Limit theory for high frequency sampled MCARMA models

Abstract

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {h_n, 2h_n, . . ., nh_n}, where h_n ↓ 0 and nh_n → ∞ as n → ∞, or at a constant time grid where h_n = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.