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?