Clearly you don't live in San Francisco, New York, or any of the many other desirable places where prices have gone up at a compound rate of 6 percent or more for decades. Nor do you live in Detroit, where folks would be delighted if their houses appreciated at all. In high COL areas with constrained available land, the pressure on prices results from concentration of high tech and high finance professionals plus demand from overseas investors worried about keeping their money safe. More demand than supply. Very little correlation with the general rate of inflation except as a contributing factor. Inflation can't help Detroit, where supply dwarfs demand. So much supply that a lot of it is left to rot, like leftover unsold produce.
I have done very well investing in Bay Area real estate, far better than the rate of inflation. Shiller is an overly coiffed academic with a product to sell. Interesting theory, but not accurate. All else is NOT equal.