  We study the long-term spectral and timing behaviour of LS I +61$^{\circ}$303
in hard X-rays (15--150 keV) using $\sim$10 years of survey data from the
$Swift$ Burst Alert Telescope (BAT) monitor. We focus on the detection of long
periodicities known to be present in this source in multiple wavelengths. We
clearly detect three periods: the shorter one at 26.48 days is compatible with
the orbital period of the system; the second, longer, periodicity at 26.93
days, is detected for the first time in X-rays and its value is consistent with
an analogous temporal feature recently detected in the radio and in the
gamma-ray waveband, and we associate it with a modulation caused by a
precessing jet in this system. Finally, we find also evidence of the long-term
periodicity at $\sim$1667 d, that results compatible with a beat frequency of
the two close, and shorter, periodicities. We discuss our results in the
context of the multi-band behaviour of the physical processes of this source.
