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GARCH for Irregularly Spaced Data: The ACD-GARCH Model

  • Eric Ghysels
  • Joanna Jasiak

We develop a class of ARCH models for series sampled at unequal time intervals set by trade or quote arrivals. Our approach combines insights from the temporal aggregation for GARCH models discussed by Drost and Nijman (1993) and Drost and Werker (1994), and the autoregressive conditional duration model of Engle and Russell (1996) proposed to model the spacing between consecutive financial transactions. The class of models we introduce here will be called ACD-GARCH. It can be described as a random coefficient GARCH, or doubly stochastic GARCH, where the durations between transactions determine the parameter dynamics. The ACD-GARCH model becomes genuinely bivariate when past asset return volatilities are allowed to affect transaction durations and vice versa. Otherwise the spacings between trades are considered exogenous to the volatility dynamics. This assumption is required in a two-step estimation procedure. The bivariate setup enables us to test for Granger causality between volatility and intra-trade durations. Under general conditions we propose several GMM estimation procedures, some having a QMLE interpretation. As illustration we present an empirical study of the IBM 1993 tick-by-tick data. We find that volatility of IBM stock prices Granger causes intra-trade durations. We also find that the persistence in GARCH drops dramatically once intra-trade durations are taken into account. Nous développons une classe de modèles ARCH pour les séries temporelles échantillonnées à intervalles inégaux comme des observations liées à des transactions de marché. Notre approche est fondée sur la méthode d'aggrégation temporelle pour les modèles ARCH de Drost et Nijman (1993) et de Drost et Werker (1994), et d'autre part sur le modèle autorégressif des moyennes conditionnelles des durées entre les transactions financières de Engle et Russell (1996). La classe de modèles présentée ici est nommée ACD-GARCH. Ce type de modèles peut être défini comme un GARCH aux coefficients aléatoires où la durée entre les transactions détermine la dynamique des paramètres. Le ACD-GARCH devient un modèle bivarié quand sa formation admet les interactions entre les volatilités des rendements passés et les durées, et vice-versa. Sinon, la série de durées est considérée exogène par rapport au processus de volatilité. Cette condition est préalable à l'estimation du modèle ACD-GARCH en deux étapes. La spécification bivariée nous permet de tester l'existence de la causalité de type Granger entre les volatilités et les durées. Sous conditions générales, diverses procédures d'estimation par la méthode de moments généralisés sont considérées, dont certaines fournissent les estimateurs, à la fois de type GMM et de type QMLE. Pour ce qui est des applications, nous présentons une étude empirique basée sur les données de transactions du titre IBM en 1993. Nos résultats indiquent que la volatilité des rendements sur les prix d'actions de IBM cause, au sens de Granger, les durées entre les transactions. Nous observons aussi que la persistance du processus GARCH diminue fortement quand on introduit les durées dans la formulation du modèle.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 97s-06.

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Length: 31 pages
Date of creation: 01 Feb 1997
Date of revision:
Handle: RePEc:cir:cirwor:97s-06
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  1. Diebold & Lopez, . "Modeling Volatility Dynamics," Home Pages _062, University of Pennsylvania.
  2. Bollerslev, T. & Ghysels, E., 1994. "Periodic Autoregressive Conditional Heteroskedasticity," Cahiers de recherche 9408, Universite de Montreal, Departement de sciences economiques.
  3. Ghysels, E. & Jasiak, J., 1994. "Stochastic Volatility and time Deformation: an Application of trading Volume and Leverage Effects," Cahiers de recherche 9403, Universite de Montreal, Departement de sciences economiques.
  4. Drost, F.C. & Werker, B.J.M., 1994. "Closing the GARCH gap : Continuous time GARCH modeling," Discussion Paper 1994-2, Tilburg University, Center for Economic Research.
  5. Eric Ghysels & Christian Gouriéroux & Joanna Jasiak, 1995. "Market Time and Asset Price Movements Theory and Estimation," CIRANO Working Papers 95s-32, CIRANO.
  6. Newey, Whitney K & West, Kenneth D, 1987. "A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix," Econometrica, Econometric Society, vol. 55(3), pages 703-08, May.
  7. Nijman, T.E. & Palm, F.C., 1987. "Predictive accuracy gain from disaggregate sampling in ARIMA-models," Research Memorandum FEW 273, Tilburg University, School of Economics and Management.
  8. Ghysels, E. & Harvey, A. & Renault, E., 1996. "Stochastic Volatility," Cahiers de recherche 9613, Universite de Montreal, Departement de sciences economiques.
  9. Marcel Boyer, 1997. "Competition and Access in Telecoms: ECPR, Global Price Cap, and Auctions," CIRANO Working Papers 97s-03, CIRANO.
  10. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-54, July.
  11. Drost, Feike C & Nijman, Theo E, 1993. "Temporal Aggregation of GARCH Processes," Econometrica, Econometric Society, vol. 61(4), pages 909-27, July.
  12. Nijman, T.E. & Palm, F.C., 1984. "Missing observations in the dynamic regression model," Other publications TiSEM 4d689d7c-4d89-4ab6-b8c3-f, Tilburg University, School of Economics and Management.
  13. Donald W.K. Andrews & Christopher J. Monahan, 1990. "An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator," Cowles Foundation Discussion Papers 942, Cowles Foundation for Research in Economics, Yale University.
  14. Drost, F.C. & Nijman, T.E., 1993. "Temporal aggregation of GARCH processes," Other publications TiSEM 0642fb61-c7f4-4281-b484-4, Tilburg University, School of Economics and Management.
  15. Eric Ghysels & Christian Gouriéroux & Joanna Jasiak, 1995. "Trading Patterns, Time Deformation and Stochastic Volatility in Foreign Exchange Markets," CIRANO Working Papers 95s-42, CIRANO.
  16. Robert F. Engle & Jeffrey R. Russell, 1994. "Forecasting Transaction Rates: The Autoregressive Conditional Duration Model," NBER Working Papers 4966, National Bureau of Economic Research, Inc.
  17. Cheung, Yin-Wong & Ng, Lilian K., 1996. "A causality-in-variance test and its application to financial market prices," Journal of Econometrics, Elsevier, vol. 72(1-2), pages 33-48.
  18. 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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