The Annals of Statistics

Asymptotic nonequivalence of GARCH models and diffusions

Yazhen Wang

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This paper investigates the statistical relationship of the GARCH model and its diffusion limit. Regarding the two types of models as two statistical experiments formed by discrete observations from the models, we study their asymptotic equivalence in terms of Le Cam's deficiency distance. To our surprise, we are able to show that the GARCH model and its diffusion limit are asymptotically equivalent only under deterministic volatility. With stochastic volatility, due to the difference between the structure with respect to noise propagation in their conditional variances, their likelihood processes asymptotically behave quite differently, and thus they are not asymptotically equivalent. This stochastic nonequivalence discredits a general belief that the two types of models are asymptotically equivalent in all respects and warns against the common financial practice that applies statistical inferences derived under the GARCH model to its diffusion limit.

Article information

Ann. Statist., Volume 30, Number 3 (2002), 754-783.

First available in Project Euclid: 6 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62B15: Theory of statistical experiments
Secondary: 90A09 90A20 90A16 62M99: None of the above, but in this section

Black-Scholes comparison of experiments conditional variance deficiency distance ARCH financial modeling likelihood process stochastic differential equation stochastic volatility


Wang, Yazhen. Asymptotic nonequivalence of GARCH models and diffusions. Ann. Statist. 30 (2002), no. 3, 754--783. doi:10.1214/aos/1028674841.

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