constancy is responsible for the higher mean-squared state extraction error associated with the Hamilton model.
This paper has been largely methodological, and numerous additional methodological issues are currently under investigation, including formal asymptotic distribution theory, elimination of the linear approximation employed in solving the first-order conditions, model specification tests, and analytic determination of ergodic probabilities. We shall not dwell on those issues here; instead, we shall briefly discuss two potentially fruitful areas of application.
The first concerns exchange rate dynamics Engel and Hamilton ( 1990) have suggested that exchange rates may follow a switching process. We agree. But certainly, it is highly restrictive to require constancy of the transition probabilities. Rather, they should be allowed to vary with fundamentals, such as relative money supplies, relative real outputs, interest rate differentials, and so forth. Moreover, Mark ( 1992) produces useful indexes of fundamentals, which may be exploited to maintain parsimony. We shall provide a detailed report on this approach in a future paper.
The second concerns aggregate output dynamics. Diebold Rudebusch and Sichel ( 1993) have found strong duration dependence in postwar US contractions. That is, the longer a contraction persists, the more likely it is to end. That suggests allowing the transition probabilities in a Markov switching model of aggregate output dynamics to depend on length-to-date of the current regime, which can readily be achieved by expanding the state space of the process.12
The general form of the maximum expected complete-data likelihood estimators for the 2k transition probability function parameters,and , is given in Section 3.2. Here we include the explicit expressions for the cases of k = 2 and k = 3, which are of particular interest in applied work. Due to space limitations, it is understood that in the expressions that follow all smoothed probabilities are conditional on y + ̱T and x + ̱T given θ(j-1), and that transition probabilities , and their derivatives are evaluated at and , respectively.____________________