Multiperiod Predictions from Stochastic Difference Equations by Bayesian Methods

initial conditions, and given a prior density (possibly diffuse) of its parameters, this paper
obtains the predictor of the time series k periods into the future with minimum mean
squared error. Completely analytical solution is given for predictions from the first-order
univariate system, and, in the general higher-order multivariate case, for k up to 5.
1. INTRODUCTION
EXISTINGMETHODS of parameter estimation, be they versions of least squares,
maximum likelihood, or Bayes, that have been applied to systems of dynamic
econometric equations to produce forecasts were not designed for the purpose of
forecasting. In this paper, it is argued that these estimation methods may be
inadequate if the resulting estimates are to be used to make ex ante predictions
for more than one period ahead, and if the accuracy of the predictions is measured,
as it usually is, by the mean squared errors. A formulation of the multiperiod
prediction problem is presented. It will then become clear that the same set of
parameter estimates cannot be optimal in making predictions for different time
periods into the future, when optimality is defined by minimum mean squared
errors in small samples.