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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.

Suggested Citation

  • Eric Ghysels & Joanna Jasiak, 1997. "GARCH for Irregularly Spaced Data: The ACD-GARCH Model," CIRANO Working Papers 97s-06, CIRANO.
  • Handle: RePEc:cir:cirwor:97s-06

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    References listed on IDEAS

    1. 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.
    2. 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.
    3. Drost, Feike C & Nijman, Theo E, 1993. "Temporal Aggregation of GARCH Processes," Econometrica, Econometric Society, vol. 61(4), pages 909-927, July.
    4. 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.
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    6. Ghysels, E. & Gourieroux, C. & Jasiak, J., 1995. "Market Time and Asset Price Movements: Theory and Estimation," Cahiers de recherche 9536, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    7. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    8. Diebold & Lopez, "undated". "Modeling Volatility Dynamics," Home Pages _062, University of Pennsylvania.
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    10. Bollerslev, Tim & Ghysels, Eric, 1996. "Periodic Autoregressive Conditional Heteroscedasticity," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(2), pages 139-151, April.
    11. Nijman, Theo E & Palm, Franz C, 1990. "Predictive Accuracy Gain from Disaggregate Sampling in ARIMA Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(4), pages 405-415, October.
    12. Newey, Whitney & West, Kenneth, 2014. "A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 33(1), pages 125-132.
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    15. Drost, Feike C. & Werker, Bas J. M., 1996. "Closing the GARCH gap: Continuous time GARCH modeling," Journal of Econometrics, Elsevier, vol. 74(1), pages 31-57, September.
    16. Eric Ghysels & Christian Gouriéroux & Joanna Jasiak, 1995. "Trading Patterns, Time Deformation and Stochastic Volatility in Foreign Exchange Markets," CIRANO Working Papers 95s-42, CIRANO.
    17. 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. BAUWENS, Luc & VEREDAS, David, 1999. "The stochastic conditional duration model: a latent factor model for the analysis of financial durations," CORE Discussion Papers 1999058, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Wing Lon NG, 2004. "Duration and Order Type Clusters," Econometric Society 2004 Far Eastern Meetings 730, Econometric Society.
    3. GIOT, Pierre, 1999. "Time transformations, intraday data and volatility models," CORE Discussion Papers 1999044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Gerhard, Frank & Hautsch, Nikolaus, 2002. "Volatility estimation on the basis of price intensities," Journal of Empirical Finance, Elsevier, vol. 9(1), pages 57-89, January.
    5. Wing Lon NG, 2004. "Duration and Order Type Clusters," Econometric Society 2004 Australasian Meetings 272, Econometric Society.
    6. BAUWENS, Luc & GIOT, Pierre, 1998. "Asymmetric ACD models: introducing price information in ACD models with a two state transition model," CORE Discussion Papers 1998044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

    More about this item


    Heteroskedasticity; Market Activity; Tick-by-Tick Data; Volatility; Causality; Duration Models; Hétéroscédasticité; activité de marché; volatilité; causalité; modèles de durées;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • D41 - Microeconomics - - Market Structure, Pricing, and Design - - - Perfect Competition
    • G1 - Financial Economics - - General Financial Markets


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