The way I think about it as that these laws only need to affect some families that are on the margin. Are there some families that are just on the fence and tossing up whether to have a third child? It seems likely to me. As long as this cost is a significant one, it just needs to move the needle for these groups to show up in aggregate effects. Lots of people will make the same choice either way, whether it's to have a third child or not have a third child. Our predictions are for people right on the margin.
>Controlling for income is really really hard since basically everything about lifestyle has some relation to income.
We use county-by-year fixed effects to take out the average effect of birth rates in that county and year, whatever form that variation takes. So if it's just that Cook County in Illinois got richer one year and so had more children, this will be absorbed by the fixed effects, and our car seat impacts are considered relative to this baseline. We also control for birth rates for that number of children and age combination, so we're comparing essentially two families with a six year old and a three year old, but in one state/year they both have to be in car seats, and in the other one only one child has to.
Especially when you consider that something like a state law mandating car seats is only gonna exert upward pressure on the cost of a kid for people with higher incomes. Basically the poors are gonna pile into whatever vehicle they have and the law can fuck off. Nobody says "gee we shouldn't have another kid because we'll need X, Y and Z' (one of those being a bigger car in this case). That's just not how people make decisions at that level. If you want a kid you get a kid and you roll with it. You know it's gonna be expensive but you just deal.
Controlling for income is really really hard since basically everything about lifestyle has some relation to income. Controlling for income with sufficient accuracy that you can tease out an effect that itself correlates with income is gonna be neigh on impossible when you're looking for something that's a 2nd order effect. This is like trying to predict trout populations in Canada by chemically analyzing reindeer piss in Russia. You can do it but it's an exercise in measurement and it's gonna have zero practical utility since sooooo many other things will have affects that dwarf those of what you're measuring.
I think it's far more likely that rear seats have shrunk and car seats have grown because fertility rates have declined and regulators and manufacturers are therefore emboldened to do things that would be laughable in a world where three kid households were more common.