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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.

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Bibliographic Info

Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 2001 with number 76.

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Date of creation: 01 Apr 2001
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Handle: RePEc:sce:scecf1:76

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

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References

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  1. 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.
  2. Neil Shephard, 2005. "Stochastic volatility," Economics Series Working Papers 2005-W17, University of Oxford, Department of Economics.
  3. 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.
  4. White, Halbert, 1982. "Instrumental Variables Regression with Independent Observations," Econometrica, Econometric Society, vol. 50(2), pages 483-99, March.
  5. Drost, F.C. & Nijman, T.E., 1990. "Temporal Aggregation Of Garch Processes," Papers 9066, Tilburg - Center for Economic Research.
  6. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
  7. 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.
  8. Fiorentini,G. & Calzolari,G. & Panattoni,L., 1995. "Analytic Derivatives and the Computation of Garch Estimates," Papers 9519, Centro de Estudios Monetarios Y Financieros-.
  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.
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Cited by:
  1. Hwang. S. & Pedro L. Valls Pereira, 2003. "Small Sample Properties of GARCH Estimates and Persistence," Finance Lab Working Papers flwp_48, Finance Lab, Insper Instituto de Ensino e Pesquisa.
  2. Eric Hillebrand & Gunther Schnabl, 2004. "The Effects of Japanese Foreign Exchange Intervention, GARCH Estimation and Change Point Detection," Money Macro and Finance (MMF) Research Group Conference 2004 7, Money Macro and Finance Research Group.
  3. 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.
  4. 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.
  5. 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.
  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. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 121-138.
  8. Henrik Amilon, 2002. "A Score Test for Discreteness in GARCH Models," Research Paper Series 76, Quantitative Finance Research Centre, University of Technology, Sydney.

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