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
- 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'
- 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?
