Let's assume we have 2 variables and test each of them for a unit root with the ADF test. When plotting the data, we can see that it has some up/down movements but is overall clearly trending upwards.
At level, variable A is stationary with "no constant", "constant" and "constant+trend" (p<0.05). Variable B is only stationary with "constant+trend" (p<0.05).
At first differences, both variables are stationary at "no constant", "constant" and "constant+trend" (p<0.05).
Is it o.k. to conclude that Variable A and B are both integrated of the same order I(1), or do we have to assume that Variable B is I(0) with respect to a trend and Variable B is I(1)?
I want to conduct a Johansen Cointegration test and after reading the literature and several posts, I'm still not sure if it's okay test for correlation with the data at level (and a trend).
Best Answer
Variable A seems to be stationary as whatever the specification ("no constant", "constant", "constant+trend") of the test regression you still reject the unit root hypothesis. Therefore you cannot conclude that A~I(1). Since at least one of the two variable is stationary, you cannot proceed to cointegration analysis (recall that it requires both variables to be integrated to begin with).
Note on proceeding to testing stationarity of A at first differences: you should not do that as A is already stationary in levels. If A is stationary in levels, testing for stationarity in first differences is redundant when your question is whether A should be included in cointegration analysis.