A generalized ´adaptive expectations´ formula in auto-regressive models

In the model of this paper the state of the system follows a linear auto-regressive process and is observed with noise. The decision-maker´s problem is to estimate the current state: with a payoff function quadratic in the decision variables the optimal estimator is the conditional mean, given the observations. With the use of the Kalman filter results of interest to economists are obtained. The basic result is that the optimal estimate is a convex linear combination of the current observation and the previous optimal estimate. This is a generalization of the ´adaptive expectations´ formula widely used in economics.