I have a data set for a variable, for which I have run some unit-root tests:

  • ADF (constant/without trend): t-stat=-1.0816, p-val=0.7218 - DNR
  • ADF (constant & trend): t-stat=-4.5203, p-val=0.0021 - REJECT @5% level
  • PP (constant/without trend): t-stat=-1.3507, p-val=0.6044 - DNR
  • PP (constant & trend): t-stat=-3.6030, p-val=0.0334 - REJECT @5% level

(Note I have also run all of the above tests with the first-differenced values, and they ALL reject the null of the presence of a unit root.)

As I understand, I fail-to-reject the null on both the ADF & PP tests with ONLY a constant (no trend), but statistically reject the null for both tests when a trend is included.

What should I conclude about the data series - is this indicative of any statistical property I may be ignoring/oblivious to? What does this indicate as to the presence of a unit-root in the series given the contradictory results of the tests?

EDIT: Just have run into the opposite problem on another data series: ADF & PP (no trend) Rejects the null but ADF & PP WITH trend Does Not Reject the null.

What would this case of the problem mean?

I feel like I'm missing something - the Alternative Hypotheses?


  • $\begingroup$ You should consider if the series under study "needs" a trend term in the test regression, i.e., whether, under the alternative, stationarity around a linear time trend or stationarity around some mean seems more appealing. $\endgroup$ Commented Feb 25, 2019 at 6:39
  • $\begingroup$ @ChristophHanck I'm a bit ignorant regarding the alternatives, how would I go about doing what you suggest? $\endgroup$ Commented Feb 25, 2019 at 6:49
  • $\begingroup$ I hope that this answer might help: stats.stackexchange.com/questions/173040/… $\endgroup$ Commented Feb 28, 2019 at 7:18

2 Answers 2


Unit-root tests should always return the "non-stationarity result" in series with trend. Indeed, the possibility of simplification from ARIMA(p,d,q) to ARMA(p,q) is one of the main reasons that lead to performing the test in the first place

  • 2
    $\begingroup$ would you be able to clarify/elaborate? I'm unsure of what you mean with regards to the question above. $\endgroup$ Commented Feb 25, 2019 at 8:51

We can compare two series to show that it is necessary to include a trend in your ADF test, and that the ADF test with a trend is thus more reliable.

Take the first (unit root) series. This series is trending because of the drift term, $\phi_0$:

$$y_t = \phi_0 + y_{t-1} + \epsilon_t$$

And then take a second series which is trending, but it does not have a unit root:

$$y_t = ct + \epsilon_t$$

We therefore include a trend in the ADF test to distinguish between those two.


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