Home prices rose a blistering 18.8 percent in 2021, and rent has climbed 17.6 percent nationwide over the last year, industry data shows. But those prices, the result of a severe supply shortage fueled by municipal government restrictions across the country, haven’t fully shown up in inflation figures because leases are typically annual.

“The housing shortage is going to push up the overall [consumer price index] to uncomfortable positions for the Federal Reserve,” National Association of Realtors chief economist Lawrence Yun said. “Consequently, this high inflation that we have is certainly not transitory, and it’s going to remain stubbornly high through the end of the year.”

https://www.politico.com/news/2022/03/18/housing-costs-infla...

Also, the CPI DOES include cost of housing - https://www.bls.gov/cpi/factsheets/owners-equivalent-rent-an...