Model verification for Lévy-driven Ornstein–Uhlenbeck processes with estimated parameters

When an Ornstein–Uhlenbeck (or CAR(1)) process is observed at discrete times 00, hh, 2h2h, ……[T/h]h[T/h]h, the unobserved driving process can be approximated from the observed process. Approximated increments of the driving process are used to test the assumption that the process is Lévy-driven. Asymptotic behavior of the test statistic at high sampling frequencies is developed in Abdelrazeq et al. (2014) assuming that the model parameters a,σa,σ are known. Here we explore the performance of the test statistic when the model coefficient aa is unknown and must be estimated. The parameter σσ can be assumed to be one. We will show the consistency and asymptotic normality of our proposed estimator and then demonstrate its effect on the asymptotic behavior of the test statistic. Performance of the proposed test with estimated aa is illustrated through simulation.