I have been working on and reading about the ur.df test in R, so I know what the tests mean, what the hypotheses are and how to read the critical values. (For instance I read this nice piece.) However, it is still not completely clear to me how to interpret all the tests. Especially for my specific case, in the link I put, it shows in all cases "fail to reject", when the interpretation is more clear, I think.
First case
First of all, when I execute the ur.df test with trend, I get the following:
Value of test-statistic is: -4.0881 5.7431 8.6016
Critical values for test statistics:
1pct 5pct 10pct
tau3 -4.38 -3.60 -3.24
phi2 8.21 5.68 4.67
phi3 10.61 7.24 5.91
Meaning that we reject all hypotheses at a 5% significance level. Now I am wondering how to interpret this.
From tau3, we know that there is no unit root, since we reject $H_0: \gamma=0$.
From phi3 we reject $\gamma=trend=0$, we already have by tau3 that $\gamma\neq0$. But now can we say that there is trend or not? Because rejecting $\gamma=trend=0$ means that either $\gamma \neq 0$ and/or $trend \neq 0$, therefore we can not directly say that there is trend or not right?
From phi2 we reject $\gamma=trend=drift=0$, but then the same question arises?
Hence, I just want to know if it is correct that in this case we cannot really say something on the trend or drift?
Second case The output is as follows:
Value of test-statistic is: -3.5593 6.1884 6.3397
Critical values for test statistics:
1pct 5pct 10pct
tau3 -4.38 -3.60 -3.24
phi2 8.21 5.68 4.67
phi3 10.61 7.24 5.91
Thus from tau3: we do not reject $\gamma=0$, therefore there is a unit root (non stationary).
From phi3: we do not reject $\gamma=trend=0$, therefore no unit root and no trend.
But then from phi2: we reject $\gamma=trend=drift=0$. Does this mean that there is only drift, as we concluded from tau3 and phi3 that there is no unit root and no trend? Or is it not correct that we take the previous results from tau3 and phi3 into account when interpreting the hypothesis from phi2?
Finally, is it better to look at a plot of the object you want to test on, whether to use the none/trend/drift specification? Or is the "correct" specification more determined by the outcomes of all ur.df outputs? For instance, if the object already shows a clear upward trend, we should use the trend specification, while if there is no trend or constant visible in the plot, then there is no need to go for a different specification than none?
Thanks in advance!
