Jarque-Bera normality test for the driving Lévy process of a discretely observed univariate SDE

In this paper, we study the Jarque-Bera test for a class of univariate parametric stochastic differential equations (SDE) dX_t = b(X_t, α)dt + dZ_t, constructed based on the sample observed at discrete time points t_iⁿ = ih_n, i = 1, 2,..., n, where Z is a nondegenerate Lévy process with finite moments and h is a sequence of positive real numbers with nh_n → ∞ and nh_n² → 0 as n → ∞. It is shown that under proper conditions, the Jarque-Bera test statistic based on the Euler residuals can be used to test for the normality of the unobserved Z and the proposed test is consistent against the presence of any nontrivial jump components. Our result indicates that the Jarque-Bera test is easy to implement and asymptotically distribution-free with no fine-tuning parameters. Simulation results to validate the test are given for illustration.