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The Selection of ARIMA Models with or without Regressors

Author

Listed:
  • Søren Johansen

    (Department of Economics, University of Copenhagen and CREATES, University of Aarhus)

  • Marco Riani

    (Dipartimento di Economia, Universita di Parma)

  • Anthony C. Atkinson

    (Department of Statistics, London School of Economics, UK)

Abstract

We develop a Cp statistic for the selection of regression models with stationary and nonstationary ARIMA error term. We derive the asymptotic theory of the maximum likelihood estimators and show they are consistent and asymptotically Gaussian. We also prove that the distribution of the sum of squares of one step ahead standardized prediction errors, when the parameters are estimated, differs from the chi-squared distribution by a term which tends to infinity at a lower rate than X (2/n). We further prove that, in the prediction error decomposition, the term involving the sum of the variance of one step ahead standardized prediction errors is convergent. Finally, we provide a small simulation study. Empirical comparisons of a consistent version of our Cp statistic with BIC and a generalized RIC show that our statistic has superior performance, particularly for small signal to noise ratios. A new plot of our time series Cp statistic is highly informative about the choice of model. On the way we introduce a new version of AIC for regression models, show that it estimates a Kullback-Leibler distance and provide a version for small samples that is bias corrected. We highlight the connections with standard Mallows Cp.

Suggested Citation

  • Søren Johansen & Marco Riani & Anthony C. Atkinson, 2012. "The Selection of ARIMA Models with or without Regressors," Discussion Papers 12-17, University of Copenhagen. Department of Economics.
  • Handle: RePEc:kud:kuiedp:1217
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    File URL: http://www.econ.ku.dk/english/research/publications/wp/dp_2012/1217.pdf
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    References listed on IDEAS

    as
    1. Marc K. Francke & Siem Jan Koopman & Aart F. de Vos, 2010. "Likelihood functions for state space models with diffuse initial conditions," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(6), pages 407-414, November.
    2. Peide Shi & Chih-Ling Tsai, 2004. "A Joint Regression Variable and Autoregressive Order Selection Criterion," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(6), pages 923-941, November.
    3. Riani, Marco & Atkinson, Anthony C., 2010. "Robust model selection with flexible trimming," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3300-3312, December.
    4. Mike K. P. So & Cathy W. S. Chen & Feng-Chi Liu, 2006. "Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 55(2), pages 201-224.
    5. Hansheng Wang & Guodong Li & Chih-Ling Tsai, 2007. "Regression coefficient and autoregressive order shrinkage and selection via the lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(1), pages 63-78.
    6. Claeskens,Gerda & Hjort,Nils Lid, 2008. "Model Selection and Model Averaging," Cambridge Books, Cambridge University Press, number 9780521852258.
    7. Qiwei Yao & Peter J. Brockwell, 2006. "Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(6), pages 857-875, November.
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    More about this item

    Keywords

    AIC; ARMA models; bias correction; BIC; Cp plot; generalized RIC; Kalman filter; Kullback-Leibler distance; state-space formulation;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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