I develop an omnibus specification test for diffusion models based on the infinitesimal operator instead of the already extensively used transition density. The infinitesimal operatorbased identification of the diffusion process is equivalent to a "martingale hypothesis" for the new processes transformed from the original diffusion process. The transformation is via the celebrated "martingale problems". My test procedure is to check the "martingale hypothesis" via a multivariate generalized spectral derivative based approach which enjoys many good properties. The infinitesimal operator of the diffusion process enjoys the nice property of being a closed-form expression of drift and diffusion terms. This makes my test procedure capable of checking both univariate and multivariate diffusion models and particularly powerful and convenient for the multivariate case. In contrast checking the multivariate diffusion models is very difficult by transition density-based methods because transition density does not have a closed-form in general. Moreover, different transformed martingale processes contain different separate information about the drift and diffusion terms and their interactions. This motivates us to suggest a separate inference-based test procedure to explore the sources when rejection of a parametric form happens. Finally, simulation studies are presented and possible future researches using the infinitesimal operator-based martingale characterization are discussed
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