Maximum Likelihood Estimation of State-Space Models

The use of the Kalman filter for estimation purposes is not always an easy task despite the obvious advantages in many situations of the state-space representation. This is in part due to the fact that the computation of the corresponding score (gradient of the log-likelihood) is sometimes complicated and numerically challenging. We present an algorithm for the exact construction of the score that is written in the form of explicit and ready to be used Markov type matrix recursions that are very efficient from the memory management point of view. As an example, this general result is applied to the estimation of ARMA models in state-space representation