ABSTRACT
This chapter discusses the autoregressive moving average model, (ARMA)(p, q), with zero. In order to use the Kalman filter to calculate the exact value of -2 In likelihood for given values of the ARMA coefficients, with or without missing observations, it is necessary to calculate the proper initial state covariance matrix. Kalman's recursive optimal estimation procedure for state space models in discrete time was generalized by Kalman and Bucy (1961) to continuous time state space models. By integrating a continuous time state space representation over time intervals, a discrete time state space representation is developed at unequally spaced observation times. Continuous time models beyond the AR( 1) model are significantly more complicated. Pandit and Wu (1983) discuss the sampling of continuous time processes at equally spaced intervals. If a continuous time ARMA(p,q) process with q < p is sampled at equally spaced time intervals, the result is a discrete time ARMA(p,p-1) process.
