The Kalman filter formula, given by the linear recursive algorithm, is usually used for estimation of the time-varying parameter model. The filtering formula, introduced by Kalman (I960) and Kalman and Bucy (1961), requires the initial state variable. The obtained state estimates are influenced by the initial value when the initial variance is not too large. To avoid the choice of the initial state variable, in this paper we utilize the diffuse prior for the initial density. Moreover, using the Gibbs sampler, random draws of the state variables given all the data are generated, which implies that random draws are generated from the fixed-interval smoothing densities. Using the EM algorithm, the unknown parameters included in the system are estimated. As an example, we estimate a traditional consumption function for both the U.S. and Japan.

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