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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?

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Your reasoning is correct: if both series are I(1), and the residuals from their so-called "long-run relationship" are I(0), one can conclude that the series are cointegrated. The way you did it follows the "Engle-Granger two step approach", where a unit root test is applied on the residuals.

Note that this approach, the residual based approach, was improved by Philips and Ouliaris (PO), who suggest a very similar procedure, and their test is recommended instead of the two-step approach you use. But note that in any case, be it using the ADF on residuals, the PO test, or the Johansen test, you will have to specify a parameter including the lags/bandwidth for auto-correlation, so in any case what you call "a subjective choice" has to be taken, although you can base it on stat tests or information criteria such as AIC/BIC. That results might differ according to the number of lags is a unfortunate but pretty common fact with time series...

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    $\begingroup$ Thanks for the answer. My time series are annual 19 observations so I don t think Johansen cointegration test would help me as it specifies in the results that the critical values are calculated for 20 observations. Please tell me how to choose the number of lags in Eviews if you know based om Aic and bic and where do I specify them, probably in the lagged dependent regressors box in Johasen window. $\endgroup$ – Raluca Feb 21 '14 at 15:31

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