‘Modern’ Phillips curve theories predict inflation is an integrated, or near integrated, process. However, inflation appears bounded above and below in developed economies and so cannot be ‘truly’ integrated and more likely stationary around a shifting mean. If agents believe inflation is integrated as in the ‘modern’ theories then they are making systematic errors concerning the statistical process of inflation. An alternative theory of the Phillips curve is developed that is consistent with the ‘true’ statistical process of inflation. It is demonstrated that United States inflation data are consistent with the alternative theory but not with the existing ‘modern’ theories.