Are real estate prices a leading indicator of rents, and therefore of consumer price inflation?

Residential real estate prices are now rising very fast in many OECD countries (we look at the United States, United Kingdom, Germany and France). It seems reasonable to think that a rise in real estate prices is passed on to a rise in rents, and thus ultimately to a rise in consumer prices. We seek to determine whether this transmission between residential real estate prices – > rents – > consumer prices actually occurs. We see that a rise in real estate prices leads to a rise in rents 2 to 2.5 years later in the United States, Germany and France, but not in the United Kingdom.