https://www.zillow.com/research/birth-rates-home-values-20165/
There is always a question of causality, but I think we can discern high housing costs does, in fact, lead to reduced fertility:
“Children are expensive. The USDA estimates that the cost of raising a child is $233,000 as of 2017. But what may not be immediately obvious is that 1/3rd of this figure is attributed to housing alone. Indeed, one study has found that a 10 percent rise in housing prices leads to a 1.3 percent drop in birth rates.”
Granted, the mechanisms at play are extremely complex. High housing costs prevent labor from settling in the most productive locations, slowing growth. People tend to have fewer children when the economy is weaker.
h/t symplektisk