A nonparametric method for asymmetrically extending signal extraction filters

Two important problems in the X-11 seasonal adjustment methodology are the construction of standard errors and the handling of the boundaries. We adapt the ‘implied model approach’ of Kaiser and Maravall to achieve both objectives in a nonparametric fashion. The frequency response function of an X-11 linear filter is used, together with the periodogram of the differenced data, to define spectral density estimates for signal and noise. These spectra are then used to define a matrix smoother, which in turn generates an estimate of the signal that is linear in the data. Estimates of the signal are provided at all time points in the sample, and the associated time-varying signal extraction mean squared errors are a by-product of the matrix smoother theory. After explaining our method, it is applied to popular nonparametric filters such as the Hodrick–Prescott (HP), the Henderson trend, and ideal low-pass and band-pass filters, as well as X-11 seasonal adjustment, trend, and irregular filters. Finally, we illustrate the method on several time series and provide comparisons with X-12-ARIMA seasonal adjustments.