<p>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,</p>
<p><span class="math-container">$$x=\hat{\alpha_0}+\hat{\alpha_1}z+\hat{u}$$</span></p>
<p>Because <span class="math-container">$\hat{u}$</span> are residuals, we know that <span class="math-container">$Cov(\hat{u},z)=0$</span>. Then consider,</p>
<p><span class="math-container">$$x=\hat{x}+\hat{u}$$</span></p>
<p>It is the case that <span class="math-container">$Cov(\hat{x},\hat{u})=0$</span> because <span class="math-container">$\hat{x}=\hat{\alpha_0}+\hat{\alpha_1}z$</span>.</p>
<p>Let us now consider how the second stage equation of estimation is created. The theoretical equation is,</p>
<p><span class="math-container">$$y=\beta_0 +\beta_1x+e$$</span>
Let us plug in <span class="math-container">$x=\hat{x}+\hat{u}$</span>.
<span class="math-container">$$y=\beta_0 +\beta_1(\hat{x}+\hat{u})+e$$</span>
<span class="math-container">$$y=\beta_0 +\beta_1\hat{x}+\beta_1\hat{u}+e$$</span>
<span class="math-container">$$y=\beta_0 +\beta_1\hat{x}+\eta$$</span></p>
<p>The error term is <span class="math-container">$\eta=\beta_1\hat{u}+e$</span>. The exclusion/exogeneity assumption of IV is <span class="math-container">$Cov(z,e)=0$</span>, thus <span class="math-container">$Cov(\hat{x},e)=0$</span>. Also, by construction of OLS, <span class="math-container">$Cov(\hat{x},\hat{u})=0$</span>. We thus have <span class="math-container">$Cov(\hat{x},\eta)=0$</span> and estimation of the second stage by OLS is consistent.</p>
<p>To bring this back to your question, the first stage of OLS imposes that <span class="math-container">$Cov(\hat{x},u)=0$</span> for the sample used in estimation of the first stage. If you estimate the first stage using all observations, then <span class="math-container">$Cov(\hat{x},u)=0$</span> 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.</p>
<p>If you estimate the first stage using the same subsample as used in the second stage, then <span class="math-container">$Cov(\hat{x},u)=0$</span> in the subsample used in the second stage. Estimates are always consistent.</p>

CC BY-SA 4.0
2023-03-01T07:39:56.877Z
2023-03-01T07:39:56.877Z
Michael Gmeiner
<p>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.</p>
<p><span class="math-container">$$
y = \beta_{0} + \beta_{1} x + e
$$</span></p>
<p><span class="math-container">$$
x= \alpha_{0} + \alpha_{1} z + u
$$</span></p>
<p>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 &gt; N - K so that I have more complete than incomplete observations in the data.</p>
<p>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?</p>
<p>Any help would be greatly appreciated.</p>

<econometrics><applied-econometrics><instrumental-variable>
IV Regression with More Observations for First Stage than Second Stage
Student in need
# IV Regression with More Observations for First Stage than Second Stage

(Asked by: username_0 on 2023-02-17)

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.

**Tags:** econometrics, applied-econometrics, instrumental-variable

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## 1 answers

### Answer by username_1 on 2023-03-01. Score: 2

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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