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Shift current is a bulk photovoltaic effect (BPVE) that arises in noncentrosymmetric materials and results in the generation of a direct current under continuous wave illumination. It holds promise for next generation solar cells, but the role of electron-hole interactions is just beginning to be explored, with some theories suggesting it will suppress the shift current and others seeing an enhancement. Here, we calculate the shift current including excitonic effects using a new ab initio approach based on a perturbative sum-over-states formalism in the exciton basis, where the exciton states are obtained within the ab initio GW plus Bethe Salpeter equation (GW-BSE) approach. We show that the sum-over-states formalism gives results identical to a real-time nonequilibrium Green's function approach within the adiabatic GW approximation. We apply this method to calculate the shift current of the ferroelectric perovskite BaTiO3 including quasiparticle and exciton effects and see that electron-hole interactions give rise to both a red shift and enhancement of the shift current near the band edge. Our calculation demonstrate a rigorous and efficient method for studying excitonic effects in the shift current.