The exact likelihood for a state space model with stochastic inputs ☆

In this work, we derive exact and approximate expressions for the conditional mean and variance of the initial state of a state space model, allowing for unit roots and stochastic inputs. These results provide adequate initial conditions to compute the exact likelihood using the Kalman filter. The exact conditional moments are the best choice when the stochastic structure of the inputs is known. If this is not the case, the approximate expressions are a good alternative, as some simulation results illustrate.