The macroblog explains why the Bureau of Labor Statistics uses the rental equivalence approach to measure the cost of shelter rather than house prices.
What is the cost of owning a home to someone who already owns a home? Well, it's the opportunity cost of living in your home rather than renting it out. So, in this sense, there is no "mismeasurement"...
However, house prices may be a leading indicator of inflation.
I think it entirely possible that assets can provide a leading signal of a generalized, monetary inflation. But admittedly, this is mostly an article of faith and not science--the statistical link between these cost-of-living measures and asset prices is very tenuous. And economists have not been able to map the link that leads from excess money growth to a generalized inflation. So until that day, I think one who ignores asset prices when looking for signs of rising inflation does so at some peril.But I am reminded of Japan. For all its asset bubble in the 1990s, CPI inflation there peaked out around 3-4 percent, not particularly high. Of course, what followed was deflation.
Once you have an asset bubble, it's bad if it is followed by inflation, but it may be worse if it's not. Of course, if you deny that there is a bubble, then there's nothing to worry about.