I sometimes agree with this theory, however in practice the past few years have shown us that interest rate increases have little effect after all. People were predicting that interest rates would crash the housing market and nothing significant happened.

All of the input costs to new construction go up over time (labor, materials, licenses, services) and new home pricing has an anchoring effect on the existing home market.

Citation:

https://fred.stlouisfed.org/series/MSPUS (review max, then switch to 10yr)