Simultaneous equation model

[Marry Lee]

Hello,

I was studying simultaneous equation models and got curious about something, I would appreciate your help.

Consider that theory suggests that two choices are jointly determined, so we have the following econometric model

y1 = a1 + b1 y2 + c1 z1 + dz2 + u1
y2 = a2 + b2 y1 + c2 z1 + u2

So here the second equation is identified if d is different from 0.

Imagine that z1 is a vector of variables that determine both y1 and y2, and imagine we are only interested in how z1 affects y2. In this case, we will estimate the following model:

 y2 = alpha + beta z1 + e

So, my question here is, in this model is y1 included in the error term e and since z1 also determines y1 we will be having an endogeneity problem?
If z1 was not correlated with y1 would there be any other problems with estimating this last model?


Adding the context:
I was reading this paper, and the author mentioned the following

A number of other variables that may be correlated with return migration behavior like work effort and earnings, fluency in German, the location of the spouse and children, and remittance behavior are not used in the specification because these are choice variables jointly determined with the return migration decision. For instance, an immigrant with stronger return intentions might work harder, be less willing to invest in learning German, more likely to keep his children in his home country and remit. The goal of the specification chosen is to uncover the exogenous underlying forces that determine the return migration decision and that would also determine related choices like remittance decision.

Reference:

Kırdar, M. G. (2009). Labor market outcomes, savings accumulation, and return migration. Labour Economics, 16(4), 418-428.


Thank you!