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Multimodality and the GARCH Likelihood

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  • Jurgen A. Doornik and Marius Ooms

Abstract

We investigate several aspects of GARCH(p,q) models which are relevant for empirical applications. In particular, we note that the inclusion of a dummy variable as regressor can lead to multimodality in the GARCH likelihood. This makes standard inference on the estimated coefficient impossible. Next, we investigate the implementation of different restrictions on the GARCH parameter space. We present a small refinement to the Nelson-Cao (1992) conditions for a GARCH(2,q) model, and show how these can be implemented by parameter transformations. We argue that these conditions are also too restrictive, and consider restrictions which are formulated in terms of the unconditional variance. These are easier to work with and understand. Finally, we show that multimodality is a real concern for models of the pounds exchange rate, and should be taken account of, especially when p >= 2.

Suggested Citation

  • Jurgen A. Doornik and Marius Ooms, 2001. "Multimodality and the GARCH Likelihood," Computing in Economics and Finance 2001 76, Society for Computational Economics.
  • Handle: RePEc:sce:scecf1:76
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    References listed on IDEAS

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    1. Drost, Feike C & Nijman, Theo E, 1993. "Temporal Aggregation of GARCH Processes," Econometrica, Econometric Society, vol. 61(4), pages 909-927, July.
    2. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 2002. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 69-87, January.
    3. Bollerslev, Tim & Engle, Robert F. & Nelson, Daniel B., 1986. "Arch models," Handbook of Econometrics,in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 49, pages 2959-3038 Elsevier.
    4. White, Halbert, 1982. "Instrumental Variables Regression with Independent Observations," Econometrica, Econometric Society, vol. 50(2), pages 483-499, March.
    5. Fiorentini, Gabriele & Calzolari, Giorgio & Panattoni, Lorenzo, 1996. "Analytic Derivatives and the Computation of GARCH Estimates," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(4), pages 399-417, July-Aug..
    6. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    7. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford.
    8. Nelson, Daniel B & Cao, Charles Q, 1992. "Inequality Constraints in the Univariate GARCH Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(2), pages 229-235, April.
    9. 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.
    10. repec:bla:restud:v:65:y:1998:i:3:p:361-93 is not listed on IDEAS
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    Cited by:

    1. B. D. McCullough & H. D. Vinod, 2003. "Verifying the Solution from a Nonlinear Solver: A Case Study," American Economic Review, American Economic Association, vol. 93(3), pages 873-892, June.
    2. Eric Hillebrand & Gunther Schnabl, 2003. "The Effects of Japanese Foreign Exchange Intervention: GARCH Estimation and Change Point Detection," Departmental Working Papers 2003-09, Department of Economics, Louisiana State University.
    3. Kwami Adanu, 2006. "Optimizing the Garch Model–An Application of Two Global and Two Local Search Methods," Computational Economics, Springer;Society for Computational Economics, vol. 28(3), pages 277-290, October.
    4. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 121-138.
    5. Soosung Hwang & Pedro L. Valls Pereira, 2006. "Small sample properties of GARCH estimates and persistence," The European Journal of Finance, Taylor & Francis Journals, vol. 12(6-7), pages 473-494.
    6. Henrik Amilon, 2002. "A Score Test for Discreteness in GARCH Models," Research Paper Series 76, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Amilon, Henrik, 2003. "GARCH estimation and discrete stock prices: an application to low-priced Australian stocks," Economics Letters, Elsevier, vol. 81(2), pages 215-222, November.
    8. Laurini, M. P. & Portugal, M. S., 2003. "Long Memory int the R$/US$ Exchange Rate: A Robust Analysis," Finance Lab Working Papers flwp_50, Finance Lab, Insper Instituto de Ensino e Pesquisa.
    9. repec:wsi:ceprxx:v:02:y:2013:i:02:n:s1793969013500076 is not listed on IDEAS

    More about this item

    Keywords

    GARCH; EGARCH; multimodality; dummy variable; parameter space;

    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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