Step two in the Engle Granger method (include constant?)

[skifozoa]

Whithin the Engle Granger method to test cointegration...

Say you have estimated your static model Y = a + b X and have saved your residuals in RES. Now The following step would be to estimate D(Y) = D(Y(-1)), D(X), D(X(-1)), ... , RES(-1). Do you also include a constant to this equation?

In my coursenotes it says yes, in some slides I can find on the internet it says no, in other slides it says yes again, personally I would think not.



This is why I would not do it, assume following data generating process:

Y = alpha + beta Y(-1) + gamma0 X + gamma1 X(-1) + whitenoise

the static equation estimates Y = alpha/(1-beta) + (gamma0 + gamma1)/(1-beta) X

The ECM model is given by D(Y) = gamma0 * D(X) - (1-beta) [ Y(-1) - alpha/(1-beta) - (gamma0+gamma1)/(1-beta) X(-1) ].

RES(-1) is an estimater for [part within brackets]

so we now estimate (in step two):

D(Y) = gamma0 * D(X) - (1-beta) * RES(-1)

so adding a constant to this is not conform to the underlying data generating process.



Is my reasoning correct ? Or is there somewhere along the way reason to add a constant to the second estimation as well ?
Thank you for your responses