I am trying to figure out whether I can trust OLS results in this situation.

I have possibly simultaneous equations that follow the models:

$ y = k_1 + \gamma_1 x + \beta_1 z + u_1 $

$ z = k_2 + \beta_2 x + ... + u_2 $

where $ z $ is exogenous in the first equation, but partially determined by x. $u_1 , u_2$ are the error terms in the equations (random, mean zero and normal distr).

I am interested in consistent estimation of $ \beta_1 $ through OLS: the impact of z on y.

do I have OLS biased estimates of beta_1 if I introduce x in the first equation?

what if instead I also have a third equation:

$ x = k_3 + \beta_2 z + ... + u_3 $

I am trying to understand whether the equation for y can be estimated through OLS despite the endogeneity of z-x equations. Additionally, in the remote case it is a consistent estimator, the appearance of y in any of the two equations (the second and third) would bias the estimator right?



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