Estimation error and the specification of unobserved component models

The paper deals with tile problem of identifying stochastic unobserved two-component models, as in seasonal adjustment or trend-cycle decompositions. Solutions based on the properties of the component estimation error are considered, and analytical expressions for the variances and covariances of the errors in the final, preliminary, and concurrent estimators are obtained for any admissible decomposition. These expressions are straightforwardly derived from the ARIMA model for the observed series, It is shown that, in all cases, the estimation error variance is minimized at a canonical decomposition (i.e., at a decomposition with one of the components noninvertible), and a procedure to determine that decomposition is presented. On occasion, however, the most precise final estimator is obtained at a canonical decomposition different from the one that yields the most, precise preliminary estimator.