Limit experiments of GARCH

Boris Buchmann and Gernot Müller

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GARCH is one of the most prominent nonlinear time series models, both widely applied and thoroughly studied. Recently, it has been shown that the COGARCH model (which was introduced a few years ago by Klüppelberg, Lindner and Maller) and Nelson’s diffusion limit are the only functional continuous-time limits of GARCH in distribution. In contrast to Nelson’s diffusion limit, COGARCH reproduces most of the stylized facts of financial time series. Since it has been proven that Nelson’s diffusion is not asymptotically equivalent to GARCH in deficiency, in the present paper, we investigate the relation between GARCH and COGARCH in Le Cam’s framework of statistical equivalence. We show that GARCH converges generically to COGARCH, even in deficiency, provided that the volatility processes are observed. Hence, from a theoretical point of view, COGARCH can indeed be considered as a continuous-time equivalent to GARCH. Otherwise, when the observations are incomplete, GARCH still has a limiting experiment, which we call MCOGARCH, which is not equivalent, but nevertheless quite similar, to COGARCH. In the COGARCH model, the jump times can be more random than for the MCOGARCH, a fact practitioners may see as an advantage of COGARCH.

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Bernoulli, Volume 18, Number 1 (2012), 64-99.

First available in Project Euclid: 20 January 2012

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COGARCH Le Cam’s deficiency distance random thinning statistical equivalence time series


Buchmann, Boris; Müller, Gernot. Limit experiments of GARCH. Bernoulli 18 (2012), no. 1, 64--99. doi:10.3150/10-BEJ328.

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  • [1] Aït-Sahalia, Y., Mykland, P.A. and Zhang, L. (2011). Ultra high frequency volatility estimation with dependent microstructure noise. J. Econometrics 160 160–175.
  • [2] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
  • [3] Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Economy 81 637–659.
  • [4] Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedasticity. J. Econometrics 31 307–327.
  • [5] Brown, L.D. and Low, M. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398.
  • [6] Brown, L.D., Wang, Y. and Zhao, L.H. (2003). On the statistical equivalence at suitable frequencies of GARCH and stochastic volatility models with the corresponding diffusion model. Statist. Sinica 13 993–1013.
  • [7] Carter, A.V. (2007). Asymptotic approximation of nonparametric regression experiments with unknown variances. Ann. Statist. 35 1644–1673.
  • [8] Dalalyan, A.S. and Reiss, M. (2006). Asymptotic statistical equivalence for scalar ergodic diffusions. Probab. Theory Related Fields 134 248–282.
  • [9] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Berlin: Springer.
  • [10] Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the United Kingdom inflation. Econometrica 50 987–1007.
  • [11] Falk, M., Hüsler, J. and Reiss, R.D. (1994). Laws of Small Numbers: Extremes and Rare Events. Basel: Birkhäuser.
  • [12] Fasen, V., Klüppelberg, C. and Lindner, A. (2006). Extremal behavior of stochastic volatility models. In Stochastic Finance (A. Shiryaev, M.D.R. Grossinho, P. Oliviera, M. Esquivel, eds.) 107–155. New York: Springer.
  • [13] Grama, I.O. and Neumann, M.H. (2006). Asymptotic equivalence of nonparametric autoregression and nonparametric regression. Ann. Statist. 34 1701–1732.
  • [14] Hubalek, F. and Posedel, P. (2010). Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models. Quant. Finance. To appear.
  • [15] Jacod, J., Klüppelberg, C. and Müller, G. (2010). Testing for non-correlation or for a functional relationship between price and volatility jumps. Preprint, Univ. Paris VI and Technische Univ. München.
  • [16] Kallsen, J. and Vesenmayer, B. (2009). COGARCH as a continuous time limit of GARCH(1, 1). Stochastic Process. Appl. 119 74–98.
  • [17] Klüppelberg, C., Lindner, A. and Maller, R. (2004). A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour. J. Appl. Probab. 41 601–622.
  • [18] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. New York: Springer.
  • [19] Le Cam, L. and Yang, G.L. (1990). Asymptotics in Statistics. Some Basic Concepts. New York: Springer.
  • [20] Maller, R., Müller, G. and Szimayer, A. (2008). GARCH modelling in continuous time for irregularly spaced time series data. Bernoulli 14 519–542.
  • [21] Merton, R. (1973). The theory of rational option pricing. Bell J. Econom. Management Sci. 4 141–183.
  • [22] Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535–543.
  • [23] Nelson, D.B. (1990). ARCH models as diffusion approximations. J. Econometrics 45 7–38.
  • [24] Nussbaum, M. (1996). Asymtotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399–2430.
  • [25] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. New York: Springer.
  • [26] Reiss, R.-D. (1993). A Course on Point Processes. New York: Springer.
  • [27] Strasser, H. (1985). Mathematical Theory of Statistics. Berlin: de Gruyter.
  • [28] Wang, Y. (2002). Asymptotic nonequivalence of GARCH models and diffusions. Ann. Statist. 30 754–783.