# 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).

• My guess is that the proposed linear projection is hypothetical, to demonstrate the nature of the bias if you actually knew $q$ and its associations with the included predictors. Please add more information from the book, or a direct reference to the book (with a web link, if possible), if that's not how the matter is pursued later in the text.
– EdM
Commented Feb 11, 2023 at 20:38
• Thanks for your reply. References added. The book denotes $q$ an unobservable. I think that you are right, otherwise it would impossible to run that linear projection Commented Feb 11, 2023 at 20:45
• I don't think $q$ is "unobserved", it just gets "omitted". You should include more excerpts after (4.23) to see what the author tries to convey. Commented Feb 11, 2023 at 22:06
• @JohnM. In my opinion, "linear projection" of the response variable vector $y$ onto the space spanned by columns of $X$ is a geometric way of reflecting the nature of "linear regression". So loosely speaking, yes, they are the same. It is not very standard though, using "linear projection" with the explicit model $q = \delta_0 + \delta_1 x_1 + \cdots + \delta_Kx_K + r$ (the thing I feel uncomfortable is the error term $r$, when the term "linear projection" is used, "$r$" is usually suppressed because that's what "projection" really means). Commented Feb 12, 2023 at 3:19
• (continued) However, the point made by the author is still unambiguous because he placed down an explicit regression model. Commented Feb 12, 2023 at 3:21

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.

• Clear! The very last point, why the book talks about linear projection, and not a simple linear regression? Are they the same? Commented Feb 11, 2023 at 23:03
• @JohnM. I think that the distinction is between a "linear projection" of a random variable upon other random variables and the estimate of that projection that would be accomplished by a linear regression. If you don't have the values of $q$ you can at least imagine the "projection" even if you have no data for the "regression."
– EdM
Commented Feb 12, 2023 at 2:30