Inconsistency of Forbidden Regression Estimator I am trying to prove the inconsistency of the estimator of the  in the following model:
1st stage: 
2nd stage: 
When the population model is  
and .
I know this is a forbidden regression because of the non-linear nature of the population model, but I'm not managing to prove this estimator is inconsistent.
Any thoughts on how to do this or where to find such proof in detail?
Thanks in advance.
 A: Describe the stages as follows:
1st Stage: $Z=X\pi + V \Rightarrow \hat{Z} = X\hat{\pi}$
2nd Stage: $Y=\hat{Z'}\hat{Z} \alpha + U$.
Because the first stage is OLS, $ \hat{\pi} = (X'X)^{-1}X'Z$, and therefore $\hat{Z} = X(X'X)^{-1}X'Z$. The second stage is also OLS: $$\hat{\alpha} = ((\hat{Z}'\hat{Z})'(\hat{Z}'\hat{Z}))^{-1}(\hat{Z}'\hat{Z})'Y$$
Note that $$ \hat{Z}'\hat{Z}=Z'X(X'X)^{-1}X'X(X'X)^{-1}X'Z = Z'X(X'X)^{-1}X'Z$$ which is symmetric. Therefore:
$$\hat{\alpha} = [Z'X(X'X)^{-1}X'ZZ'X(X'X)^{-1}X'Z]^{-1}Z'X(X'X)^{-1}X'Z(Z'Z\alpha + U)$$ which in trun, can be written as:
$$\hat{\alpha} = \left [\frac{Z'X}{n}\left(\frac{X'X}{n}\right)^{-1}\frac{X'Z}{n}\frac{Z'X}{n}\left(\frac{X'X}{n}\right)^{-1}\frac{X'Z}{n}\right]^{-1}\frac{Z'X}{n}\left(\frac{X'X}{n}\right)^{-1}\frac{X'Z}{n}(\frac{Z'Z}{n}\alpha + \frac{U}{n})$$
Taking the plim:
$$  Plim\hat{\alpha} =  ( [\frac{Z'X}{n}(\frac{X'X}{n})^{-1}\frac{X'Z}{n}\frac{Z'X}{n}(\frac{X'X}{n})^{-1}\frac{X'Z}{n}]^{-1}\frac{Z'X}{n}(\frac{X'X}{n})^{-1}\frac{X'Z}{n}\frac{Z'Z}{n}\alpha $$
But: $$  \frac{Z'Z}{n} \overset{p}{\rightarrow}  Q_{zz}$$
$$\frac{Z'X}{n} \overset{p}{\rightarrow} Q_{xz}'$$
$$\frac{X'X}{n} \overset{p}{\rightarrow} Q_{xx}$$
$$\frac{X'Z}{n} \overset{p}{\rightarrow} Q_{xz}$$
So:
$$ Plim \hat{\alpha} = \left ( Q_{xz}'Q_{xx}^{-1}Q_{xz}Q_{xz}'Q_{xx}^{-1}Q_{xz}' \right )^{-1} Q_{xz}'Q_{xx}^{-1}Q_{xz}Q_{zz} \alpha \neq \alpha$$ and the estimator is inconsistent.
