Robust Standard Errors for Control Function Approach? I need to know how to find robust S.E. for the CF approach to endogeneity. 

Consider the model: 
$$y_i=X_i\beta_1 + W_i\beta_2+\epsilon_i$$

Assume: 
$$E[X_i\epsilon_i]=0$$
$$E[W_i\epsilon_i] \neq 0$$

Thus, $W_i$ is endogenous. 

Now, let: 
$$E[Z_iW_i] \neq 0$$
$$E[Z_i\epsilon_i]=0$$ 

The control function approach: 


$W_i = \gamma_1 X_i + \gamma_2 Z_i + \phi_i  $
$\epsilon_i = \alpha \phi_i + \chi$ 


now we replace $\epsilon_i$ in our original equation: 

$$y_i=X_i\beta_1 + W_i\beta_2+ \phi_i \alpha + \chi$$

where now we have $E[W_i \chi]=0$

Intuition: I think I used the series of linear projections to 'control' for the endogenous portion of $W_i$. 

EDIT: I originally typed this incorrectly. I have changed the relevant orthogonality condition. Here is the intuition behind the (correct) orthogonality condition $E[W_i \chi]=0$:

Since $W_i$ is a linear function of $X_i , Z_i$, and both are themselves orthogonal  to $\chi$, we obtain the given orthogonality condition. 

Okay - the question. I think that $\hat \beta_{2.CF} \equiv \hat \beta_{2.OLS} \equiv \frac{cov(W_i,Y_i)}{Var(W_i)}$

If this is correct, do I just use the R.S.E. form that I use in OLS if I want heteroskedasticity robust S.E. when using the C.F. approach?
123, the CF method is a specific case of when we have generated regressors. Check the appendix to chapter 6, pages 157-160, for the asymptotic covariance matrix of $N^{1/2}(\hat\beta_{CF}-\beta)$. The expression is truly enormous. ;)

Instead I will just cite wooldridge on page 160, 2nd paragraph:

«If $\mathbf{G}\neq \mathbf{0}$, then (...). Neither the usual 2SLS variance matrix estimator nor the heteroskedasticity-robust form is valid in this case.»The $\mathbf{G}\neq \mathbf{0}$ is the case when we have generated regressors.

Good luck ;)