This is boring, but sort of crucial so please hang on. One of the biggest conceptual issues in putting together a "cost of living" index is how you should deal with the fact that some people own houses and other people rent them.
Note that this wouldn't be a big deal in a different kind of world where homeownership was for farmers (who aren't included in the U.S. CPI anyway) and the super-rich (who are rare) and where normal people purchased housing services from professional landlords. In a world like that, the relevant consumer price would clearly be the rent and everything else related to housing would be treated the same way we treat capital goods. Every statistical agency would agree on this, and we could all go home and be happy. But that's not how the world works, so instead agencies disagree:
— In America, the Bureau of Labor Statistics has basically decided that the relevant consumer price is the rent even though most people don't rent. Seventy-three percent of the shelter component of CPI is thus what they call "Owners' Equivalent Rent of Primary Residence" which you can think of as being the rental income you forego by living in your house rather than sleeping in your car and renting the house out to someone else. This is, I think, the most correct view of the situation from the standpoint of economic theory but it's at odds with a common sense understanding of what the cost of living is.
— In the eurozone, they've decided this is silly so the Harmonized Index of Consumer Prices simply excludes owner-occupied housing altogether. That has some obvious problems on its face, and the problems get especially problematic when you consider that there's huge country-to-country variation in homeownership. In Germany, only 40 percent of people live in owner-occupied housing while in Spain it's 85 percent. So there's nothing particularly "harmonized" about the index.
— In Sweden, they calculate the cost of owner-occupied housing in terms of mortgage payments which captures the folk understanding of what the cost of living consists of. The problem here is that if you cut interest rates, that's deflationary since mortgages get cheaper! Then Lars Svensson gets into tedious disagreements with his colleagues about how you have to target the forecast, and not just do macroeconomic forecasting and policy rates on separate tracks.
The main policy implication for Europe is that, once again, European Monetary Union was not thought-through correctly. Before you establish a single agency charged with stabilizing HICP in both Germany and Spain, you ought to come up with a HICP statistic that is comparable across countries. In the US context, the main upshot is that the adequacy of the Cost of Living Adjustment for Social Security has a lot to do with the state of your housing situation. An increase in rents is a real cost to renters, but to most homeowners it represents an increase in wealth.