IDEAS home Printed from https://ideas.repec.org/a/bpj/sndecm/v22y2018i5p25n4.html
   My bibliography  Save this article

Closed-form estimators for finite-order ARCH models as simple and competitive alternatives to QMLE

Author

Listed:
  • Prono Todd

    (Federal Reserve Board, Washington, DC, USA, Phone: +(202) 973-6955)

Abstract

Strong consistency and (weak) distributional convergence to highly non-Gaussian limits are established for closed-form, two stage least squares (TSLS) estimators of linear and threshold ARCH (p) models, with special attention paid to the ARCH (1) and threshold ARCH (1) cases. Conditions supporting these results include (relatively) mild moment existence criteria that enjoy empirical support. These conditions are not shared by competing estimators like OLS. Identification of the TSLS estimators depends on asymmetry, either in the model’s rescaled errors or in the conditional variance function. Monte Carlo studies reveal TSLS estimation can sizably outperform quasi maximum likelihood (QML) in small samples and even best recently proposed two-step estimators specifically designed to enhance the efficiency of QML.

Suggested Citation

  • Prono Todd, 2018. "Closed-form estimators for finite-order ARCH models as simple and competitive alternatives to QMLE," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 22(5), pages 1-25, December.
  • Handle: RePEc:bpj:sndecm:v:22:y:2018:i:5:p:25:n:4
    DOI: 10.1515/snde-2017-0070
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/snde-2017-0070
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.

