Tuesday, February 26, 2013

Non-linear Functions of Non-Stationary Data Can be Stationary

I was at a conference the other day, and Peter Phillips made the comment that if we take the Sine or Cosine of a non-stationary time-series, then the Dickey-Fuller test will suggest that the transformed series is stationary. More specifically, this happens if the sample size is large enough.

That got me thinking, and searching, and eventually I came across a paper by Chien-Ho Wang and Robert M. de Jong (see the reference below). Indeed, they establish precisely the result that Peter was referring to.

I also came across another result relating to a non-linear transformation of a non-stationary time-series:
If εt is Gaussian white noise, and  Yt is the random walk, Yt = (ε1 + ε2 + .... + εt), then the series Xt = sgn(εt)Yt2 is stationary. On the other hand, the series |Xt| = Yt2 is non-stationary. 
(See Gourieroux and Jasiak, 1999, p.1, footnote 1.)

To illustrate these results, I've undertaken a simple simulation experiment using EViews - the workfile is available here.

The original time-series is a random walk: Yt = Yt-1 + ε; where εt ~ N[0 , σ2] ; and Y0 = 0; for t = 1, ....., 1000. This is what the series looks like:

Then I've constructed St = sin(Yt); Ct = cos(Yt); and Xt = sgn(εt)Yt2. The following graphs show the resulting time-series:

When we apply the Augmented Dickey-Fuller (ADF) test (no drift, no trend) to Yt, St, Ct, and Xt, here are the results that we obtain, using different sample sizes (T):

Clearly, although Yt is generated as a unit root (and therefore non-stationary) process, the ADF results indicate a rejection of the non-stationary null hypothesis in favour of the alternative hypothesis of stationarity, for the trigonometrically transformed data, when T is sufficiently large; and for Xt for virtually any sample size.

In response to Pedro's comment below, here are the corresponding results when the ADF test is applied with a drift in the DF regression:

When we're working with non-linear time-series, lots of interesting things can happen, as we've seen here. Perhaps I'll write some more on this topic on another occasion.


Gourieroux, C. and  J. Jasiak, 1999. Nonlinear persistence and copersistence. (Download.)

Wang, C-H. and R. M. de Jong, 2008. Unit root tests when the data are a trigonometric transformation of an integrated process. (Download.)

© 2013, David E. Giles


  1. Since the sine and cosine are bounded functions between -1 and 1, isn't this result a bit expected?
    Despite the values (in this case a time series) we pass as argument, we will always get a number between -1 and 1 so the transformed series is of course "forced" to be stationary... am I missing anything?

    1. Michele - the issue being addressed here is the behaviour of the tests in this bounded case. For further information, see Cavaliere, "Limited Time Series With a Unit Root", Econometric Theory, 21, 907-945 (2005); and Cavaliere and Xu, "Testing for Unit Roots in Bounded Time Series (2011): http://www.ucy.ac.cy/data/economics/seminar-papers/2012/Cavaliere_Xu.pdf

      Dave Giles

    2. Very interesting topic Dave, thanks! Now, I think I got it.

      The issue here is that we can express a trigonometric series like a random walk so we would like to not reject the null of unit root even if the resulting time series looks stationary.

      A related question: the definition of stationarity implies that the joint distribution of the observations is the same over time or in the case of covariance stationary series we just requires some moments to be time invarying. What I expect, in the case of double-bounded time series is that the mean stays the same but what about other moments?


    3. Michele - not sure about your last question - I need to think about it!

  2. Does the result change if you do include an intercept? I don't know, but the ADF test does not control size when one does not include the drift...

    1. Pedro - no they don't change. I've added a second table into the post above so you can see.
      Dave Giles

  3. For the series Xt, clearly it is not stationary in second order moment, though it is in the first order moment (this is natural as the sign can move up or down with the same probability so that the series is essentially centered). Perhaps with an asymmetrically distributed shock we could have problems also for stationarity in mean.


Note: Only a member of this blog may post a comment.