IDEAS home Printed from https://ideas.repec.org/p/sgo/wpaper/1612.html
   My bibliography  Save this paper

Maximum Likelihood Estimation of Autoregressive Models with a Near Unit Root and Cauchy Errors

Author

Listed:
  • Jungjun Choi

    () (School of Economics, Sogang University, Seoul)

  • In Choi

    () (School of Economics, Sogang University, Seoul)

Abstract

This paper studies maximum likelihood estimation of autoregressive models of order 1 with a near unit root and Cauchy errors. Autoregressive models with an intercept and with an intercept and a linear time trend are also considered. The maximum likelihood estimator (MLE) for the autoregressive coeffcient is n^(3/2)-consistent with n denoting the sample size and has a mixture-normal dis- tribution in the limit. The MLE for the scale parameter of Cauchy distribution is n^(1/2)-consistent and its limiting distribution is normal. The MLEs of the intercept and the linear time trend are n^(1/2)- and n^(3/2)-consistent, respectively. It is also shown that the t-statistic for a unit root based on the MLE has a standard normal distribution in the limit. In addition, finite sample properties of the MLE are compared with those of the least square estimator (LSE). It is found that the MLE is more effcient than the LSE when the errors have a Cauchy distribution or a distribution which is a mixture of Cauchy and normal distributions. It is also shown that empirical power of the MLE-based t-test for a unit root is much higher than that of the Dickey-Fuller t-test.

Suggested Citation

  • Jungjun Choi & In Choi, 2016. "Maximum Likelihood Estimation of Autoregressive Models with a Near Unit Root and Cauchy Errors," Working Papers 1612, Research Institute for Market Economy, Sogang University.
  • Handle: RePEc:sgo:wpaper:1612
    as

    Download full text from publisher

    File URL: ftp://163.239.156.99/wpaper/CJJ_RIME_2016_12.pdf
    File Function: First version, 2016
    Download Restriction: no

    References listed on IDEAS

    as
    1. Lau, Amy Hing-Ling & Lau, Hon-Shiang & Wingender, John R, 1990. "The Distribution of Stock Returns: New Evidence against the Stable Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(2), pages 217-223, April.
    2. Benoit Mandelbrot, 1967. "The Variation of Some Other Speculative Prices," The Journal of Business, University of Chicago Press, vol. 40, pages 393-393.
    3. Barry Falk & Chun-Hsuan Wang, 2003. "Testing long-run PPP with infinite-variance returns," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 18(4), pages 471-484.
    4. Rong-Mao Zhang & Ngai Hang Chan, 2012. "Maximum likelihood estimation for nearly non-stationary stable autoregressive processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(4), pages 542-553, July.
    5. Knight, Keith, 1991. "Limit Theory for M-Estimates in an Integrated Infinite Variance," Econometric Theory, Cambridge University Press, vol. 7(02), pages 200-212, June.
    6. Phillips, P.C.B., 1990. "Time Series Regression With a Unit Root and Infinite-Variance Errors," Econometric Theory, Cambridge University Press, vol. 6(01), pages 44-62, March.
    7. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(03), pages 468-497, December.
    8. D. M. Mahinda Samarakoon & Keith Knight, 2009. "A Note on Unit Root Tests with Infinite Variance Noise," Econometric Reviews, Taylor & Francis Journals, vol. 28(4), pages 314-334.
    9. Resnick, Sidney & Greenwood, Priscilla, 1979. "A bivariate stable characterization and domains of attraction," Journal of Multivariate Analysis, Elsevier, vol. 9(2), pages 206-221, June.
    10. Sung Ahn & Stergios Fotopoulos & Lijian He, 2001. "Unit Root Tests With Infinite Variance Errors," Econometric Reviews, Taylor & Francis Journals, vol. 20(4), pages 461-483.
    11. Akgiray, Vedat & Booth, G Geoffrey, 1988. "The Stable-Law Model of Stock Returns," Journal of Business & Economic Statistics, American Statistical Association, vol. 6(1), pages 51-57, January.
    12. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
    13. Koedijk, Kees G & Kool, Clemens J M, 1992. "Tail Estimates of East European Exchange Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(1), pages 83-96, January.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    autoregressive model; near unit root; Cauchy distribution; maxi- mum likelihood estimator; infi?nite variance;

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:sgo:wpaper:1612. 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: (Jung Hur). General contact details of provider: http://edirc.repec.org/data/risogkr.html .

    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 CitEc recognized a reference but did not link an item in RePEc 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 RePEc Author Service 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.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.