Piecewise quantile autoregressive modeling for nonstationary time series

Alexander Aue, Rex C.Y. Cheung, Thomas C.M. Lee, and Ming Zhong

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We develop a new methodology for the fitting of nonstationary time series that exhibit nonlinearity, asymmetry, local persistence and changes in location scale and shape of the underlying distribution. In order to achieve this goal, we perform model selection in the class of piecewise stationary quantile autoregressive processes. The best model is defined in terms of minimizing a minimum description length criterion derived from an asymmetric Laplace likelihood. Its practical minimization is done with the use of genetic algorithms. If the data generating process follows indeed a piecewise quantile autoregression structure, we show that our method is consistent for estimating the break points and the autoregressive parameters. Empirical work suggests that the proposed method performs well in finite samples.

Article information

Bernoulli, Volume 23, Number 1 (2017), 1-22.

Received: January 2014
Revised: July 2014
First available in Project Euclid: 27 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

autoregressive time series change-point genetic algorithm minimum description length principle nonstationary time series structural break


Aue, Alexander; Cheung, Rex C.Y.; Lee, Thomas C.M.; Zhong, Ming. Piecewise quantile autoregressive modeling for nonstationary time series. Bernoulli 23 (2017), no. 1, 1--22. doi:10.3150/14-BEJ671.

Export citation


  • Aue, A., Cheung, R.C.Y., Lee, T.C.M. and Zhong, M. (2014). Segmented model selection in quantile regression using the minimum description length principle. J. Amer. Statist. Assoc. 109 1241–1256.
  • Aue, A. and Horváth, L. (2013). Structural breaks in time series. J. Time Series Anal. 34 1–16.
  • Aue, A., Horváth, L. and Steinebach, J. (2006). Estimation in random coefficient autoregressive models. J. Time Series Anal. 27 61–76.
  • Aue, A. and Lee, T.C.M. (2011). On image segmentation using information theoretic criteria. Ann. Statist. 39 2912–2935.
  • Bai, J. (1998). Estimation of multiple-regime regressions with least absolutes deviation. J. Statist. Plann. Inference 74 103–134.
  • Csörgő, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analysis. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • Davis, L.D. (1991). Handbook of Genetic Algorithms. New York: Van Nostrand Reinhold.
  • Davis, R.A., Lee, T.C.M. and Rodriguez-Yam, G.A. (2006). Structural break estimation for nonstationary time series models. J. Amer. Statist. Assoc. 101 223–239.
  • Davis, R.A., Lee, T.C.M. and Rodriguez-Yam, G.A. (2008). Break detection for a class of nonlinear time series models. J. Time Series Anal. 29 834–867.
  • Hallin, M. and Jurečková, J. (1999). Optimal tests for autoregressive models based on autoregression rank scores. Ann. Statist. 27 1385–1414.
  • Hansen, M. and Yu, B. (2000). Wavelet thresholding via MDL for natural images. IEEE Trans. Inform. Theory 46 1778–1788.
  • Hughes, G.L., Subba Rao, S. and Subba Rao, T. (2007). Statistical analysis and time-series models for minimum/maximum temperatures in the Antarctic Peninsula. Proc. R. Soc. Ser. A 463 241–259.
  • Koenker, R. (2005). Quantile Regression. Econometric Society Monographs 38. Cambridge: Cambridge Univ. Press.
  • Koenker, R. and Xiao, Z. (2006). Quantile autoregression. J. Amer. Statist. Assoc. 101 980–990.
  • Koul, H.L. and Saleh, A.K.Md.E. (1995). Autoregression quantiles and related rank-scores processes. Ann. Statist. 23 670–689.
  • Lee, C.-B. (1997). Estimating the number of change points in exponential families distributions. Scand. J. Statist. 24 201–210.
  • Lee, T.C.M. (2001). An introduction to coding theory and the two-part minimum description length principle. International Statistical Review 69 169–183.
  • Paparoditis, E. and Politis, D.N. (2003). Residual-based block bootstrap for unit root testing. Econometrica 71 813–855.
  • Paparoditis, E. and Politis, D.N. (2005). Bootstrapping unit root tests for autoregressive time series. J. Amer. Statist. Assoc. 100 545–553.
  • Rissanen, J. (1989). Stochastic Complexity in Statistical Inquiry. World Scientific Series in Computer Science 15. Teaneck, NJ: World Scientific.
  • Shephard, N. and Andersen, T.G. (2009). Stochastic volatility: Origins and overview. In Handbook of Financial Time Series (T.G. Andersen et al., eds.) 233–254. Heidelberg: Springer.
  • Yao, Y.-C. (1988). Estimating the number of change-points via Schwarz’ criterion. Statist. Probab. Lett. 6 181–189.
  • Yu, K., Lu, Z. and Stander, J. (2003). Quantile regression: Applications and current research areas. The Statistician 52 331–350.