We develop and estimate a structural model of inflation that allows for a fraction of firms that use a backward-looking rule to set prices. The model nests the purely forward-looking New Keynesian Phillips curve as a particular case. We use measures of marginal cost as the relevant determinant of inflation, as the theory suggests, instead of an ad hoc output gap. Real marginal costs are a significant and quantitatively important determinant of inflation. Backward-looking price setting, while statistically significant, is not quantitatively important. Thus, we conclude that the New Keynesian Phillips curve provides a good first approximation to the dynamics of inflation.