# Conflicting results from $p$-value and $t$-value: Should I ignore the $p$-value in the ADF test?

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.

• 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? – Mike Hunter Nov 5 '15 at 10:34
• 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? – ElizaTYX Nov 6 '15 at 3:43
• I assumed the reverse wrt critical vs observed t. – Mike Hunter Nov 6 '15 at 11:11
• 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/… – ElizaTYX Nov 9 '15 at 4:03
• @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". – Richard Hardy Nov 9 '15 at 6:37

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.
• 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. – Richard Hardy Nov 5 '15 at 9:35
• 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). – Richard Hardy Nov 5 '15 at 10:45