Continuous Time ARMA Processes : Discrete Time Representation and Likelihood Evaluation

This paper explores the representation and estimation of mixed continuous time ARMA (autoregressive moving average) systems of orders $p$, $q$. Taking the general case of mixed stock and flow variables, we discuss new state space and exact discrete time representations and demonstrate that the discrete time ARMA representations widely used in empirical work, based on differencing stock variables, are members of a class of observationally equivalent discrete time ARMA($p+1$,\,$p$) representations, which includes a more natural ARMA($p$,\,$p$) representation. We compare and contrast two approaches to likelihood evaluation and computation, namely one based on an exact discrete time representation and another utilising a state space representation and the Kalman-Bucy filter. We demonstrate the value of our approach in two applications: a univariate study of the yield curve at different frequencies; and, a multivariate study of the relationship between US GDP and oil prices, taking account of the mixed frequencies with which these data are available.