In terms of the general effects that some counties are richer than others, and some years people are richer than others, this will be captured by our fixed effects that control for overall birth rates for each combination of county, year, and number of children. So all effects are measured relative to other two child families in the same county and year. This captures the large majority of other regulatory and social changes that might be happening in that county and year.

Next, we also add income by year and income by county fixed effects. So if richer people in a given county have more children relative to the average of all two child families in that county and year, we'll also control for that. If richer people in a given year have more children compared to the average of all two child families in that county and year, we'll capture that too.

The point isn't that you need to add every one of these complicated fixed effects to find our result. It shows up in simpler tests too. It's just that you can control for a lot of potential confounding effects, and it still seems to be there.