I am teaching a basic course in Econometrics, and, as usual, I use Gretl for all empirical applications. A week ago, a couple of students (Alejandra Pérez and Natividad Aguilera) discovered something weird when using the DF test in the residuals or the Engle-Granger test. They know that, although the test is the same, critical are not. Anyway, the value of the t-ratio should be the same whether you use the EG option or do the test by yourself using the
DF-test. They showed to me their example and I think they might be right. In their own words:
"Dear Sir or Madam,
We have some points that we would like to have clarified about the Engle-Granger cointegration test (coint ) and the Augmented Dickey-Fuller test (adf ) using two variables.
First, in the coint test, there is an option to allow Gretl to determine the number of lags of the dependent variable used in the adf test (from a maximum number of lags established by the user) and in the first step it reduces the sample according to this. Then the same sample is used to run the adf test on the other variable. Why Gretl does not use a sample according to the significant lags in each case/variable as if it were doing the adf test individually?
Second, once the sample is reduced in the first step, the ADF performed on residuals (second step) is done with
the same reduced sample. Inference is not drawn using the original sample size. Why?
Third, when we run the coint test it is not possible to do the test for the residuals with different deterministic components. However, it is possible that the variables with unitary root have a tendency and the residuals series not. Why is it not possible to select different deterministic components for initial adf test and the adf test on the residuals?
Thank you for your attention to this
message.
Yours faithfully,
Natividad Aguilera, economic’s master student from University of Guanajuato (UG) &
Alejandra Pérez, economic’s undergraduate student from Center for Reseach and Teaching in Economics
(CIDE)."
Many thanks for your attention. You can corroborate this using any pair of time series (the number of lags must be fixed for both variables, the ADF and the EG).