I got smth which looks like this
$y_1 = \alpha_1\cdot y_2 + \alpha_2\cdot y_3 + X\cdot\alpha_3 + u_1$
$y_2 = \beta_1\cdot y_1 + \beta_2\cdot y_3 + X\cdot\beta_3 + u_2$
$y_3 = \gamma_1\cdot y_1 + \gamma_2\cdot y_2 + X\cdot\gamma_3 + u_3$
After I put some thought into my model I could not really restrict some of the exogenous variables (which are contained in $X$) to zero nor could I kick out some of the dependencies between those endogenous variables...
The topic which my model is about will regard to the success of a student during his studies. Hence the dependent variables will be grade, semester and how many times a student revisited a exam (because of fail or to get a higher grade).
First I tried IV, 2SLS, 3SLS but since my data just do not fit the requirements of this methods this is a dead end. The reason is that a) there is no a good instrument available and b) even if I try to find the reasons to not include one of the exogenous variables in one or more equations the first stage fit is so bad that the second stages suffers a great deal from it.
The next thing I tried was to introduce a approach which allows me to not exclude any variables by including a rank-order variable (introduced by vella). This RO-Variable is supposed to account for the endogeneity by transforming the reduced form residuals such that the new error term in the structural equation is now uncorrelated with the endogenous regressors.
The reason this did not work out was that the costs of this approach lies in the high multicolinearity which renders my results useless. After I did a simulation I figured out that my results pointed towards some heavy problems which cant be overcome. I think this is a very neat solution which was the reason I took it but it just cannot work out.
So the bottom line is: I'm stuck, I don't now what I could do next and I hope for some input of you guys what I could possibly do to make this work. I just need some hints in the right direction because I feel a bit lost..