Forecasting Markov-switching dynamic, conditionally heteroscedastic processes

Recursive formulae are derived for the multi-step point forecasts and forecast standard errors of Markov switching models with ARMA(1; q) dynamics (including the fractionally integrated case) and conditional heteroscedasticity in ARCH(1) form. Hamilton’s dynamic models of switching mean and variance are also treated, in a slightly modi…ed version of the analysis. 1 Introduction Computing multi-step forecasts for nonlinear dynamic models is inherently di¢ cult because there is no natural way to compute the conditional expectation of the future path of the process. In general, substituting the expected values of future shocks into the model equation will not achieve this. The only solution to the problem with a general application appears to be Monte Carlo simulation. It has been pointed out by a number of authors (e.g. Clements and Krolzig 1997, Krolzig 2002, Blix 1999) that although Markov-switching models are nonlinear, they have features that permit an analytical solution of the forecasts, at least in simple cases. However, these applications have dealt in practice with relatively simple models, …nite order VARs with with just one switching component, such as the mean of the process. This note considers the general problem of multi-step forecasting with a Markov-switching dynamic regression model, possibly featuring conditional heteroscedasticity. We shall be interested in the point forecasts, but the calculation of forecast standard errors is of equal importance and proves to be the more computationally challenging problem. For simplicity a univariate model is treated. Thus, the framework envisaged can be written in the form yt = st + 1 X