# Omitted variable problem

I'm studying the cases in which the endogeneity problem arises in OLS regression.

Suppose we have the following population equation:

$$y=\beta_0 +\beta_1 x_1 + ... + \beta_k x_k + \gamma q + \epsilon$$

and say $$E(\epsilon | x,q)=0$$, such that: $$E(y|x,q)=\beta_0 +\beta_1 x_1 + ... + \beta_k x_k + \gamma q$$

Suppose $$q$$ is unobserved and so it goes into the error term, thus your population equation reads as

$$y=\beta_0 +\beta_1 x_1 + ... + \beta_k x_k + \nu$$ , where $$\nu=\gamma q + \epsilon$$

Then, the slides says, nothing is lost assuming that $$E(q)=0$$, because an intercept is included in the basic equation, so that $$E(\nu)=0$$.

Why is fine assuming that $$E(q)=0$$, because an intercept is included in the basic equation?

$$y=\beta_0 +\beta_1 x_1 + ... + \beta_k x_k + \gamma q +\epsilon.$$
Say that the mean value of $$q$$ is $$\bar q$$. Then centering $$q$$ around its mean gives $$q_c=q-\bar q$$. Substitute into the above and collect constant terms:
$$y=(\beta_0 + \gamma \bar q)+\beta_1 x_1 + ... + \beta_k x_k + \gamma q_c + \epsilon.$$
Any offset of the unobserved $$q$$ in this situation will be included in the intercept of a model that's based on the observed predictors. It won't affect the estimates of the coefficients for the observed predictors $$x_i$$, or the bias in the coefficient for any $$x_i$$ correlated with the unobserved $$q$$.
Two warnings. First, omitting an intercept in such a model will lead to problems. Second, omitted-variable bias can be more of a problem in other types of models, as explained here for a probit model. In OLS there is no bias in the coefficient for an observed predictor uncorrelated with the unobserved predictor. In models without an error term like $$\epsilon$$ in OLS to capture excess heterogeneity resulting from $$q$$, an unobserved/unmodeled predictor can lead to bias in coefficients for all included predictors.