Estimation error and the specification of unobserved component models

The paper deals with the problem of identifying stochastic unobserved two-component models, as in seasonal adjustment or trend-cycle decompositions. Solutions based on the properties of the unobserved component estimation error are considered, and analytical expressions for the variance of the errors in the final, preliminary, and concurrent estimators are obtained. These expressions are straightforwardly derived from the ARIMA model for the observed series. timation error variance is always minimized at a canonical decomposition (i.e., at a decomposition with one of the components noninvertible), and a simple procedure to determine that decomposition is presented. On occasion, however, the most precise final estimator may be obtained at a canonical decomposition different from the one that yields the most precise preliminary estimator. Two examples are presented. First, a simple ‘trend plus cycle’-type model is used to illustrate the derivations. The second example presents results for a class of models often encountered in actual time series.