Dominique Ladiray

A note on Musgrave asymmetrical trend-cycle filters

Abstract 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 or equal to those obtained using the least squares predictors. Finally, we derive the prior distributions under which the implied predictors are Bayes estimators.

Outline of the X-11 Method

The X-11 method is based on an iterative principle of estimation of the different components, this estimation being done at each step using appropriate moving averages. The method is designed for the decomposition and seasonal adjustment of monthly and quarterly series.

Modelling of the Easter Effect

X-11-ARIMA and X-12-ARIMA propose different models for correcting the Easter effect based on an estimate of the irregular component. The proposed models and the methods used are sometimes quite different in the two softwares, which is why we have not integrated them directly into the seasonal adjustment example of Chapter 4.

The Various Tables

This chapter presents a complete and detailed example of seasonal adjustment with the X-11 method. The series that is used in this example is a monthly series; it is in such cases that the softwares’ options are most numerous and complex. The series studied X t is the monthly index of industrial production in France between October 1985 and March 19951. The series is represented in the top panel of Figure 4.1, which gives a decomposition plot. The seasonal factors S t (Table D10) are graphed in the third panel, and, in the case at hand, the trading-day factors D t (Table C18) are provided in the fourth panel. These two sets of factors are used to compute the seasonally adjusted series A t (Table D11), which is shown with the original series in the top panel, and with the trend-cycle C t (Table D12) in the second panel. Finally, the irregular component I t (Table D13) is graphed in the bottom panel. It is obtained by removing the trend-cycle from the seasonally adjusted series.

Brief History of Seasonal Adjustment

It is common today to decompose an observed time series X t into several components, themselves unobserved, according to a model such as:

Nurse workload and inexperienced medical staff members are associated with seasonal peaks in severe adverse events in the adult medical intensive care unit: A seven-year prospective study

X_t = T_t + C_t + S_t + I_t ,

Business Cycle Extraction of Euro-Zone GDP: Direct versus Indirect Approach

where T t ,C t ,S t and I t designate, respectively, the trend, the cycle, the seasonality and the irregular components. This is an old idea, and it is doubtless to astronomy that one should turn to find its origin1.