<p>I would suspect that if  <span class="math-container">$X_1$</span> is independent of <span class="math-container">$X_3$</span>, then it must imply <span class="math-container">$E[X_1*X_3]$</span> is equal to <span class="math-container">$E[X1] * E[X_3]$</span>, given this you can conclude that the Covariance is 0. <span class="math-container">$Cov(X_1,X_3) = E[X_1*X_3] - E[X_1]* E[X_3]$</span>. If the Covariance is 0 between these two variables then so is the correlation. Thus I would believe that removing <span class="math-container">$X_3$</span> from the regression and having it's effect enter the error term would not result in having <span class="math-container">$γ_1$</span> be biased as the <span class="math-container">$X_3$</span> in the residual term does not have correlation with <span class="math-container">$γ_1$</span>. </p>

<p>Are you looking for a rigorous mathematical proof?</p>

CC BY-SA 4.0
2018-12-07T04:32:59.850Z
2018-12-19T18:17:27.990Z
2018-12-19T18:17:27.990Z
Aram
<p>Your approach leads to a biased estimator. Nevertheless, there exists a way to obtain an unbiased estimator for <span class="math-container">$\beta_1$</span>. We asssume all the regularity conditions and exogeneity of regressors with respect to the error <span class="math-container">$\varepsilon$</span>.</p>

<p>1) Center the variables on their sample means, <span class="math-container">$ y_c = y- \bar y, X_{c1} = X_1 - \bar X_1$</span>, etc.</p>

<p>2) Specify the regression</p>

<p><span class="math-container">$$y_c = X_{c1}\delta + u$$</span></p>

<p>Then </p>

<p><span class="math-container">$$\hat \delta_{OLS} = \left(X'_{c1} X_{c1}\right)^{-1}X'_{c1}y_c $$</span></p>

<p><span class="math-container">$$=\left(X'_{c1} X_{c1}\right)^{-1}X'_{c1}X_{c1}\beta_1 + \left(X'_{c1} X_{c1}\right)^{-1}X'_{c1}X_{c2}\beta_2 \\+ \left(X'_{c1} X_{c1}\right)^{-1}X'_{c1} X_{c3}\beta_3 + \left(X'_{c1} X_{c1}\right)^{-1}X'_{c1}\varepsilon_c$$</span></p>

<p>Simplifying and taking expectations conditional on <span class="math-container">$X_{c1}$</span> we get </p>

<p><span class="math-container">$$\mathbb E(\hat \delta_{OLS} \mid X_{c1})= \beta_1 + \left(X'_{c1} X_{c1}\right)^{-1}X'_{c1}\mathbb E(X_{c2} \mid X_{c1})\beta_2  \\+\left(X'_{c1} X_{c1}\right)^{-1}X'_{c1} \mathbb E(X_{c3}\mid X_{c1})\beta_3 + \left(X'_{c1} X_{c1}\right)^{-1}X'_{c1}\mathbb E(\varepsilon_c\mid X_{c1})$$</span></p>

<p>By the regressor independence and exogeneity assumptions, as well as due to the centering we have</p>

<p><span class="math-container">$$\mathbb E(X_{c2} \mid X_{c1}) = \mathbb E(X_{c2} ) = 0$$</span>
<span class="math-container">$$\mathbb E(X_{c3} \mid X_{c1}) = \mathbb E(X_{c3} ) = 0$$</span>
<span class="math-container">$$\mathbb E(\varepsilon_c \mid X_{c1})  = 0$$</span></p>

<p>So we are left with </p>

<p><span class="math-container">$$\mathbb E(\hat \delta_{OLS} \mid X_{c1})= \beta_1 \implies \mathbb E(\hat \delta_{OLS}) = \beta_1$$</span></p>

CC BY-SA 4.0
2018-12-19T23:21:03.047Z
2018-12-19T23:21:03.047Z
Alecos Papadopoulos
<p>Suppose we have a model that is:</p>

<p><span class="math-container">$$
y = X_1\beta_1 + X_2\beta_2 + X_3\beta_3 + \varepsilon
$$</span></p>

<p>where <span class="math-container">$X_1$</span> is independent of <span class="math-container">$X_2$</span> and <span class="math-container">$X_3$</span> but <span class="math-container">$X_2$</span> and <span class="math-container">$X_3$</span> are correlated with each other.</p>

<p>Suppose we don't have data on <span class="math-container">$X_3$</span> so we omit it and run the regression:</p>

<p><span class="math-container">$$
y = X_1\gamma_1 + X_2\gamma_2 + \nu
$$</span></p>

<p>Econometric theory on OVB states that <span class="math-container">$\gamma_2$</span> will be biased.</p>

<p>My questions:  </p>

<ol>
<li><p>Will <span class="math-container">$\gamma_1$</span> be biased in this case as well? I ran a simulation with some data I created and I do get a biased result.  </p></li>
<li><p>If it is unbiased then how do I prove it? I use FWL theorem to find <span class="math-container">$\gamma_1$</span> and show that its expectation equals <span class="math-container">$\beta_1$</span> but get stuck here:</p></li>
</ol>

<p><span class="math-container">$$
\hat{\gamma_1} = \beta_1 + (X_1'M_{X_2}X_1)^{-1}(X_1'M_{X_2}X_3\beta_3) + (X_1'M_{X_2}X_1)^{-1}(X_1'M_{X_2}\varepsilon)
$$</span></p>

<p>The third term will become 0 when we take the expectation but not sure how to make the second term 0.</p>

CC BY-SA 4.0
2018-11-30T11:05:40.787Z
Are you sure you've done your simulation correctly? If X1 and X3 are independent then gamma1 shouldn't be biased since Cov(X1,X3)=0. This also implies that Cov(X1Mx2X3)=0. So the the second term in part two of your question goes to zero when X1 and X3 are independent of one another.
Andrew M
CC BY-SA 4.0
2018-12-07T17:37:21.370Z
$\gamma_1$ won't be biased, given $\mathbb{C}(X_1, X_2) = 0 = \mathbb{C}(X_1, X_3)$
Pedro Cavalcante
<econometrics>
Does omitted variable bias 'contaminate' coefficients for regressors not correlated with the omitted variable?
Odysseus56
# Does omitted variable bias 'contaminate' coefficients for regressors not correlated with the omitted variable?

