Housing construction is a lagging indicator because of the regulatory and capital requirements to build a whole bunch of homes. So what we see right now is that demand is growing because the economy is growing again, but supply has not yet caught up. Result: fewer affordable houses.
In my neighborhood a new row house development was just breaking ground when the shit hit the fan in 2008, and of course they stopped. Well, they're building fast today, and selling every unit, but they're still not done. It just takes a while to build houses.
The only connection is Millenials is that they are the youngest generation currently looking to buy a home, so they are more price-sensitive than Gen Xers or Boomers.