# Slope Coefficient on Exogenous Variables in 2SLS and Directly Regressing the instrument

Suppose we have $$k_1$$ exogenous variables $$X_1$$, $$k_2$$ endogenous variables $$X_2$$ and $$k_2$$ instruments $$Z$$ and the model $$Y=X_1\beta_1+X_2\beta_2+e$$. Let the 2SLS estimates be $$(\hat\beta_1,\hat\beta_2).$$If we directly run a regression with the instruments instead $$Y=X_1\alpha_1+Z\alpha_2+e$$, then the claim is that $$\hat\alpha_1=\hat\beta_1$$ i.e. the estimated coefficient on the exogenous variables are the same.

I'm having trouble formally proving the result as well as understanding the intuition. I tried applying the Frisch–Waugh–Lovell theorem so that $$\hat\alpha_1=(X_1'M_ZX_1)'(X_1'M_ZY)$$ where $$M_Z=(I-Z(Z'Z)^{-1}Z')$$ and then tried showing that $$\hat\beta_1$$ will be equal to that. However, applying the FWL theorem in the 2SLS context, $$\hat\beta_1=(X_1'M_{\hat{X_2}}X_1)'(X_1'M_{\hat{X_2}}Y)$$. This would require $$M_{\hat{X_2}}=M_Z$$. However, I can't see why the two should be equal. Furthermore, that would imply that the residuals from regressing $$X_1$$ on $$Z$$ and $$X_1$$ on $$\hat{X}_2$$ would be the same which doesn't seem to make sense either since the predicted value $$\hat{X}_2$$ uses both $$X_1$$ and $$Z$$. What am I missing here?

The OLS regression of $$Y$$ on $$X$$ decomposes dependent variable $$Y$$ into orthogonal components $$Y = \hat{Y} + e$$ in $$\mathbb{R}^n$$, where $$\hat{Y} = X \hat{\beta}$$ is the fitted value and $$e$$ the residuals.
It's a basic fact of linear algebra that this orthogonal decomposition remains the same, for all regressor matrix $$X$$ with the same column space. The OLS $$\hat{\beta}$$ may change, but $$\hat{Y}$$ remains the same.
Suppose $$X = [X_1 \; Z]$$ and you replace $$Z$$ by $$\tilde{Z}$$ so that the regressors in $$\tilde{Z}$$ has the same span as those in $$Z$$, i.e. $$Z = \tilde{Z} W$$ for some invertible $$W$$. Then \begin{align*} Y &= X_1 \hat{\beta}_1 + Z \hat{\beta}_2 + e \\ &= X_1 \hat{\beta}_1 + \tilde{Z} W \hat{\beta}_2 + e. \end{align*} Uniqueness of the orthogonal decomposition therefore tells you that regressing $$Y$$ on $$[X_1 \; \tilde{Z}]$$ gives OLS estimate $$[\hat{\beta}_1 \; W \hat{\beta}_2].$$ Notice that the coefficients for $$X_1$$ from the two regressions remains the same---$$\hat{\beta}_1$$.
The 2SLS example is a special case, where $$Z$$ consists of instruments and $$\tilde{Z} = \hat{X}_2$$ are the fitted values from the first stage regression of $$X_2$$ on $$Z$$. Note that $$\tilde{Z}$$ and $$Z$$ has the same column space if the model is exactly identified---in your notation, $$k_2$$ instruments for $$k_2$$ endogenous variables.
The statement is no longer true when model is over-identified---when the column space of $$\tilde{Z}$$ is a proper subspace of that of $$Z$$.