從時點與量解析月變動率和年變動率反向變動的矛盾

The Timing andMagnitude Relationships BetweenMonth-to-Month Changes and Year-to-Year Changes That Make Comparing ThemDifficult

當一個經濟月指標出現數個月劇烈收縮後開始復甦時，發佈的年成長率和月成長率可能會出現相互矛盾的訊號－－年成長率呈現降低而月成長率則出現增加， 或反之亦然。這種現象係季節調整專家眾所周知的， 如參閱Shiskin （1957）。在本文中， 我們重新深入分析、闡明、並解釋出現這種潛在矛盾的現象。例如因為與上年同月的差分是當前水準值與之前11 個月差分的總和， 且全年的加總有一個5.5 個月的相平移， 年差分落後月差分5.5 個月。本文先對全年加總作增益及相函數的基本理論推導， 然後輔以實際資料解析。

When a monthly economic indicator series contracts sharply for a fewmonths and then starts to recover, the published annual and monthly growth rates can give conflicting signals: the annual growth rate can indicate a decrease and the monthly growth rate an increase or vice versa. This is well known to the seasonal adjustment community, see, for example, Shiskin (1957). In this paper, we revisit, illustrate and then explain this potential for conflict more analytically. For example, the annual differences lag the monthly differences by five and a half months because the same-month-year-ago difference is the sum of the current and eleven preceding monthly differences, and the annual sum has a phase shift of five and a half months. Illustrative examples are followed by an elementary formal mathematical derivation using the gain and phase functions of the annual sum.