    File URL: https://libkey.io/10.1515/snde-2017-0070?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kuersteiner, Guido M., 2002. "Efficient Iv Estimation For Autoregressive Models With Conditional Heteroskedasticity," Econometric Theory, Cambridge University Press, vol. 18(3), pages 547-583, June.
    2. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 95-115, May.
    3. Hall, Peter & Yao, Qiwei, 2003. "Inference in ARCH and GARCH models with heavy-tailed errors," LSE Research Online Documents on Economics 5875, London School of Economics and Political Science, LSE Library.
    4. Hafner, Christian M. & Linton, Oliver, 2017. "An Almost Closed Form Estimator For The Egarch Model," Econometric Theory, Cambridge University Press, vol. 33(4), pages 1013-1038, August.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. Bouchaud,Jean-Philippe & Potters,Marc, 2003. "Theory of Financial Risk and Derivative Pricing," Cambridge Books, Cambridge University Press, number 9780521819169, September.
    7. Igor Vaynman & Brendan K. Beare, 2014. "Stable Limit Theory for the Variance Targeting Estimator," Advances in Econometrics, in: Essays in Honor of Peter C. B. Phillips, volume 33, pages 639-672, Emerald Group Publishing Limited.
    8. Arthur Lewbel, 1997. "Constructing Instruments for Regressions with Measurement Error when no Additional Data are Available, with an Application to Patents and R&D," Econometrica, Econometric Society, vol. 65(5), pages 1201-1214, September.
    9. Jianqing Fan & Lei Qi & Dacheng Xiu, 2014. "Quasi-Maximum Likelihood Estimation of GARCH Models With Heavy-Tailed Likelihoods," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(2), pages 178-191, April.
    10. Jondeau, Eric & Rockinger, Michael, 2003. "Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1699-1737, August.
    11. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    12. Hill, Jonathan B., 2010. "On Tail Index Estimation For Dependent, Heterogeneous Data," Econometric Theory, Cambridge University Press, vol. 26(5), pages 1398-1436, October.
    13. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 5-59.
    14. Weiss, Andrew A., 1986. "Asymptotic Theory for ARCH Models: Estimation and Testing," Econometric Theory, Cambridge University Press, vol. 2(1), pages 107-131, April.
    15. Peter Hall & Qiwei Yao, 2003. "Inference in Arch and Garch Models with Heavy--Tailed Errors," Econometrica, Econometric Society, vol. 71(1), pages 285-317, January.
    16. Carrasco, Marine & Chen, Xiaohong, 2002. "Mixing And Moment Properties Of Various Garch And Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 18(1), pages 17-39, February.
    17. 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. Todd Prono, 2016. "Closed-Form Estimation of Finite-Order ARCH Models: Asymptotic Theory and Finite-Sample Performance," Finance and Economics Discussion Series 2016-083, Board of Governors of the Federal Reserve System (U.S.).
    2. Aguilar, Mike & Hill, Jonathan B., 2015. "Robust score and portmanteau tests of volatility spillover," Journal of Econometrics, Elsevier, vol. 184(1), pages 37-61.
    3. Kristensen Dennis & Rahbek Anders, 2009. "Asymptotics of the QMLE for Non-Linear ARCH Models," Journal of Time Series Econometrics, De Gruyter, vol. 1(1), pages 1-38, April.
    4. Eric Beutner & Julia Schaumburg & Barend Spanjers, 2024. "Bootstrapping GARCH Models Under Dependent Innovations," Tinbergen Institute Discussion Papers 24-008/III, Tinbergen Institute.
    5. Sébastien Laurent & Luc Bauwens & Jeroen V. K. Rombouts, 2006. "Multivariate GARCH models: a survey," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(1), pages 79-109.
    6. Zhu, Ke & Ling, Shiqing, 2013. "Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models," MPRA Paper 51509, University Library of Munich, Germany.
    7. Hill, Jonathan B. & Prokhorov, Artem, 2016. "GEL estimation for heavy-tailed GARCH models with robust empirical likelihood inference," Journal of Econometrics, Elsevier, vol. 190(1), pages 18-45.
    8. Pan, Jiazhu & Wang, Hui & Tong, Howell, 2008. "Estimation and tests for power-transformed and threshold GARCH models," Journal of Econometrics, Elsevier, vol. 142(1), pages 352-378, January.
    9. 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.
    10. Audrone Virbickaite & M. Concepción Ausín & Pedro Galeano, 2015. "Bayesian Inference Methods For Univariate And Multivariate Garch Models: A Survey," Journal of Economic Surveys, Wiley Blackwell, vol. 29(1), pages 76-96, February.
    11. Degiannakis, Stavros & Xekalaki, Evdokia, 2004. "Autoregressive Conditional Heteroskedasticity (ARCH) Models: A Review," MPRA Paper 80487, University Library of Munich, Germany.
    12. Xibin Zhang & Maxwell L. King, 2013. "Gaussian kernel GARCH models," Monash Econometrics and Business Statistics Working Papers 19/13, Monash University, Department of Econometrics and Business Statistics.
    13. Luger, Richard, 2012. "Finite-sample bootstrap inference in GARCH models with heavy-tailed innovations," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3198-3211.
    14. Emma M. Iglesias & Garry D. A. Phillips, 2012. "Estimation, Testing, and Finite Sample Properties of Quasi-Maximum Likelihood Estimators in GARCH-M Models," Econometric Reviews, Taylor & Francis Journals, vol. 31(5), pages 532-557, September.
    15. Chen, Min & Zhu, Ke, 2015. "Sign-based portmanteau test for ARCH-type models with heavy-tailed innovations," Journal of Econometrics, Elsevier, vol. 189(2), pages 313-320.
    16. Yuanyuan Zhang & Rong Liu & Qin Shao & Lijian Yang, 2020. "Two‐Step Estimation for Time Varying Arch Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(4), pages 551-570, July.
    17. Conrad, Christian & Mammen, Enno, 2016. "Asymptotics for parametric GARCH-in-Mean models," Journal of Econometrics, Elsevier, vol. 194(2), pages 319-329.
    18. Javed Farrukh & Podgórski Krzysztof, 2017. "Tail Behavior and Dependence Structure in the APARCH Model," Journal of Time Series Econometrics, De Gruyter, vol. 9(2), pages 1-48, July.
    19. Li, Qi & Yang, Jian & Hsiao, Cheng & Chang, Young-Jae, 2005. "The relationship between stock returns and volatility in international stock markets," Journal of Empirical Finance, Elsevier, vol. 12(5), pages 650-665, December.
    20. Issler, João Victor, 1999. "Estimating and forecasting the volatility of Brazilian finance series using arch models (Preliminary Version)," FGV EPGE Economics Working Papers (Ensaios Economicos da EPGE) 347, EPGE Brazilian School of Economics and Finance - FGV EPGE (Brazil).

    More about this item

    Keywords

    ARCH; closed form estimation; heavy tails; threshold ARCH; instrumental variables; regular variation; two stage least squares;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

    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:bpj:sndecm:v:22:y:2018:i:5:p:25:n:4. 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: Peter Golla (email available below). General contact details of provider: https://www.degruyter.com .

    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.