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House prices actually have nothing to do with CPI. "Shelter" is calculated using "Owners Equivalent Rent": https://www.bls.gov/cpi/factsheets/owners-equivalent-rent-an...

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I think, all things being equal, higher home prices should lead to higher rents, since at the margin people on the verge of buying a home would be more likely to choose to keep renting when prices are higher, thus increasing demand for rental units.

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