IDEAS home Printed from https://ideas.repec.org/p/crs/wpaper/2009-19.html
   My bibliography  Save this paper

Properties of the QMLE and the Weighted LSE for LARCH(q) Models

Author

Listed:
  • Christian FRANCQ

    (Crest)

  • Jean-Michel ZAKOIAN

    (Crest)

Abstract

This paper considers a class of finite-order autoregressive linear ARCH models. The model captures theleverage effect, allows the volatility to be zero and to reach its minimum for non-zero innovations, and isappropriate for long-memory modeling when infinite orders are allowed. However, the quasi-maximum likelihoodestimator is, in general, inconsistent. A self-weighted least-squares estimator is proposed and is shownto be asymptotically normal. A score test for conditional homoscedasticity and diagnostic portmanteau testsare developed. Their performance is illustrated via simulation experiments. It is also investigated whetherstock market returns exhibit some of the characteristic features of the linear ARCH model.

Suggested Citation

  • Christian FRANCQ & Jean-Michel ZAKOIAN, 2009. "Properties of the QMLE and the Weighted LSE for LARCH(q) Models," Working Papers 2009-19, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2009-19
    as

    Download full text from publisher

    File URL: http://crest.science/RePEc/wpstorage/2009-19.pdf
    File Function: Crest working paper version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Berkes, István & Horváth, Lajos & Kokoszka, Piotr, 2003. "Asymptotics For Garch Squared Residual Correlations," Econometric Theory, Cambridge University Press, vol. 19(4), pages 515-540, August.
    2. Paul Doukhan & Patrice Bertail & Philippe Soulier, 2006. "Dependence in Probability and Statistics," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00268232, HAL.
    3. Francq, Christian & Makarova, Svetlana & Zakoi[diaeresis]an, Jean-Michel, 2008. "A class of stochastic unit-root bilinear processes: Mixing properties and unit-root test," Journal of Econometrics, Elsevier, vol. 142(1), pages 312-326, January.
    4. W. K. Li & T. K. Mak, 1994. "On The Squared Residual Autocorrelations In Non‐Linear Time Series With Conditional Heteroskedasticity," Journal of Time Series Analysis, Wiley Blackwell, vol. 15(6), pages 627-636, November.
    5. Giraitis, Liudas & Surgailis, Donatas, 0. "ARCH-type bilinear models with double long memory," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 275-300, July.
    6. Robinson, P. M., 1991. "Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression," Journal of Econometrics, Elsevier, vol. 47(1), pages 67-84, January.
    7. Giraitis, Liudas & Leipus, Remigijus & Robinson, Peter M. & Surgailis, Donatas, 2004. "LARCH, leverage, and long memory," LSE Research Online Documents on Economics 294, London School of Economics and Political Science, LSE Library.
    8. A. I. McLeod & W. K. Li, 1983. "Diagnostic Checking Arma Time Series Models Using Squared‐Residual Autocorrelations," Journal of Time Series Analysis, Wiley Blackwell, vol. 4(4), pages 269-273, July.
    9. Ling, Shiqing, 2007. "Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models," Journal of Econometrics, Elsevier, vol. 140(2), pages 849-873, October.
    10. Beran, Jan, 2006. "On location estimation for LARCH processes," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1766-1782, September.
    11. Liudas Giraitis, 2004. "LARCH, Leverage, and Long Memory," Journal of Financial Econometrics, Oxford University Press, vol. 2(2), pages 177-210.
    12. Giraitis, Liudas & Robinson, Peter & Surgailis, Donatas, 2000. "A model for long memory conditional heteroscedasticity," LSE Research Online Documents on Economics 2103, London School of Economics and Political Science, LSE Library.
    13. Enrique Sentana, 1995. "Quadratic ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 62(4), pages 639-661.
    14. Paul Doukhan & Patrice Bertail & Philippe Soulier, 2006. "Dependence in Probability and Statistics," Post-Print hal-00268232, HAL.
    15. Francq, Christian & Roy, Roch & Zakoian, Jean-Michel, 2005. "Diagnostic Checking in ARMA Models With Uncorrelated Errors," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 532-544, June.
    16. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    17. Higgins, Matthew L & Bera, Anil K, 1992. "A Class of Nonlinear ARCH Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 33(1), pages 137-158, February.
    18. Horváth, Lajos & Kokoszka, Piotr, 2001. "LARGE SAMPLE DISTRIBUTION OF WEIGHTED SUMS OF ARCH(p) SQUARED RESIDUAL CORRELATIONS," Econometric Theory, Cambridge University Press, vol. 17(2), pages 283-295, April.
    19. Christie, Andrew A., 1982. "The stochastic behavior of common stock variances : Value, leverage and interest rate effects," Journal of Financial Economics, Elsevier, vol. 10(4), pages 407-432, December.
