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

Question

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?

Answer 1

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.

This can be extended to a subset of regressors.

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 $Z$.