Abstract Recent observations of the periodic fast radio burst source 180916.J0158+65 (FRB 180916) find small linear polarization position angle swings during and between bursts, with a burst activity window that becomes both narrower and earlier at higher frequencies. Although the observed chromatic activity window disfavors models of periodicity in FRB 180916 driven solely by the occultation of a neutron star by the optically thick wind from a stellar companion, the connection to theories where periodicity arises from the motion of a bursting magnetar remains unclear. In this Letter, we show how altitude-dependent radio emission from a magnetar, with bursts emitted from regions that are asymmetric with respect to the magnetic dipole axis, can lead to burst activity windows and polarization consistent with the recent observations. In particular, the fact that bursts arrive systematically earlier at higher frequencies disfavors theories where the FRB periodicity arises from forced precession of a magnetar by a companion or fallback disk, but is consistent with theories where periodicity originates from a slowly rotating or freely precessing magnetar. Several observational tests are proposed to verify/differentiate between the remaining theories, and pin down which theory explains the periodicity in FRB 180916.