Friday, September 12, 2014

Unit Root Tests and Seasonally Adjusted Data

We all know why it's common to "seasonally adjust" economic time series data that are recorded on a monthly, quarterly, etc. basis. Students are sometimes surprised to learn that in some countries certain such time series are reported only in seasonally adjusted terms. You can't get the original (unadjusted data). This applies to some U.S. economic data, for example.

Does this matter?

Statistical agencies across the world typically use one of two statistical procedures to seasonally adjust their data. These are the TRAMO/SEATS method, or some version of the Census X method - e.g., Census X-11, Census X-12, etc. If you're an EViews user, then you have access to both of the Census and TRAMO/SEATS procedures, including the X-13ARIMA-SEATS procedure which is the latest adjustment method being used by the U.S. Census Bureau, Statistics Canada, and the Bank of England.

If you're not too sure what's going on "behind the scenes" when these seasonal adjustment methods are used, take a look at this earlier post. You can also find a good description here.

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.

Other, related, studies from this period include those of Franses (1991), and Olekalns (1994). The latter author shows that seasonal adjustment using frequency domain band-pass filters or by the use of dummy variables, also impacts adversely on the finite-sample power of the ADF test.

More recent contributions to this topic include those of del Barrio Castro and Osborn (2004, 2006, 2014); del Barrio Castro et al. (2013a, b); and Bell (2011). The first group of these papers focuses specifically on the issues associated with the non-invertible moving average process that is induced in the data when standard seasonal adjustment procedures are used, and the ADF or PP tests are used.

Some important results that emerge include:

  • "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."

  • (del Barrio Castro and Osborn, 2014, p.16.)

    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!

    Postscript: A word of warning. Seasonal adjustment is intended to remove any "deterministic seasonality" from the original time series. There is another, quite different type of seasonality that may be of concern - "stochastic seasonality". The latter refers to the possibility that the data may be non-stationary, with unit roots at the "seasonal frequencies", rather just at the usual "zero frequency" (which is what the ADF and similar tests are designed to deal with).

    In this post I've completely ignored the possibility of such stochastic seasonality, but it's something that's worth taking up in a subsequent post.


    Bell, W. R., 2011. Unit root properties of seasonal adjustment and related filters. Research Report Series (Statistics #2010-08), revised, Research and Methodology Directorate, U.S. Census Bureau.

    del Barrio Castro, T. and D. R. Osborn, 2004. The consequences of seasonal adjustment for periodic autoregressive processes. Econometrics Journal, 7, 307-321.

    del Barrio Castro, T. and D. R. Osborn, 2006. A random walk through seasonal adjustment: Noninvertible moving averages and unit root tests. Mimeo., Economics, School of Social Sciences, University of Manchester.

    del Barrio Castro, T. and D. R. Osborn, 2014. The distribution of unit root test statistics after seasonal adjustment. Mimeo., Economics, School of Social Sciences, University of Manchester.

    del Barrio Castro, T., P. M. M. Rodrigues, and A. M. R. Taylor, 2013a. The impact of persistent cycles on zero frequency unit root tests, Econometric Theory, 29, 1289-1313.

    del Barrio Castro, T., P. M. M. Rodrigues, and A. M. R. Taylor, 2013b. On the behaviour of Phillips-Perron tests in the presence of persistent cycles. CEFAGE-UE Working Paper 2013_11.

    Diebold, F. X., 1993. Discussion: Effect of seasonal adjustment filters on tests for a unit root. Journal of Econometrics, 55, 99-103.

    Franses, P. H., 1991. Moving average filters and unit roots. Economics Letters, 37, 399-403.

    Ghysels, E., 1990. Unit root tests and the statistical pitfalls of seasonal adjustment: The case of U.S. postwar real gross national product. Journal of Business and Economic Statistics, 8, 145-152.

    Ghysels, E. and P. Perron, 1993. The effect of seasonal adjustment filters on tests for a unit root. Journal of Econometrics, 55, 57-98.

    Maddala, G. S. and I-M. Kim, 1998. Unit Roots, Cointegration, and Structural Change. Cambridge University Press, Cambridge.

    Olekalns, N., 1994. Testing for unit roots in seasonally adjusted data. Economics Letters, 45, 273-279.

    © 2014, David E. Giles


    1. Dear prof.

      I understand that x-12 influence the outcome of ADF, but how about TRAMO/SEATS? Will TRAMO/SEATS influence ADF or ACF/PACF?
      In other words, should the differencing be taken before TRAMO/SEATS?

      Thank you

      1. Yes, TRAMO/SEATS will also impact on unit root tests.


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