についてDave Gilesがまとめている

One of the consequences of these standard seasonal adjustment procedures is that they introduce a moving average (MA) component into the data. Indeed, that's part of the point of using them - to smooth out the series and eliminate the repetitive seasonal component. However, there's a twist. The MA effect that's introduced is "non-invertible", and this has some pretty important implications.
A similar situation arises when we aggregate "flow" time series over time. That is, when we aggregate a series such as monthly imports into quarterly imports; or convert quarterly GDP into annual GDP. This is something that's been discussed previously in this blog. In that case there can also be serious implications when the data are used subsequently for modelling purposes.
Here, let's look at just one consequence of the non-invertible MA effect that's introduced when we seasonally adjust a time series - the effect that this has on testing for unit roots in the data.
There's actually quite an extensive literature dealing with the effect that seasonal adjustment has on standard tests for unit roots. A short, but very clear discussion of the early part of this literature is provided by Maddala and Kim (1998, pp. 364-365). One important result is that, in finite samples:
"the ADF and Philliups-Perron statistics for testing a unit root will be biased towards nonrejection of the unit root null if filtered data are used."
Here, the term "filtered" refers to "seasonally adjusted", using a filter such as that found in the Census method, and the studies that these authors are referring to include those of Ghysels (1990), Ghysels and Perron (1993), and Diebold (1993). In other words, these tests lack power when applied to seasonally adjusted data. The usual asymptotic properties of the ADF and Phillips-Perron (PP) tests are unaffected.
実際のところ、季節調整が標準的な単位根検定に与える影響を扱った研究は数多く存在する。初期の研究における簡潔だが極めて明快な議論は、Maddala and Kim(1998, pp. 364-365)において提供された。一つの重要な結果は、有限のサンプルにおいては:


ここで「フィルタリングされた」という用語は、センサス法などに見られるフィルタを用いて「季節調整された」ことを指しており、著者たちが言及している研究は、Ghysels (1990)Ghysels and Perron (1993)、およびDiebold (1993)などである。換言すれば、これらの検定は季節調整されたデータに適用すると検定力を失う。ただし、ADFとフィリップス=ペロン検定の通常の漸近的な特性は影響されない。

この後Gilesは、より最近の研究についても触れ、そのうちのdel Barrio Castro and Osborn (2014)(p.16)から以下の結論を引用している。

  • The invertibility assumption is not crucial in that the use of a sufficiently high order of augmentation does, indeed, deliver the usual ADF asymptotic distributions.
  • However, the order of augmentation required can be very large, due to both non-invertibility and the length of the two-sided filter used in adjustment.
  • Further, the high orders required to deliver good size lead to substantial power losses for adjusted data compared with direct testing on the unadjusted series.
  • The PP test requires consistent estimation of the long-run variance, with consistency requiring the kernel employed to take account of the long MA component arising from the use of the X-11 seasonal adjustment filter.


  • 十分に高次の拡張を用いれば、実際のところ、通常のADFの漸近的な分布が現れる、という点で、不可逆性という条件は致命的なものではない。
  • しかし、その際に要求される拡張の次数は極めて大きなものとなり得る。それは、不可逆性と、調整に用いられる両面フィルタの長さという、二つの要因による。
  • また、検定量を適切な大きさにするために必要とされる高次の次数は、未調整データを直接検定する場合に比べ、調整データの検定力を顕著に弱める。
  • フィリップス=ペロン検定は長期の分散の一致性を持つ推計値を必要とし、一致性はX-11季節調整フィルタを用いることにより生じる長期の移動平均項を織り込むためのカーネルを必要とする。


At the start of post I noted that sometimes only the seasonally adjusted data are available - the original time series isn't published. I think you now the answer to the question that followed: "Does this matter?"

That answer is "heck, yes!"

If all that we have are the seasonally adjusted data, and we then use these data to construct models based on what we conclude from standard unit root and cointegration tests, that's one thing. The whole exercise is based on the seasonally adjusted data.

However, what if we're limited to having only the seasonally adjusted data, but we want to make policy recommendations that are cast in terms of the actual data? If our ability to determine the stationarity of the data, and hence construct appropriate models, is marred by the seasonal adjustment process, we may then give policy advice that's based on false premises. (You were hoping to keep your job, right?)

So, the take-away message from this post is a simple one. Standard methods of seasonal adjustment can change the characteristics of your time series data in complex ways. In general, if you are testing for unit roots and cointegration then it's better to use the original, unadjusted, data if they're available. If you must use seasonally adjusted data for such testing, be careful!
しかし、季節調整済データしか無いという制約下において、実際のデータと関連付けられた政策提言をしたい場合はどうか? もしデータの定常性を決める我々の能力、延いては適切なモデルを構築する我々の能力が季節調整過程によって損なわれるならば、誤った前提に基づく政策提言を行うことになってしまうかもしれない(その結果仕事を続けられなくなるかも?)。