Heckman Selection (Selection vs. Outcome equation)

Question

A common example of the Heckman selection model involves wages, which are only observed if an individual chooses to participate in the labor force. The first stage probit model dependent variable is a labor force participation indicator. The second stage models wages (w), which are only observed conditional on working, so a correction term is added.

I’ve read a couple working papers recently that employ the Heckman model, but the selection and outcome equations seem less directly related. I’m wondering if that creates any issues. For instance, what if the second-stage here was total individual spending based on consumption data, and the first stage modeled labor force participation. Are there situations where something like this might be reasonable?

I personally haven’t worked with the Heckman model in quite some time and any clarification is much appreciated.

Answer 1

Remember that what you want to fix with Heckman's procedure is sample selection bias. For example, you observe wages only for people who work, so your selection equation should model an individual's decision rule of "working vs. not working." If you only observe college grades for people who actually went to college, your selection equation should capture the "selection into college" (which would include the variables influencing the admission committee, for instance).

Long story short: your selection equation should definitely be "directly related" to the outcome equation. If they're not, how do you expect to correct the sampling bias?