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?


1 Answer 1


Start with

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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.