以模型為基礎降低殘差季節自我相關度的季節調整

An AlternativeModel-Based Seasonal Adjustment That Reduces Residual Seasonal Autocorrelation

當擴展Wiener-Kolmogorov （WK） 訊號抽出濾器到非平穩數序列及干擾時， 它對於高斯過程具有均方誤差（MSE） 最小的特性。然而， WK 訊號估計的隨機動態性質卻經常與目標過程大相逕庭。使用這廣為周知的濾器， 它可能會在季節調整後時間序列的譜函數中產生凹點。這些凹點符合落後12 期的負自我相關（或負季節自我相關）， 亦即存在全年隨機週期的現象。所謂的「平方根」WK 濾器是由Wecker （1979） 在分析平穩的訊號和干擾數列時提出的，它能確保訊號估計與原始的數列有相同的隨機動態， 亦即消除了譜凹點。這說明了一個不同的統計原理： 我們不只想要估計量能精準貼近目標值，我們同時也希望估計量的動態也能很貼近目標的動態。MSE 標準忽略訊號抽出在這方面的問題， 然而「動態匹配」濾器雖然會增加額外的MSE，它考慮並解決這個問題。本文對於季節調整後序列發生負季節自我相關這個現象提出一個實證研究，並提供符合理想訊號動態特徵且同時符合非平穩時間序列有限樣本下濾器的矩陣公式。我們將這些濾器應用到88個時間序列，大幅降低出現負季節自我相關的頻率。

The Wiener-Kolmogorov (WK) signal extraction filter, extended to handle nonstationary signal and noise, has minimum Mean Squared Error (MSE) for Gaussian processes. However, the stochastic dynamics of the signal estimate typically differ from that of the target. The use of such filters, although widespread, has been observed to produce dips in the spectrum of the seasonal adjustments of seasonal time series. These spectral troughs correspond in practice to negative autocorrelations at lag 12 (or negative seasonal autocorrelation), a phenomenon corresponding to an annual stochastic cycle. So-called “square root” WK filters were introduced by Wecker (1979) in the case of stationary signal and noise, to ensure that the signal estimate shared the same stochastic dynamics as the original signal, and thereby remove spectral dips. This represents a different statistical philosophy: not only do we want to closely estimate a target quantity, but we desire that the dynamics of our estimate closely resemble those of the target. The MSE criterion ignores this aspect of the signal extraction problem, whereas the “dynamic matching” filters account for this issue at the cost of accruing additional MSE. This paper provides empirical documentation of the occurrence of negative seasonal autocorrelation in seasonally adjusted data, and provides matrix formulas for filters that match the dynamics of the desired signal, and are appropriate for finite samples of nonstationary time series. We apply these filters to 88 time series to produce seasonal adjustments that have greatly reduced incidences of negative seasonal autocorrelation.