# What (and how) to control for in adjusted wage-gap regressions?

Background: The unconditional wage gap between some group and the rest is often estimated using a regression of the form $$w_{it} = \alpha + g_{it}'\theta +\epsilon_{it}$$ where $$w_{it}$$ is log of real wage, $$g_{it}$$ the binary group indicator, and $$\epsilon_{it}$$ an iid error term. If omitted productivity variables are correlated with $$g_{it}$$, then $$\theta$$ might instead serve as a proxy for these omitted variables.

The adjusted wage gap is usually estimated by including a set of control $$X$$ capturing productivity differences between individuals. It seems to me there is no consensus on what variables should be included in $$X$$ and how these should be included. I have seen hugely different specifications and little discussion on what is correct. Two things are particularly confusing to me

Questions

1. Is there an argument to be made to include only few, exogenous controls to avoid the issue of bad controls in $$X$$? In particular, I would worry that some controls like occupation or industry are examples of 'colliders'

1. I have seen versions where $$X$$ is just included as additional controls $$w_{it} = \alpha + g_{it}'\theta + X'\beta + \epsilon_{it}$$ and I have seen versions using saturated models with interactions of the group variable with the additional controls $$w_{it} = \alpha + g_{it}'\theta + X'\beta + \sum \gamma_j x_{it}'g_{it}' + \epsilon_{it}$$ I think the first version is closer to what I would think of adjusted wage gap. Indeed, I am not sure how I would interpret $$\theta$$ in the saturated model?