Jarque-Bera Normality Test for the Driving Lévy Process of a Discretely Observed Univariate SDE

We study the validity of the Jarque-Bera test for a class of univariate parametric stochastic differential equations (SDE) dXt = b(Xt, α)dt+ dZt observed at discrete time points t n i = ihn, i = 1, 2, . . . , n, where Z is a nondegenerate Lévy process with finite moments, and nhn → ∞ and nhn → 0 as n → ∞. Under appropriate conditions it is shown that Jarque-Bera type statistics based on the Euler residuals can be used to test the normality of the unobserved Z, and moreover, that the proposed test is consistent against presence of any nontrivial jump component. Our result therefore provides a very easy and asymptotically distribution-free test procedure without any fine-tuning parameter. Some illustrative simulation results are given to reveal good performance of our test statistics. Running head: Normality Test for SDE. The AMS Mathematics Subject Classifications (2000): 62M02, 62F05.