Omitted variables problem I'm studying the omitted variables problem.
My model is:
$E[y|x_1,x_2,...,x_k,q]=\beta_0+\beta_1x_1+...+\beta_k x_k + \gamma q$

From the first equation, I write the population model as
$y= \beta_0+\beta_1x_1+...+\beta_k x_k + \gamma q + \nu$
where, $\nu$ is the error, and I assumed $E[\nu |x_1,...,x_k,q]=0$
One way to handle this unobservability is to put $q$ in the error term. Without any loss of generality, I assume $E[q]=0$. By doing this, the population model reads as
$y= \beta_0+\beta_1x_1+...+\beta_k x_k + u$
where $u \equiv \gamma q + \nu$, and $E[u]=0$
If $q$ is correlated with one of the regressors, the endogeneity problem arises.
Then, the book says


where equation $(4.19)$ : $y= \beta_0+\beta_1x_1+...+\beta_k x_k + \gamma q + \nu$
How can I run a linear projection (that I guess is the same of a linear regression) on $q$, if $q$ is unobserved?
The book's reference is : Wooldridge - econometric analysis of cross section and panel data, last edition (pages 65-66).
 A: Section 4.1 of the Wooldridge text describes what's being considered:

The correlation of explanatory variables with unobservables is often due to self-selection: if agents choose the value of [independent variable] $x_j$, this might depend on factors ($q$) that are unobservable to the analyst. A good example is omitted ability in a wage equation, where an individual’s years of schooling are likely to be correlated with unobserved ability. We discuss the omitted variables problem in detail in Section 4.3.

In that context, the "linear projection" in Equation 4.23 (from Section 4.3) is a hypothetical projection. (It might only be considered a linear "regression" if you had actual values to work with.) If you somehow knew how $q$ is associated with the $x_j$, then you could perform the substitutions shown to come up with the critical result: if $q$ is uncorrelated with $x_j$, then in linear regression it introduces no bias into the estimate of the corresponding $\beta_j$; if $q$ is correlated with $x_K$ then (Equation 4.24):
$$\text{plim  } \hat\beta_K = \beta_K + \gamma \left[\text{Cov}(x_K,q)/\text{Var}(x_K) \right] $$
where $\text{plim  } \hat\beta_K$ is the limit in probability of the coefficient estimate in the model omitting $q$, and $\beta_K$ is the true coefficient. In practice, if you can make reasonable assumptions about that correlation, it allows you to gauge the nature of the bias.
