I can't find anything on this topic on the internet, so I'm trying here. I need to estimate parameters of multiple equation model by two stage least-square method.
Variables are Y1, Y2, Y3, X1, X2, X3. Y1 depends on Y2, Y3 and X1 so, as dependent variable I choose Y1, as regressors I choose Y2, Y3 and X1, and as instruments I choose X1, X2, X3. AND.
As far as parameters are concerned everything is fine. The problem is with either t-student values or p values which indicate if the variable is relevant or not.
Strictly speaking, it shows huge p values, which are much different than they are when computing two-stage method step-by-step (First least-square method to compute theoretical values of Y2 and Y3 dependent on X1, X2, X3, then least-square method of Y1 dependent on Y2^, Y3^, X1).
Anyone knows why is that? And which results are true.
In order to get the correct indicator for inference (p=value or t-statistics) you need to adjust the Covariance Matrix when using the "manual" estimation. The reason is that when using OLS in the second stage it does not take into account that the degrees fo freedom are different, because of the estimation in the first stage. See here and example in stata.