The traditional manner (in Economics at least) to explain an omitted variables bias involves the consideration of a Mincer type regression:$$w_{it}=\alpha+x_{it}'\beta+\gamma E_{i}+\alpha_{i}+\epsilon_{it}$$ where the LHS denotes wage of individual i at time $t$, $x_{it}'$ denotes a vector of controls , $E_{i}$ denotes education levels and $\alpha_{i}$ denotes individual specific heterogeneity, like ability, which is probably correlated with Education. As a result, if we do not “control” for ability in some manner, we obtain biased estimates of $\gamma$.
Now, I have come across some readings, in particular related to “bad controls.” What these readings point towards is that inclusions as controls of variables that could potentially themselves be outcome variables can lead to biases in parameters of interest.
Using such a line of reasoning, even if we did have a measure of ability, including it in a regression would point to this problem, because I can think of many reasons why education levels are a function of ability (the nobel prize winning model by Spence points towards exactly this hypothesis).
In an omitted variables scenario, we assume that a problem can exist if :
• $cov(.)$ between the included regressor(s) and the exluded one(s) is non 0
• The excluded regressor(s) are relevant.
This leads me to my question. If the omitted variable is supsected of having a non zero cov(.) with the included one, there are two possible scenarios:
One of the causes the other, leading to a dependence between the two
The two are caused by a third variable.
Case 2 seems to be fine, as long as this third variable is not important in determining $w_{it}$ . But Case 1 is definitely problematic. It seems to me that there may be a tradeoff between correcting for an omitted variables bias problem,and a bad control problem. How can one reconcile this?