AC losses of stabilized multifilamentary MgB <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> tapes in applied and self magnetic field were studied systematically at different temperatures, self and external ac fields (les0.3 T), and frequency (3-2000 Hz) to assess the contribution of the magnetic sheath and its interplay with the superconducting filaments. The self-field losses were dominated by the hysteretic losses of the ferromagnetic Ni sheath while the contribution from the superconducting filaments was negligible. Although a considerable loss reduction was obtained with the use of NiCr alloy sheath, the resulting losses were still significantly higher than that of the superconductor as the NiCr sheath was weakly magnetic after thermal processing. In applied ac field, a complex loss behavior was found with loss contributions from the magnetic hysteresis of the sheath, the critical state of the superconducting filaments, and the coupling current across the stabilizing matrix inside the filaments. A long time-constant coupling current was greatly enhanced by the co-existence in the inner matrix of a highly magnetic iron buffer layer and a very low resistive stabilizing copper core. Compared to the normal state losses above the critical temperature T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</sub> , the overall losses in the superconducting state were increased at low frequencies by the coupling current loss but reduced at high frequencies due to the magnetic shielding by the coupling current.