Journal of Official Statistics, Vol.12, No.4, 1996. pp. 329347

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Derivation and Properties of the X11ARIMA and Census X11 Linear Filters

This study extends previous analyses by including all the possible combinations of trend-cycle and seasonal filters available in the X11ARIMA with and without extrapolations (in this latter case, they are equivalent to those of X11). The corresponding “cascade” filters for historical and current observations are discussed, including the combined filters for the residuals. It is shown that the variance amplification and phase shift of the standard concurrent filters fall between those corresponding to the combination of longer and shorter trend-cycle and seasonal moving averages. The variance amplification of the shorter filter is the largest whereas the corresponding phase shifts are near zero at low frequencies. The contrary is observed for the longer concurrent filter. The use of ARIMA extrapolations produces gain functions closer to those of the symmetric filters if the ARIMA parameter values are consistent with the filters' implicit assumptions of trend-cycle and seasonality. The presence of inconsistency highly affects the linear properties of the trend-cycle filters.

Seasonality; trend-cycle; irregular variations; symmetric cascade linear filters; asymmetric cascade linear filters.

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