Does omitted variable bias 'contaminate' coefficients for regressors not correlated with the omitted variable?

Question

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.

Answer 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?

Answer 2

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

$$\mathbb E(\hat \delta_{OLS} \mid X_{c1})= \beta_1 \implies \mathbb E(\hat \delta_{OLS}) = \beta_1$$