Can an ECM be built such that new regressors enter in Stage 2?

Question

If I have an ECM such that there are 2 cointegrated variables, $Z_t\sim I(1)$, $X_t\sim (1)$, and the equation in levels is $Z_t=a+bX_t+e_t$, $e_t\sim I(0)$, can I add other regressors to the 2nd stage regression as follows?

$$ \Delta z_t = \alpha_1 (z_{t-1}-a-bx_{t-1}) + \gamma_{11} \Delta z_{t-1} + \gamma_{12} \Delta x_{t-1} + \gamma_{13} \Delta y_{t-1} + + \gamma_{14} w_{t} +\varepsilon_{t} $$

Where $w_{t}$ is a known dummy variable and along with $y_{t-1}$ was added in the 2nd stage?

Edit: I guess since the long-term relationship is modeled by the variables $Z_t$, $X_t$, and this is modeled through the error term in the ECM, I just wonder why some short-term variables that do have an influence on $Z_t$ cant be added to the ECM.

Thanks

Answer 1

Hi: If you add them in the second stage but then you've most likely totally negated the ecm relationship. The ecm relationship (without the new variables ) follows exactly from the cointegrating relationship. This is mostly what engle and granger got the noble prize for. So, if you add other variables to the ecm, you've kind of thrown out the baby with the bath water in the sense that the ecm relationship is no longer the same as it would have been if you left the new variables out. In fact, the estimated ecm coefficients ( with those new variables in there ) may not have any meaning once you do that. They have meaning if you don't do that. So the answer is that I wouldn't do that.