The Wiener-Kolmogorov (WK) signal extraction filter, extended to handle nonstationary signal and noise, has minimum Mean Square Error (MSE) among filters that preserve the signal's initial values; however, the stochastic dynamics of the signal estimate typically differ substantially from that of the target. The use of such filters, although widespread, is observed to produce dips in the spectrum of the seasonal adjustments of seasonal time series. These spectral troughs tend to correspond to negative autocorrelations at lags 12 and 24 in practice, a phenomenon that will be called "negative seasonality." So-called "square root" WK filters were introduced by Wecker in the case of stationary signal and noise, to ensure that the signal estimate shared the same stochastic dynamics as the original signal, and thus remove the problem of spectral dips. This represents a different statistical philosophy: not only do we want to closely estimate a target quantity, but we desire that the internal properties and dynamics of our estimate closely resemble those of the target. The MSE criterion ignores this aspect of the signal extraction problem, whereas the square root WK filters account for this issue at the cost of accruing additional MSE. This paper provides empirical documentation of negative seasonality, and provides matrix formulas for square root WK filters that are appropriate for finite samples of nonstationary time series. We apply these filters to produce seasonal adjustments without inappropriate spectral troughs.
KEY WORDS: ARIMA, Seasonality, Signal Extraction, Wiener-Kolmogorov
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