# The Selection of ARIMA Models with or without Regressors

## Author Info

• Søren Johansen

()

(University of Copenhagen and CREATES)

• Marco Riani

()

(Dipartimento di Economia, Università di Parma)

• Anthony C. Atkinson

()

(Department of Statistics, London School of Economics)

## Abstract

We develop a $C_{p}$ 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 $\chi _{n}^{2}$. 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 $C_{p}$ 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 $C_{p}$ 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 $C_{p}$.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL: ftp://ftp.econ.au.dk/creates/rp/12/rp12_46.pdf

## Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2012-46.

as
in new window

 Length: 31 Date of creation: 08 Nov 2012 Date of revision: Handle: RePEc:aah:create:2012-46 Contact details of provider: Web page: http://www.econ.au.dk/afn/

## References

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

as in new window
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. repec:cup:cbooks:9780521852258 is not listed on IDEAS
3. 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.
4. 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.
5. 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.
6. 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.
Full references (including those not matched with items on IDEAS)

## Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

## Corrections

When requesting a correction, please mention this item's handle: RePEc:aah:create:2012-46. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: ()

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.