Bad controls problem: Introducing Selection Bias

[Maxence Morlet]

Dear Statalisters,

More of a theoretical econometric question than a practical question, so I don't think a data extract is needed, but would be happy to find one if anyone thinks otherwise.

Suppose we are in a case in which university degrees are randomly assigned (as in chapter 3.2.3 of Angrist and Pischke, 2009). We then compare earnings of people with a university degree and the earnings of people without, and this observed difference in outcomes will be the causal effect of a university degree on earnings by virtue of random assignment, correct?

However, one way through which individuals with university degrees tend to earn more than individuals without is because the former sort into different occupations than the latter, as permitted by their university degree. So university degree affects the choice of occupation, which generally occurs after university.

Here are my questions:

- If we have occupational dummies and add them to the regression of earnings on a dummy for university degree, then in the situation in which university degrees are randomly assigned, a simple mean comparison in earnings between individuals with and without university degrees is no longer the causal effect, correct?


- If the last statement is true, is it due to the fact that by conditioning earnings on occupational fixed-effects as well, we have introduced a selection bias equal to the difference between the counterfactual in earnings (without university degree) of someone with a university degree who chose a given occupation and the earnings of someone without a university degree who chose that same occupation?


In other words, when controlling for occupation in this context, do we introduce a new source of unobserved heterogeneity by segmenting individuals into multiple categories within each occupational stratum, even though university degree assignment is still random?

Does the selection bias arise for any another reason?


Please let me know if the question is unclear, I would be very happy to reformulate it.

[Maarten Buis]

It depends on how you think about the causal effects. I would say that occupation is one of the mechanisms through which a degree influences income. If that is the case then a simple mean comparison is the causal effect (assuming random assignment of university degrees). In other words occupation is an intervening variable rather than a confounding variable in this context. You normally don't want to control for intervening variables, because then you remove part of the effect of degree that you actually understand why it is there.

[George Ford]

I suppose the total effect can be decomposed (and probably estimated in some type of multilevel model) into (1) income effect from education due to occupation choice and (2) income from education within an occupation. Sounds kind of interesting, but may have been done.

[Maarten Buis]

I suppose the total effect can be decomposed (and probably estimated in some type of multilevel model) into (1) income effect from education due to occupation choice and (2) income from education within an occupation. Sounds kind of interesting, but may have been done.

That idea is behind the ISEI index for occupational status. I am away from my office, so I don't have the reference ready, but Google ISEI Ganzeboom if you are interested.

[George Ford]

As I suspected.