(Asked by: username_0 on 2018-11-12)

Suppose we have a model that is:

$$
y = X\_1\beta\_1 + X\_2\beta\_2 + X\_3\beta\_3 + \varepsilon
$$

where $X\_1$ is independent of $X\_2$ and $X\_3$ but $X\_2$ and $X\_3$ are correlated with each other.

Suppose we don't have data on $X\_3$ so we omit it and run the regression:

$$
y = X\_1\gamma\_1 + X\_2\gamma\_2 + \nu
$$

Econometric theory on OVB states that $\gamma\_2$ will be biased.

My questions:

1. Will $\gamma\_1$ be biased in this case as well? I ran a simulation with some data I created and I do get a biased result.
2. If it is unbiased then how do I prove it? I use FWL theorem to find $\gamma\_1$ and show that its expectation equals $\beta\_1$ but get stuck here:

$$
\hat{\gamma\_1} = \beta\_1 + (X\_1'M\_{X\_2}X\_1)^{-1}(X\_1'M\_{X\_2}X\_3\beta\_3) + (X\_1'M\_{X\_2}X\_1)^{-1}(X\_1'M\_{X\_2}\varepsilon)
$$

The third term will become 0 when we take the expectation but not sure how to make the second term 0.

**Tags:** econometrics
> **Comments:**
> **username_1**: Are you sure you've done your simulation correctly? If X1 and X3 are independent then gamma1 shouldn't be biased since Cov(X1,X3)=0. This also implies that Cov(X1Mx2X3)=0. So the the second term in part two of your question goes to zero when X1 and X3 are independent of one another.  
> **username_2**: $\gamma_1$ won't be biased, given $\mathbb{C}(X_1, X_2) = 0 = \mathbb{C}(X_1, X_3)$  

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## 2 answers

### Answer by username_3 on 2018-12-07. Score: 1

I would suspect that if $X\_1$ is independent of $X\_3$, then it must imply $E[X\_1\*X\_3]$ is equal to $E[X1] \* E[X\_3]$, given this you can conclude that the Covariance is 0. $Cov(X\_1,X\_3) = E[X\_1\*X\_3] - E[X\_1]\* E[X\_3]$. If the Covariance is 0 between these two variables then so is the correlation. Thus I would believe that removing $X\_3$ from the regression and having it's effect enter the error term would not result in having $γ\_1$ be biased as the $X\_3$ in the residual term does not have correlation with $γ\_1$.

Are you looking for a rigorous mathematical proof?


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### Answer by username_4 on 2018-12-19. Score: 0

Your approach leads to a biased estimator. Nevertheless, there exists a way to obtain an unbiased estimator for $\beta\_1$. We asssume all the regularity conditions and exogeneity of regressors with respect to the error $\varepsilon$.

1) Center the variables on their sample means, $ y\_c = y- \bar y, X\_{c1} = X\_1 - \bar X\_1$, etc.

2) Specify the regression

$$y\_c = X\_{c1}\delta + u$$

Then

$$\hat \delta\_{OLS} = \left(X'\_{c1} X\_{c1}\right)^{-1}X'\_{c1}y\_c $$

$$=\left(X'\_{c1} X\_{c1}\right)^{-1}X'\_{c1}X\_{c1}\beta\_1 + \left(X'\_{c1} X\_{c1}\right)^{-1}X'\_{c1}X\_{c2}\beta\_2 \\+ \left(X'\_{c1} X\_{c1}\right)^{-1}X'\_{c1} X\_{c3}\beta\_3 + \left(X'\_{c1} X\_{c1}\right)^{-1}X'\_{c1}\varepsilon\_c$$

Simplifying and taking expectations conditional on $X\_{c1}$ we get

$$\mathbb E(\hat \delta\_{OLS} \mid X\_{c1})= \beta\_1 + \left(X'\_{c1} X\_{c1}\right)^{-1}X'\_{c1}\mathbb E(X\_{c2} \mid X\_{c1})\beta\_2 \\+\left(X'\_{c1} X\_{c1}\right)^{-1}X'\_{c1} \mathbb E(X\_{c3}\mid X\_{c1})\beta\_3 + \left(X'\_{c1} X\_{c1}\right)^{-1}X'\_{c1}\mathbb E(\varepsilon\_c\mid X\_{c1})$$

By the regressor independence and exogeneity assumptions, as well as due to the centering we have

$$\mathbb E(X\_{c2} \mid X\_{c1}) = \mathbb E(X\_{c2} ) = 0$$
$$\mathbb E(X\_{c3} \mid X\_{c1}) = \mathbb E(X\_{c3} ) = 0$$
$$\mathbb E(\varepsilon\_c \mid X\_{c1}) = 0$$

So we are left with

$$\mathbb E(\hat \delta\_{OLS} \mid X\_{c1})= \beta\_1 \implies \mathbb E(\hat \delta\_{OLS}) = \beta\_1$$


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