1
$\begingroup$

If you run a Engle and Granger test with the regression: $$ y_t=b_0+b_1x_t+\varepsilon_t $$ and you know that $y_t$ is integrated of order 2, what can you say about the order of integration of $x_t$, knowing that, from the Augmented DF, $\varepsilon_t$ is non-stationary (integrated of order 1)?

And what can you say about the cointegration between the two time series?

I cannot get the logic behind it. I mean, I would say that if you fail to reject the Augmented DF then you fail to reject the null hypothesis of non-cointegration between the time series and you cannot say anything about the order of integration of $x_t$. Is that correct or am I missing something?

$\endgroup$
1
  • $\begingroup$ Hi: I don't think you can say anything about the order of $x_t$. I would investigate the order of it before you continue because a lot of the econometrics such ADF and cointegration depends on both sides of an equation being of the same order. Otherwise, doing regressions of both sides doesn't make sense. Check out Hamilton for more detals on this because it's all quite fuzzy and long ago for me. Hopefully soneone else can say something more specific. $\endgroup$
    – mlofton
    Commented May 13, 2021 at 3:19

1 Answer 1

0
$\begingroup$

If we disregard estimation imprecision and assume $Y_t$ is truly I(2) and $\varepsilon_t$ is truly I(1), then $Y_t$ and $X_t$ must be cointegrated. If they were not, $\varepsilon_t$ would be at least I(2) since the I(2) process on the left hand side of the equation would need to be counterbalanced with an I(2) process on the right hand side. Since $\varepsilon_t$'s order of integration is lower, this must be the result of cointegration between $Y_t$ and $X_t$. $X_t$ must be I(2) since otherwise it could not cointegrate with $Y$.
(If we cannot disregard estimation imprecision, the question becomes more involved.)

$\endgroup$
2
  • $\begingroup$ Thank you for your answer. In this case, however, would the Engle and Grenger reject the null hypothesis of non-cointegration? I am asking this because of the generated regressors problem and I am afraid that, in cases like this one, with the Engle-Granger you fail to say that the series are cointegrated. Am I wrong? $\endgroup$
    – DanT
    Commented May 13, 2021 at 10:41
  • $\begingroup$ @DanT, Engle-Granger works directly for I(1) series so that under cointegration the ADF test rejects a unit root in the residual. For cointegrated I(2) series that yield an I(1) residual under cointegration, you can apply the ADF test on the first differences of the residual to get the standard/expected result. Not sure how the generated regressors problem relates to all this. $\endgroup$ Commented May 13, 2021 at 10:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.