Goodness-of-Fit Test in Multivariate Jump Diffusion Models

In this article, we develop a new goodness-of-fit test for multivariate jump diffusion models. The test statistic is constructed by a contrast between an “in-sample” likelihood (or a likelihood of observed data) and an“out-of-sample” likelihood (or a likelihood of predicted data). We show that under the null hypothesis of a jump diffusion process being correctly specified, the proposed test statistic converges in probability to a constant that equals to the number of model parameters in the null model. We also establish the asymptotic normality for the proposed test statistic. To implement this method, we invoke a closed-form approximation to transition density functions, which results in a computationally efficient algorithm to evaluate the test. Using Monte Carlo simulation experiments, we illustrate that both exact and approximate versions of the proposed test perform satisfactorily. In addition, we demonstrate the proposed testing method in several popular stochastic volatility models for time series of weekly S&P 500 index during the period of January 1990 and December 2014, in which we invoke a linear affine relationship between latent stochastic volatility and the implied volatility index.