Using U.S. and euro area data, this paper presents a significant and proportional relationship between money growth and subsequent inflation when accounting for equilibrium velocity movements due to inflation regimes changes. These movements, driven by money demand adjustments to low-frequency Fisherian interest rate variations, are derived from consistent U.S. and euro area money demand specifications - after contradictory coexisting results are explained. Not accounting for equilibrium velocity and interest rate movements biases cross-country and time series dynamic money growth / inflation estimated relationships, and leads to the non-proportional, non-significant, and reverse causality results found in studies that include the post-1980 period.