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I'm pretty new to the concepts of stationarity/cointegration. I am using the "urca" package in "Rstudio" to run my tests.

I have been trying to run cointegration tests, but the frustrating thing is that I haven't been able to find two series that are non-stationary, even when I try using examples cited by cointegration tutorials. My $p$-value is always too big such that I have to reject the null straight away. However, if I look at the $t$-values and compare them to the critical values, they seem to suggest otherwise.

Should I then ignore the $p$-value in the ADF test? Here are my test results. My two price series are XLE US Equity and CO1 Comdty (Brent 1st futures) from 01/01/2010 - today (5/11/2015).

Any help/elaboration will be very much appreciated, thank you!

> testXLE<-ur.df(XLE,type="drift",selectlags="AIC")
> summary(testXLE)

############################################### 
# Augmented Dickey-Fuller Test Unit Root Test # 
############################################### 

Test regression drift 


Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)

Residuals:
     Min       1Q   Median       3Q      Max 
-10.3948  -2.5809   0.6846   2.7908  10.1940 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  6.58864    3.43524   1.918   0.0596 .
z.lag.1     -0.08584    0.04533  -1.894   0.0628 .
z.diff.lag   0.05529    0.12544   0.441   0.6609  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.162 on 64 degrees of freedom
Multiple R-squared:  0.05337,   Adjusted R-squared:  0.02379 
F-statistic: 1.804 on 2 and 64 DF,  p-value: 0.1729


Value of test-statistic is: -1.8936 1.8395 

Critical values for test statistics: 
      1pct  5pct 10pct
tau2 -3.51 -2.89 -2.58
phi1  6.70  4.71  3.86

My interpretation of the results:

  • according to p-value (0.1729>0.05) do not reject null; series is stationary
  • t-value = (-1.8936>-2.89) --> do not reject null hypothesis; series is not stationary
  • t-value = (1.8395<4.71) --> do not reject a0=0 --> there is no drift

Conclusion: The series is non-stationary: Random Walk with no drift.

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  • $\begingroup$ In concluding "do not reject null hypothesis; series is not stationary" for the second test, your reading of the results as "not stationary" would hold even when the observed t-value was negative and near zero. Shouldn't the comparison be to the absolute values of the t-statistics? $\endgroup$
    – user78229
    Commented Nov 5, 2015 at 10:34
  • $\begingroup$ Hi @DJohnson, my understanding is that by the comparison of absolute values, if the |t-value|>|critical value|, you reject the null hypothesis (because the probability of wrongly rejecting the null is very low). So in this case, as |-1.8936|<|-2.89|, we do not reject null hypothesis. Is this not right? $\endgroup$
    – ElizaTYX
    Commented Nov 6, 2015 at 3:43
  • $\begingroup$ I assumed the reverse wrt critical vs observed t. $\endgroup$
    – user78229
    Commented Nov 6, 2015 at 11:11
  • $\begingroup$ Hi @DJohnson, I'm not sure if I am right, but I arrived at my conclusion taking reference to this explanation: stats.stackexchange.com/questions/24072/… $\endgroup$
    – ElizaTYX
    Commented Nov 9, 2015 at 4:03
  • $\begingroup$ @ElizaTYX, you are correct. In the ADF test, a test statistic to the left of the critical value yields rejection of $H_0$: "the process has a unit root". $\endgroup$
    – Richard Hardy
    Commented Nov 9, 2015 at 6:37

1 Answer 1

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The $p$-value in the output above corresponds to the $F$-statistic for the overall significance of the test regression. However, this is not what the ADF test is about. When interested in presence/absence of a unit root, you should look at the test statistic and compare it to the critical values of tau2. So the results are not conflicting; the two statistics are for answering two different questions.

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  • $\begingroup$ Hi @Richard, thanks for the response, but could you in layman terms explain then what it means when the p-value<0.05 in an ADF test? Normally it would mean that we are rejecting the null hypothesis, but in the scenario where the t-value indicates that the null hypothesis of a unit root existing is not rejected, what are we rejecting then? $\endgroup$
    – ElizaTYX
    Commented Nov 5, 2015 at 9:25
  • $\begingroup$ The null hypothesis associated with the $p$-value in the output above is that all regressors (except for the intercept) have zero coefficients in population. If $p$-value<0.05, then at least one of the regressors has a non-zero coefficient in population. $\endgroup$
    – Richard Hardy
    Commented Nov 5, 2015 at 9:35
  • $\begingroup$ By regressors, are we talking about the different lag terms here, since this is a unit root test and involves only one time series? Thanks again, @Richard. $\endgroup$
    – ElizaTYX
    Commented Nov 5, 2015 at 9:42
  • $\begingroup$ Yes. If you include trend in the ADF specification, it will also be one of the regressors, and I guess it should be included in the $F$-test, too (deterministic time trends are a little peculiar, so I am not completely sure). $\endgroup$
    – Richard Hardy
    Commented Nov 5, 2015 at 10:45
  • $\begingroup$ Hi @Richard, sorry to trouble again but could you verify if my interpretation of the ADF results above are correct? Another user raised that it is wrong (in the comments) and we're not too sure about it. $\endgroup$
    – ElizaTYX
    Commented Nov 9, 2015 at 5:22

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