This study analyses the long-run relationship between U.S house prices and non-housing Consumer Price Index (CPI) over the monthly period 1953 to 2016 using a quantile cointegration analysis. Our findings show evidence of instability in standard cointegration models, suggesting the possibility of structural breaks and nonlinearity in the relationship between house prices and non-housing CPI. This motivates the use of a time-varying approach, namely, a quantile cointegration analysis, which allows the cointegrating coefficient to vary over the conditional distribution of house prices and simultaneously test for the existence of cointegration at each quantile. Our results suggest that the U.S non-housing CPI and house price index series are cointegrated at lower quantiles only, with house prices over-hedging inflation at these quantiles. In addition, we also show that this result holds for higher price levels only. Using these two sets of results, we conclude that house prices act as an inflation hedge when the latter is relatively higher and the former is lower.