Forecasts of inflation in the United States since the mid eighties have had smaller errors than in the past, but those conditional on commonly used variables cannot consistently beat the ones from univariate models. This paper shows through simple modifications to the classical monetary model that something similar occurred in those major Latin American economies that achieved their own "Great Moderation." For those countries that did not attain macroeconomic stability, inflation forecasting conditional on some variables has not changed. Allowing the parameters that determine Granger causality to change when the monetary regime does, makes possible the estimation of parsimonious inflation models for all available data (eight decades for one country and five for the others). The models so obtained ouperform others in pseudo out-of-sample forecasts for most of the period under study, except in the cases when an inflation targeting policy was successfully implemented.