Lottery estimates suggest oversubscribed urban charter schools boost student achievement markedly. But these estimates needn't capture treatment effects for students who haven't applied to charter schools or for students attending charters for which demand is weak. This paper reports estimates...

Marginal outcome tests compare the expected effects of a decision on individuals who are of different races but at the same indifference point of the decision-maker. I present a simple formalization of how such tests can detect racial bias, defined as a deviation from accurate statistical...

We develop new tools for causal inference in settings where exogenous shocks affect the treatment status of multiple observations jointly, to different extents. In these settings researchers may construct treatments or instruments that combine the shocks with predetermined measures of shock...

Many studies in economics use instruments or treatments which combine a set of exogenous shocks with other predetermined variables by a known formula. Examples include shift-share instruments and measures of social or spatial spillovers. We review recent econometric tools for this setting, which...

Lottery estimates suggest oversubscribed urban charter schools boost student achievement markedly. But these estimates needn't capture treatment effects for students who haven't applied to charter schools or for students attending charters for which demand is weak. This paper reports estimates...