My problem is this: I have 2 series $y$ and $x$ that I want to verify are cointegrated. So first I verified them with ADF unit root test and revealed that one variable is integrated at first difference $I(1)$ and the other at second difference $I(2)$ using the test with an intercept! But when I run the ADF test on the variables not including an intercept (the last option – none), both series are integrated at first difference $I(1)$.
So after this I want to test for cointegration. Instead of running a Johansen test, which has some difficult options to set up (I set up option 3 and 1 2 for the lagged endogenous variables and found one cointegration equation, but I don't know if it is the correct combination for me; this decision is very subjective, I think), I want to ask you if this is correct:
Knowing that $y$ and $x$ are $I(1)$, I estimate the equation $y = a_0 + a_1 x + e$ and save the values for error variable $e$. Then I run ADF test for unit root in residual variable $e$ and find out that the error is $I(0)$ or stationary. Does this not prove that the series $y$ and $x$ are cointegrated?