Asymptotic Nonequivalence of GARCH Models and Di usions

This paper investigates the statistical relationship of the GARCH model and its di usion 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 de ciency distance. To our surprise, we are able to show that the GARCH model and its di usion limit are asymptotically equivalent only under deterministic volatility. With stochastic volatility, due to the di erence between the structure with respect to noise propagation in their conditional variances, their likelihood processes asymptotically behave quite di erently 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 nancial practice that applies statistical inferences derived under the GARCH model to its di usion limit.