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This is a very basic question, but I need help. Imagine I have a dataset for variables y, x, and z. I am running an instrumental variable regression such that y is my main outcome variable. So I am trying to estimate the following regressions.

$$ y = \beta_{0} + \beta_{1} x + e $$

$$ x= \alpha_{0} + \alpha_{1} z + u $$

I have N observations. K observations contain values for y, z and x. N-K observations contain values only for x and z, with y values being reported as NA. K > N - K so that I have more complete than incomplete observations in the data.

My question is: should I run the first stage regression with all N observations and THEN run the main equation with only K fitted values? Or should I drop the incomplete N - K observations to begin with and just focus on K complete observations for the first and second stages? Would this at all affect the way I calculate standard errors?

Any help would be greatly appreciated.

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You should completely omit the observations with missing data. The explanation takes a few lines, but the key fact of the first stage regression is that predicted values are uncorrelated with residuals. In context,

$$x=\hat{\alpha_0}+\hat{\alpha_1}z+\hat{u}$$

Because $\hat{u}$ are residuals, we know that $Cov(\hat{u},z)=0$. Then consider,

$$x=\hat{x}+\hat{u}$$

It is the case that $Cov(\hat{x},\hat{u})=0$ because $\hat{x}=\hat{\alpha_0}+\hat{\alpha_1}z$.

Let us now consider how the second stage equation of estimation is created. The theoretical equation is,

$$y=\beta_0 +\beta_1x+e$$ Let us plug in $x=\hat{x}+\hat{u}$. $$y=\beta_0 +\beta_1(\hat{x}+\hat{u})+e$$ $$y=\beta_0 +\beta_1\hat{x}+\beta_1\hat{u}+e$$ $$y=\beta_0 +\beta_1\hat{x}+\eta$$

The error term is $\eta=\beta_1\hat{u}+e$. The exclusion/exogeneity assumption of IV is $Cov(z,e)=0$, thus $Cov(\hat{x},e)=0$. Also, by construction of OLS, $Cov(\hat{x},\hat{u})=0$. We thus have $Cov(\hat{x},\eta)=0$ and estimation of the second stage by OLS is consistent.

To bring this back to your question, the first stage of OLS imposes that $Cov(\hat{x},u)=0$ for the sample used in estimation of the first stage. If you estimate the first stage using all observations, then $Cov(\hat{x},u)=0$ in the full sample, but possibly not in the subsample that would be used in the second stage. If that covariance is non-zero, then estimates of the second stage are inconsistent.

If you estimate the first stage using the same subsample as used in the second stage, then $Cov(\hat{x},u)=0$ in the subsample used in the second stage. Estimates are always consistent.

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