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.
"Asymptotic nonequivalence of GARCH models and diffusions." Ann. Statist. 30 (3) 754 - 783, June 2002. https://doi.org/10.1214/aos/1028674841