In U.S. data, inflation and output are negatively related in the long run. A Bayesian VAR with stochastic trends generalized to be piecewise linear provides robust reduced-form evidence in favor of a threshold level of trend inflation of around 4%, below which potential output is independent of trend inflation, and above which, instead, potential output is negatively affected by trend inflation. Moreover, this negative relationship is quite substantial: above the threshold every percentage point increase in trend inflation is related to about 1% decrease in potential output per year. A New Keynesian model generalized to admit time-varying trend inflation and estimated via particle filtering provides theoretical foundations to this reduced-form evidence. The structural long-run Phillips Curve implied by the estimated New Keynesian model is not statistically different from the one implied by the reduced-form piecewise linear BVAR model.