A note on Musgrave asymmetrical trend-cycle filters

In this paper, we derive the implicit forecasts in the asymmetrical trend-cycle averages used in the X-11 seasonal adjustment method. We give an algorithm to calculate them, and we study their statistical properties. We express the forecasts as Stein estimators. We derive expressions for their bias, variance, covariances and prediction mean squared errors. We show that the prediction mean squared errors of the implied predictors are always smaller than or equal to those obtained using the least squares predictors. Finally, we derive the prior distributions under which the implied predictors are Bayes estimators.