# IV Regression with More Observations for First Stage than Second Stage

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.

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.