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Multimodality and the GARCH Likelihood

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  • Jurgen A. Doornik

    (Nuffield College)

  • Marius Ooms

    (Free University Amsterdam)

Abstract

We investigate several aspects of GARCH 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 and 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 pound/dollar exchange rate, and should be taken account of, especially when p>=2.

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Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 0798.

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Date of creation: 01 Aug 2000
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Handle: RePEc:ecm:wc2000:0798

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  1. 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..
  2. Drost, F.C. & Nijman, T.E., 1993. "Temporal aggregation of GARCH processes," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153273, Tilburg University.
  3. Neil Shephard, 2005. "Stochastic volatility," Economics Series Working Papers 2005-W17, University of Oxford, Department of Economics.
  4. 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.
  5. 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.
  6. 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 229-35, April.
  7. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 1994. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(4), pages 371-89, October.
  8. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
  9. White, Halbert, 1982. "Instrumental Variables Regression with Independent Observations," Econometrica, Econometric Society, vol. 50(2), pages 483-99, March.
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Cited by:
  1. 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.
  2. Henrik Amilon, 2002. "A Score Test for Discreteness in GARCH Models," Research Paper Series 76, Quantitative Finance Research Centre, University of Technology, Sydney.
  3. Eric Hillebrand & Gunther Schnabl, 2004. "The Effects of Japanese Foreign Exchange Intervention: GARCH Estimation and Change Point Detection," International Finance 0410008, EconWPA.
  4. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 121-138.
  5. Kwami Adanu, 2006. "Optimizing the Garch Model–An Application of Two Global and Two Local Search Methods," Computational Economics, Society for Computational Economics, vol. 28(3), pages 277-290, October.
  6. 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.
  7. 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.
  8. 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.

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