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Contemporaneous Aggregation of GARCH Processes

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  • Paolo Zaffaroni
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    Abstract

    We study the impact of large cross-sections of contemporaneous aggregation of GARCH processes and of dynamic GARCH factor models. The results crucially depend on the shape of the cross-sectional distribution of the GARCH coefficients and on the cross-sectional dependence properties of the rescaled innovation. The aggregate maintains the core nonlinearity of a volatility model, uncorrelation in the levels but autocorrelation in the squares, when the rescaled innovation is common across units. The nonlinearity is, however, lost at the aggregate level, when the rescaled innovation is orthogonal across units. This is not a consequence of the usual result of a vanishing importance of purely idiosyncratic risk as, under appropriate conditions, this is simply not fully diversifiable in arbitrary large portfolios. Non-GARCH memory properties arise at the aggregate level. Strict stationarity, ergodicity and finite kurtosis might fail for the aggregate despite the micro GARCH do satisfy these properties. Under no conditions aggregation of GARCH induces long memory conditional heteroskedasticity.

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    Bibliographic Info

    Paper provided by Suntory and Toyota International Centres for Economics and Related Disciplines, LSE in its series STICERD - Econometrics Paper Series with number /2000/378.

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    Date of creation: Jan 2000
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    Handle: RePEc:cep:stiecm:/2000/378

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    Web page: http://sticerd.lse.ac.uk/_new/publications/default.asp

    Related research

    Keywords: Contemporaneous aggregation; GARCH; conditionally heteroskedastic factor models; common and idiosyncratic risk; nonlinearity; memory;

    References

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    1. Diebold, Francis X & Nerlove, Marc, 1989. "The Dynamics of Exchange Rate Volatility: A Multivariate Latent Factor Arch Model," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 4(1), pages 1-21, Jan.-Mar..
    2. 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.
    3. Gary Chamberlain & Michael Rothschild, 1982. "Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets," NBER Working Papers 0996, National Bureau of Economic Research, Inc.
    4. Nijman, T. & Sentana, E., 1994. "Marginalization and Contemporaneous Aggregation in Multivariate Garch Proceses," Papers 9419, Centro de Estudios Monetarios Y Financieros-.
    5. Sentana, E., 1997. "The Relation Between Conditionally Heteroskedastic Factor Models amd Factor GARCH Models," Papers 9719, Centro de Estudios Monetarios Y Financieros-.
    6. Meddahi, N & Renault, E., 1996. "Aggregations and Marginalization of Garch and Stochastic Volatility Models," Papers 96.433, Toulouse - GREMAQ.
    7. Drost, F.C. & Nijman, T.E., 1990. "Temporal aggregation of GARCH processes," Discussion Paper 1990-66, Tilburg University, Center for Economic Research.
    8. Torben G. Andersen & Tim Bollerslev, 1996. "Heterogeneous Information Arrivals and Return Volatility Dynamics: Uncovering the Long-Run in High Frequency Returns," NBER Working Papers 5752, National Bureau of Economic Research, Inc.
    9. Robinson, P. M., 1991. "Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression," Journal of Econometrics, Elsevier, vol. 47(1), pages 67-84, January.
    10. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 74(1), pages 3-30, September.
    11. Peter M Robinson & Paolo Zaffaroni, 1997. "Modelling Nonlinearity and Long Memory in Time Series - (Now published in 'Nonlinear Dynamics and Time Series', C D Cutler and D T Kaplan (eds), Fields Institute Communications, 11 (1997), pp.61-170.)," STICERD - Econometrics Paper Series /1997/319, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    12. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
    13. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
    14. 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.
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    Cited by:
    1. Gómez-Déniz, E., 2004. "A note on mixture prior distributions with applications in actuarial statistic/Sobre las Distribuciones a Priori Mixtas con Aplicaciones en la Estadística Actuarial," Estudios de Economía Aplicada, Estudios de Economía Aplicada, vol. 22, pages 372 (15 pá, Agosto.
    2. Beran, Jan & Schützner, Martin & Ghosh, Sucharita, 2010. "From short to long memory: Aggregation and estimation," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2432-2442, November.
    3. Zaffaroni, Paolo, 2007. "Aggregation and memory of models of changing volatility," Journal of Econometrics, Elsevier, vol. 136(1), pages 237-249, January.
    4. Sentana, E., 2000. "Factor Representing Portfolios in Large Asset Markets," Papers 0001, Centro de Estudios Monetarios Y Financieros-.

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