Procedures of Parameters' estimation of AR(1) models into lineal state-space models

The objective of this paper is to study how algorithms of optimization affect the parameters-estimation of Autoregressive AR(1)Models. In our research we have represented the AR(1) models in linear state space form and applied the Kalman Filters to estimate the different unknown parameters of the model. Many methods have been proposed by researchers for the estimation of the parameters in the case of the linear state space models. In our work we have emphasized on the estimation through the Maximum Likelihood (ML). Statisticians have used many algorithms to optimise the likelihood function and they have proposed many filters; publishing their results in many papers. In spite of the fact that this field is so extended, we have emphasized our study in the financial field. Two quasi-Newton algorithms: Berndt, Hall, Hall, and Hausman (BHHH) and Broyden-Fletcher-Goldfarb-Shanno (BFGS), and the Expectation-Maximization (EM) algorithm have been chosen for this study. A practical study of these algorithms applied to the maximization of likelihood by means of the Kalman Filter have been done. The results are focused on efficiency in time of computation and precision of the unknown parameters estimation. A simulation study has been carried out, using as true values the parameters of this model published in the literature, in order to test the efficiency and precision of our implemented algorithms.