A long time ago when I was at the RBNZ, I showed that the typical positive correlation between money and prices over the business cycle (deviations of money and CPI from trend) has broken down in New Zealand. I am sure the same is true in other inflation-targeting countries.
Successful inflation targeting rendered inflation stationary while money, which is not targeted by the CB, continued to fluctuate, hence no correlation. In economic theory, excess money growth fuels asset price inflation. People buy houses, stocks, etc. Housing is the dominant asset market in New Zealand. Here is a graph of the rate of growth of housing prices and the M2 growth lagged one year. Monthly data are from the RBNZ website (I calculated the quarterly data as monthly averages).
Visually, there is a theory-consistent positive correlation. There is a trend in both growth rates. It seems that this positive correlation between trending housing price inflation and lagged money growth may imply that money affects house price growth in addition to other micro and macro shocks (e.g., building restrictions, land shortages, zoning policies, population growth, etc.).