In this thesis we consider a state space model for bivariate observations of count data. The approach used to solve the non analytical integrals that appears as the solution of the resulting non-Gaussian filter is a natural extension of the methodology advocated by Durbin and Koopman (DK). In our approach the aproximated Gaussian Model (AGM), has a diagonal Covariance matrix, while in the original DK, this is a full matrix. This modification make it possible to use univariate Kalman recursoes to construct the AGM, resulting in a computationally more efficient solution for the estimation of a Bivariate Poisson model. This also facilitates the use of exact initialization of those recursions. The state vector is specified using the structural approach, where the state elements are components which have direct interpretation, such as trend and seasonals. In our bivariate set up the dependence between the bivariate vector of time series is accomplished by use of common components which drive both series. We present both simulation and real life examples illustrating the use of our model.