 1988. “The Akaike Information Criterion in Threshold Modelling: Some Empirical Evidences.” In Nonlinear Time Series and Signal Processing, Lecture Notes in Control and Information Sciences, vol. 106, Berlin Heidelberg: Springer, pp. 88–96, URL in Google Scholar

Liu, J., S. Wu, and J. V. Zidek. 1997. “On Segmented Multivariate Regression,” Statistica Sinica 7: 497–525.Search in Google Scholar

Luukkonen, R., P. Saikkonen, and T. Teräsvirta. 1988a. “Testing Linearity Against Smooth Transition Autoregressive Models.” Biometrika 75: 491–499.10.1093/biomet/75.3.491Search in Google Scholar

Luukkonen, R., P. Saikkonen, and T. Teräsvirta. 1988b. “Testing Linearity in Univariate Time Series Models.” Scandinavian Journal of Statistics 15: 161–175.Search in Google Scholar

McQuarrie, A., R. Shumway, and C.-L. Tsai. 1997. “The Model Selection Criterion AICu.” Statistics & Probability Letters 34: 285–292.10.1016/S0167-7152(96)00192-7Search in Google Scholar

Pitarakis, J.-Y. 2006. “Model Selection Uncertainty and Detection of Threshold Effects.” Studies in Nonlinear Dynamics & Econometrics 10.10.2202/1558-3708.1256Search in Google Scholar

Psaradakis, Z., M. Sola, F. Spagnolo, and N. Spagnolo. 2009. “Selecting Nonlinear Time Series Models Using Information Criteria.” Journal of Time Series Analysis 30: 369–394.10.1111/j.1467-9892.2009.00614.xSearch in Google Scholar

Schwarz, G. 1978. “Estimating the Dimension of A Model.” The Annals of Statistics 6: 461–464.10.1214/aos/1176344136Search in Google Scholar

Shibata, R. 1986. “Consistency of Model Selection and Parameter Estimation.” Journal of Applied Probability 23: 127–141.10.2307/3214348Search in Google Scholar

Smith, A., P. A. Naik, and C.-L. Tsai. 2006. “Markov-switching Model Selection using Kullback–Leibler Divergence.” Journal of Econometrics 134: 553–577.10.1016/j.jeconom.2005.07.005Search in Google Scholar

Teräsvirta, T. 1994. “Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models.” Journal of the American Statistical Association 89: 208–218.Search in Google Scholar

Tong, H. 1983. Threshold Models in Non-Linear Time Series Analysis. Lecture Notes in Statistics, No. 21. Springer-Verlag.10.1007/978-1-4684-7888-4Search in Google Scholar

Tong, H. 1990. Non-linear Time Series: A Dynamical System Approach. Oxford University Press.Search in Google Scholar

Tong, H., and K. S. Lim. 1980. “Threshold Autoregression, Limit Cycles and Cyclical Data.” Journal of the Royal Statistical Society. Series B (Methodological) 245–292.10.1142/9789812836281_0002Search in Google Scholar

van Dijk, D., T. Teräsvirta, and P. H. Franses. 2002. “Smooth Transition Autoregressive Models – A Survey of Recent Developments.” Econometric Reviews 21: 1–47.10.1081/ETC-120008723Search in Google Scholar

Wen, M.-J., and Y.-H. Tu. 2001. “Modified WIC for Order Selection in Autoregressive Model.” Technical Report No. 40, National Cheng-Kung University, Institute of Statistics.Search in Google Scholar

Wong, C. S., and W. K. Li. 1998. “A Note on the Corrected Akaike Information Criterion for Threshold Autoregressive Models.” Journal of Time Series Analysis 19: 113–124.10.1111/1467-9892.00080Search in Google Scholar

Wu, T.-J., and A. Sepulveda. 1998. “The Weighted Average Information Criterion for Order Selection in Time Series and Regression Models.” Statistics & Probability Letters 39: 1–10.10.1016/S0167-7152(98)00003-0Search in Google Scholar

Supplemental Material:

The online version of this article (DOI: 10.1515/snde-2015-0026) offers supplementary material, available to authorized users.

Published Online: 2015-12-9
Published in Print: 2016-6-1

©2016 by De Gruyter

Downloaded on 10.12.2023 from
Scroll to top button