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I am required to perform unit root testing on a given time series. The output obtained in Stata is somewhat confusing me. To the best of my knowledge I am obtaining two conflicting results, Stata indicating that the time series fits a unit root process while also seemingly saying that the coefficient is significantly different from zero, hence contradicting the idea that the time series indeed follows a unit root process. This is the output obtained in Stata:

. dfuller series3, noconstant regress lags(0)

Dickey-Fuller test for unit root                   Number of obs   =        48

                           ---------- Interpolated Dickey-Fuller ---------
              Test         1% Critical       5% Critical      10% Critical
            Statistic         Value              Value            Value

------------------------------------------------------------------------------
 Z(t)              8.943            -2.623            -1.950            -1.609
------------------------------------------------------------------------------
 D.series3 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
 series3 |
     L1. |   .1314633   .0147005     8.94   0.000     .1018897    .1610369
-----------------------------------------------------------------------------

Clearly the test statistic is indicating that the null hypothesis is accepted and hence the time series fits a unit root process. However, the p-value obtained for L1 is 0.000 indicating that it is significantly different from zero (0.1314633 cannot be treated as being 0). I read somewhere that this might be related to the Augmented Dickey Fuller test itself; however, I cannot seem to find this link again. I also ran the Augmented Dickey Fuller test using lags as well as the Philips-Perron test and obtained similar results.

Would anyone have any idea why the ADF is giving such results and what might be the cause of this anomaly?

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You'r test indicates that the series does follow a unit root. Note that the DF-test is one sided so you should not think in absolute values. You'r test statistic is 8.943 which clearly is higher than -1.950 so you cannot reject the null of a unit root in the series. That being said I think that you will have very low power because you only have 48 observations. When you have so few observations it will look like the series does not mean revert when in fact, with more observations, it will be mean reverting. Second I would try to augment the ADF regression with a trend and more lags. Autocorrelation in the error term will invalidate the DF test! Hence use the ADF test with more lags to account for the autocorrelation.

1) Look at the ACF to determine approximately how many lags you need to include. 2) Remove insignificant lags 3) When all included lags are significant look at the t-value of the variable of interest. If its lower than -1.950 then you do not have a unit root, if its higher then you do have a unit root (which you in fact do have). As mentioned, you'r problem is with the low observations and the lack of lags and probably also a time trend in the regression.

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  • $\begingroup$ That being said I would rather use the more efficient DF-GLS test which is available in Stata! $\endgroup$ – Plissken Jul 11 '14 at 13:21
  • $\begingroup$ +1, nice answer. The DF-GLS command has also the nice property that it automatically selects the lag length via the Schwert criterion. $\endgroup$ – Andy Jul 11 '14 at 14:30
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Clearly the test statistic is indicating that the null hypothesis is accepted and hence the time series fits a unit root process.

How is that clear to you? To me it's clear that the test statistics is beyond 1% critical value, i.e. $H_0$ has to be rejected, see Stata's dfuller command's help with examples

UPDATE: thanks @whuber, fixed typo in link

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  • $\begingroup$ the test statistics is 8.943 which is well above either of the three critical test values, indicating that the null hypothesis should be accepted...the null hypothesis is rejected given that the test statistic is smaller than the critical values $\endgroup$ – Zborvo Apr 14 '14 at 17:42
  • $\begingroup$ @Zborvo, "the null hypothesis is rejected given that the test statistic is smaller than the critical values" - where did you get this from? are you sure you mean critical value and not p-value? $\endgroup$ – Aksakal Apr 14 '14 at 17:51
  • $\begingroup$ Your link goes right back to this question itself rather than to the Stata docs--something probably went wrong when you tried to paste the link. Perhaps you wanted to direct readers to stata.com/manuals13/tsdfuller.pdf? Note that this is a one-sided test: perhaps you might want to change your assessment of the output in light of that? $\endgroup$ – whuber♦ Apr 14 '14 at 17:52
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    $\begingroup$ @Aksakal given that the test statistics gave a value of 8.943 i would expect the corresponding p-value to be greater than 0.05, and not 0.000 as obtained given that it is larger than all 3 critical values. Had the test statistics been smaller than -2.623 i would have expected the p-value to be less than 0.01, but given it is larger than the 0.1 critical value, the p-value for the test statistic should be larger than 0.1 when it is not. I hope that my concern is coming across clearly! $\endgroup$ – Zborvo Apr 14 '14 at 18:00
  • $\begingroup$ @Zborvo, "i would expect the corresponding p-value to be greater than 0.05, and not 0.000 as obtained given that it is larger than all 3 critical values." That's what I don't understand: why do you expect p-value be greater than 0.05 when the test statistic is greater than the critical value? Where did you get this intuition from? The high test statistics indicate rejection of $H_0$, which in turn mean its very low probability, i.e. p-value. $\endgroup$ – Aksakal Apr 14 '14 at 20:25
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It would seem that a possible result for this would be the inadequacy of either autoregressive or unit root processes to model the given data. This is only a conclusion I have come to based on consultation with my lecturer as well as comparing the data's behaviour under further analysis. Any comments or criticism is more than welcome.

Indeed, for the given data, when using the Expert Modeller in SPSS Holt's Exponential Smoothing was chosen as the model which fit the given data best with a normalized BIC of 6.857. When fitting an ARIMA model, ARIMA(2,2,0) was the best fitting model and gave a BIC of 6.856. Both ARIMA(2,2,0) and the Holt's Exponential Smoothing gave a similar BIC and had fitted values which were very close to the observed data values.

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