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I have data with 4 variables that are rather trending upwards. They are mostly stationary at level, but only when I include a constant and trend into the ADF unit root test. They are not stationary without a constant or with a constant in the regression.

After differencing, all variables are stationary and thus I(1) (no constant, constant, constant and trend)

When conducting the Engle-Granger cointegration test with Gretl, should I include any constant, constant and trend into the test or rather go with "without constant"?

It seems when I choose "without constant" my results confirm the other Johansen Cointegration results while by including a constant or trend, my results conflict with criterion (a)

From Gretl:

 There is evidence for a cointegrating relationship if:
 (a) The unit-root hypothesis is not rejected for the individual variables, and
 (b) the unit-root hypothesis is rejected for the residuals (uhat) from the 
cointegrating regression.

If the results are lower 0.1 for the unit root tests, I can hardly confirm cointegration, as I find it with the Johansen Cointegration test, where I do include a constant and trend.

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First, when the series are trending upwards, they are not stationary. What you mean then is "trend stationary" and I(0), which is not the same as stationary. Having said that, if your variables are all I(0) in levels (with a linear trend term), unit roots and cointegration don't make sense.

With respect to the Engle-Granger part: You would have to be more specific in which of the two steps you want to insert the terms. Or to put it differently: In the second step where you test the residuals no deterministic terms should be included anymore, only in the first step. But gretl does the right thing there anyway.

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