Hybrid halide perovskites exhibit nearly 20% power conversion efficiency, but the origin of their high efficiency is still unknown. Here, we compute the shift current, a dominant mechanism of the bulk photovoltaic (PV) effect for ferroelectric photovoltaics, in CH₃NH₃PbI₃ and CH₃NH₃PbI(3-x)Cl(x) from first-principles. We find that these materials give approximately three times larger shift current PV response to near-IR and visible light than the prototypical ferroelectric photovoltaic BiFeO₃. The molecular orientations of CH₃NH₃⁺ can strongly affect the corresponding PbI₃ inorganic frame so as to alter the magnitude of the shift current response. Specifically, configurations with dipole moments aligned in parallel distort the inorganic PbI₃ frame more significantly than configurations with near-net-zero dipole, yielding a larger shift current response. Furthermore, we explore the effect of Cl substitution on shift current and find that Cl substitution at the equatorial site induces a larger response than does substitution at the apical site.