Wednesday, February 10, 2021
Social scientists often possess fragmented information about subsets and aspects of the complex causal processes they study. Research on police-civilian interactions, for example, is complicated not only by undocumented interactions, but inconsistent recording of events within documented interactions. These data constraints can lead to a proliferation of incompatible analytic approaches relying on contradictory unstated assumptions, impeding scientific progress on important questions like the severity of racial bias in policing. Nonparametric sharp bounds, or the tightest possible range of answers consistent with available data, offer a path forward: claims outside the bounds can be immediately rejected, and claims inside the bounds must explicitly justify the additional assumptions that enable tightening. However, we show proving sharpness is NP-hard for broad classes of data constraints and causal quantities, rendering this approach computationally infeasible for even moderately sized causal processes. We present an efficient spatial branch-and-bound procedure with a theoretical guarantee that we term "ε-sharpness," indicating the worst-case looseness factor of the relaxed bounds relative to the (unknown) completely sharp bounds. The procedure is guaranteed to attain complete sharpness with sufficient computation time. We present results on asymptotic validity of and conservative statistical inference for ε-sharp bounds. The technique is illustrated using simulations using common research designs in the study of policing.