The all-gap Phillips curve (PC) explains inflation by expected inflation and an activity variable such as output or the unemployment rate, but with both inflation and the activity variable measured relative to their stochastic trends and thus as gaps. We study this relationship with minimal auxiliary assumptions and under rational expectations (RE). We show restrictions on an unobserved-components model that identify the Phillips curve parameters, first with an autonomous output gap and second with output and inflation gaps following a VAR. For the US, UK, and Canada both cases yield all-gap PCs with slopes of the expected signs,
but there is little support for the restrictions implied by RE.