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On Periodic Autogressive Conditional Heteroskedasticity

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  • Tim Bollerslev
  • Eric Ghysels

    ()

Abstract

Asset returns exhibit clustering of volatility throughout the year. This paper proposes a class of models featuring periodicity in conditional heteroskedasticity. The periodic structures in GARCH models share many properties with periodic ARMA processes studied by Gladyshev (1961), Tiao and Grupe (1980) and others. We describe the relation between periodic GARCH processes and time-invariant (seasonal) GARCH processes. Besides the periodic GARCH or P-GARCH process, we also discuss P-IGARCH, PI-GARCH, P-ARCH-M and P-EGARCH processes. Extensions to multivariate ARCH processes are studied as well. Moreover, we also consider periodicity in the common persistence of volatility for several series. A quasi-maximum likelihood estimator following Bollerslev and Wooldridge (1992) is defined and a LM test for periodicity derived from it. The models are applied to several asset pricing series. Dans cette étude, nous proposons une classe de processus ARCH périodiques. Cette structure est semblable à celle des processus linéaires périodiques. Les procésus P-ARCH partagent beaucoup de similarités avec les processus périodiques linéaires0501s ont aussi, à cause des non linéarités, des caractéristiques spécifiques. Nous étudions de façon analytique les pertes d'efficacité en terme de prévisions dues à des erreurs de spécifications lorsque les données suivent un processus P-ARCH et qu'un modèle ARCG (saisonnier) est estimé. Le papier inclut également une étude de Monte Carlo qui complémente les résultats théoriques et une appliction au taux de change DM - livre Sterling. Plusieurs extensions, telles que P-EGARCH et P-IGARCH, sont aussi proposées.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 94s-03.

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Date of creation: 01 Sep 1994
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Handle: RePEc:cir:cirwor:94s-03

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Keywords: Volatility clustering; Seasonality; Periodic structures; ARCH; GARCH; P GARCH; Exchange rates; Persistance dans la volatilité ; Structures périodiques ; Taux de change;

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References

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  1. Drost, Feike C & Nijman, Theo E, 1993. "Temporal Aggregation of GARCH Processes," Econometrica, Econometric Society, vol. 61(4), pages 909-27, July.
  2. 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.
  3. Baillie, R.T. & Bollerslev, R.T., 1990. "Prediction In Dynamic Models With Time Dependent Conditional Variances," Papers 8815, Michigan State - Econometrics and Economic Theory.
  4. Ghysels, E., 1993. "A Time Series Model with Periodic Stochastic Regime Switching," Cahiers de recherche 9314, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  5. Francis X. Diebold & Marc Nerlove, 1986. "The dynamics of exchange rate volatility: a multivariate latent factor ARCH model," Special Studies Papers 205, Board of Governors of the Federal Reserve System (U.S.).
  6. 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.
  7. Osborn, Denise R, 1988. "Seasonality and Habit Persistence in a Life Cycle Model of Consumptio n," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 3(4), pages 255-66, October-D.
  8. 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.
  9. Lars Peter Hansen & Thomas J. Sargent, 1993. "Recursive linear models of dynamic economies," Proceedings, Federal Reserve Bank of San Francisco, issue Mar.
  10. Ghysels, E., 1990. "On The Economic And Econometrics Of Seasonality," Cahiers de recherche 9028, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  11. Bollerslev, Tim & Engle, Robert F, 1993. "Common Persistence in Conditional Variances," Econometrica, Econometric Society, vol. 61(1), pages 167-86, January.
  12. Tim Bollerslev & Jeffrey M. Wooldridge, 1988. "Quasi-Maximum Likelihood Estimation of Dynamic Models with Time-Varying Covariances," Working papers 505, Massachusetts Institute of Technology (MIT), Department of Economics.
  13. Ghysels, E. & Jasiak, J., 1994. "Stochastic Volatility and time Deformation: An Application of trading Volume and Leverage Effects," Cahiers de recherche 9403, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  14. Eric Ghysels & Andrew Harvey & Éric Renault, 1995. "Stochastic Volatility," CIRANO Working Papers 95s-49, CIRANO.
  15. Lamoureux, Christopher G & Lastrapes, William D, 1990. "Persistence in Variance, Structural Change, and the GARCH Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(2), pages 225-34, April.
  16. Bollerslev, Tim & Engle, Robert F & Wooldridge, Jeffrey M, 1988. "A Capital Asset Pricing Model with Time-Varying Covariances," Journal of Political Economy, University of Chicago Press, vol. 96(1), pages 116-31, February.
  17. Ghysels, Eric, 1994. "On the Periodic Structure of the Business Cycle," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(3), pages 289-98, July.
  18. Ghysels, E., 1991. "On Scoring Asymmetric Periodic Probability Models of Turning-Point Forecasts," Cahiers de recherche 9130, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  19. Tim Bollerslev & Robert J. Hodrick, 1992. "Financial Market Efficiency Tests," NBER Working Papers 4108, National Bureau of Economic Research, Inc.
  20. Ghysels, E. & Hall, A., 1993. "On Periodic Time Series and Testing the Unit Root Hypothesis," Cahiers de recherche 9333, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  21. Engle, Robert F & Lilien, David M & Robins, Russell P, 1987. "Estimating Time Varying Risk Premia in the Term Structure: The Arch-M Model," Econometrica, Econometric Society, vol. 55(2), pages 391-407, March.
  22. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-70, March.
  23. Gallant, A Ronald & Rossi, Peter E & Tauchen, George, 1992. "Stock Prices and Volume," Review of Financial Studies, Society for Financial Studies, vol. 5(2), pages 199-242.
  24. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
  25. Osborn, Denise R & Smith, Jeremy P, 1989. "The Performance of Periodic Autoregressive Models in Forecasting Seasonal U. K. Consumption," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(1), pages 117-27, January.
  26. Wooldridge, Jeffrey M., 1990. "A Unified Approach to Robust, Regression-Based Specification Tests," Econometric Theory, Cambridge University Press, vol. 6(01), pages 17-43, March.
  27. Nelson, Daniel B., 1990. "Stationarity and Persistence in the GARCH(1,1) Model," Econometric Theory, Cambridge University Press, vol. 6(03), pages 318-334, September.
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