    20. Giraitis, Liudas & Robinson, Peter M. & Surgailis, Donatas, 2000. "A model for long memory conditional heteroscedasticity," LSE Research Online Documents on Economics 299, London School of Economics and Political Science, LSE Library.
    21. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127.
    22. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Francq, Christian & Zakoïan, Jean-Michel, 2010. "Inconsistency of the MLE and inference based on weighted LS for LARCH models," Journal of Econometrics, Elsevier, vol. 159(1), pages 151-165, November.
    2. repec:hal:journl:peer-00732536 is not listed on IDEAS
    3. Liudas Giraitis & Donatas Surgailis & Andrius Škarnulis, 2015. "Integrated ARCH, FIGARCH and AR Models: Origins of Long Memory," Working Papers 766, Queen Mary University of London, School of Economics and Finance.
    4. Liudas Giraitis & Donatas Surgailis & Andrius Škarnulis, 2015. "Integrated ARCH, FIGARCH and AR Models: Origins of Long Memory," Working Papers 766, Queen Mary University of London, School of Economics and Finance.
    5. Klar, B. & Lindner, F. & Meintanis, S.G., 2012. "Specification tests for the error distribution in GARCH models," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3587-3598.
    6. Iqbal Owadally, 2014. "Tail risk in pension funds: an analysis using ARCH models and bilinear processes," Review of Quantitative Finance and Accounting, Springer, vol. 43(2), pages 301-331, August.
    7. Dennis Kristensen, 2009. "On stationarity and ergodicity of the bilinear model with applications to GARCH models," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(1), pages 125-144, January.
    8. Li, Muyi & Li, Wai Keung & Li, Guodong, 2015. "A new hyperbolic GARCH model," Journal of Econometrics, Elsevier, vol. 189(2), pages 428-436.
    9. Ke Zhu, 2016. "Bootstrapping the portmanteau tests in weak auto-regressive moving average models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(2), pages 463-485, March.
    10. Giraitis, Liudas & Leipus, Remigijus & Robinson, Peter M. & Surgailis, Donatas, 2004. "LARCH, leverage, and long memory," LSE Research Online Documents on Economics 294, London School of Economics and Political Science, LSE Library.
    11. W. K. Li & Shiqing Ling & Michael McAleer, 2001. "A Survey of Recent Theoretical Results for Time Series Models with GARCH Errors," ISER Discussion Paper 0545, Institute of Social and Economic Research, Osaka University.
    12. Ieva Grublytė & Donatas Surgailis & Andrius Škarnulis, 2017. "QMLE for Quadratic ARCH Model with Long Memory," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(4), pages 535-551, July.
    13. Javier Haulde & Morten Ørregaard Nielsen, 2022. "Fractional integration and cointegration," CREATES Research Papers 2022-02, Department of Economics and Business Economics, Aarhus University.
    14. Liudas Giraitis & Remigijus Leipus & Peter M Robinson & Donatas Surgailis, 2003. "LARCH, Leverage and Long Memory," STICERD - Econometrics Paper Series 460, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    15. Liudas Giraitis, 2004. "LARCH, Leverage, and Long Memory," Journal of Financial Econometrics, Oxford University Press, vol. 2(2), pages 177-210.
    16. Zhu, Ke, 2012. "A mixed portmanteau test for ARMA-GARCH model by the quasi-maximum exponential likelihood estimation approach," MPRA Paper 40382, University Library of Munich, Germany.
    17. Ana Filipa Carvalho & Jose Sa da Costa & Jose Assis Lopes, 2006. "A systematic modelling strategy for futures markets volatility," Applied Financial Economics, Taylor & Francis Journals, vol. 16(11), pages 819-833.
    18. Francq, Christian & Makarova, Svetlana & Zakoi[diaeresis]an, Jean-Michel, 2008. "A class of stochastic unit-root bilinear processes: Mixing properties and unit-root test," Journal of Econometrics, Elsevier, vol. 142(1), pages 312-326, January.
    19. Martinez Oscar & Olmo Jose, 2012. "A Nonlinear Threshold Model for the Dependence of Extremes of Stationary Sequences," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 16(3), pages 1-39, September.
    20. Hentschel, Ludger, 1995. "All in the family Nesting symmetric and asymmetric GARCH models," Journal of Financial Economics, Elsevier, vol. 39(1), pages 71-104, September.
    21. Tim Bollerslev, 2008. "Glossary to ARCH (GARCH)," CREATES Research Papers 2008-49, Department of Economics and Business Economics, Aarhus University.

    More about this item

    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:crs:wpaper:2009-19. See general information about how to correct material in RePEc.

    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 bibliographic 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.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Secretariat General (email available below). General contact details of provider: https://edirc.repec.org/data/crestfr.html .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.