In accordance with Wooldridge's text, here is a classic bivariate wage equation:
$$
\textrm{wage} = \beta_{0} + \beta_{1} \textrm{education} + u,
$$
where we relate wage to observed education and other unobserved factors in $u$. These factors might include labor force experience, tenure, and innate ability. This will not hold if education and $u$ are correlated. Education is likely correlated with experience and one’s innate work ethic. If we were to estimate the foregoing equation, we would likely observe a positive relationship, even though it may be partly explained by unobserved factors contained (assumed to be fixed) in $u$. Intuition would suggest that individuals with more education might also have more experience. This would bias our estimate of $\beta_{1}$. Thus, if $\beta_{1} > 0$, it may be overstated in this regression if we leave out a measure of experience or tenure. In non-econometric jargon, the coefficient we estimate on education to predict wage might be "doing some of the work" for these ignored factors (e.g., experience, tenure, work ethic, ability, etc.). Once we account for some of them, then the effect of education is likely to be less pronounced.
- Aside: I'm sure a case could be made, or even demonstrated with sample data, that education and experience may also be somewhat negatively correlated.
Given a random sample, this simple linear regression equation implies that $\textrm{E}[u_{i}|x_{i}] = 0$ for all $i = 1, 2, … n$. Put in words, the error has an expected value of zero given any value of the explanatory variable. Put in more simplified words, the other factors affecting wage are not related to education on average. We must account for these other (ignored) factors to understand what the 'true effect' of the return to education is on wage.
Suppose we estimate a new equation
$$
\textrm{wage} = \beta_{0} + \beta_{1} \textrm{education} + \beta_{2} \textrm{experience} + u,
$$
where we now include experience explicitly. Now, we will be able to measure the effect of education on wage, holding experience fixed. Put differently, we can assess the wage differential for two individuals separated by one year of education, and who both have the same amount of labor force experience.
The equation including labor force experience is more likely to have a ceteris paribus interpretation because we "controlled for" another factor that simultaneously affects the dependent variable. We pulled this factor out of $u$ and included it explicitly, since we know it would not satisfy the zero conditional mean assumption if contained in the error term. In a bivariate case (wage regressed on education, only) where we ignore other factors, we put the variable experience into the error term, and assume it is uncorrelated with education, which is a very tenuous assumption to make.
This also becomes clearer when you make your way to the next chapter of Wooldridge's text. Here is a quote from Chapter 3 of Introductory Econometrics: A Modern Approach:
The primary drawback in using simple regression analysis for empirical work is that it is very difficult to draw ceteris paribus conclusions about how x affects y: the key assumption—that all other factors affecting y are uncorrelated with x—is often unrealistic....Because multiple regression models can accommodate many explanatory variables that may be correlated, we can hope to infer causality in cases where simple regression analysis would be misleading (p. 68).
To go back to your concern, I hope you see why assuming these 'other factors' that affect wage and are also correlated with education should not be ignored. You might have asked yourself, are we done after including experience in the equation? Not quite. In our second equation with two explanatory variables, we assume $\textrm{E}[u|\textrm{education}, \textrm{experience}] = 0$, which implies that other factors affecting wage are, on average, unrelated to education and labor force experience. What about individual work ethic? As you can see, establishing causality is difficult if these variables (also correlated with education) are unobserved in practice.
- I encourage you to also view the following post, which offers another intuitive explanation for what happens when an omitted factor is correlated with your explanatory variable.
This is the primary motivation for multiple regression analysis. I hope this becomes more clear as you move into chapter 3 of the text. In sum, we can't assume, as we did in the first equation, that these "ignored factors" (e.g., experience) are "fixed" in $u$ and simultaneously uncorrelated with education.
