Assignment at the Frontier: Identifying the Frontier Structural Function and Bounding Mean Deviations

This paper analyzes a model in which an outcome equals a frontier function of inputs minus a nonnegative unobserved deviation. Inputs may be endogenous (statistically dependent on the deviation). If zero lies in the support of the deviation given inputs -- an assumption we term assignment at the frontier -- then the frontier is identified by the supremum of the outcome at those inputs, obviating the need for instrumental variables. We then consider estimation in the presence of random error that is mean-independent of inputs. Motivated by the assignment at the frontier assumption, we regularize estimation by requiring the fitted deviation's distribution to maintain a minimum probability mass in a neighborhood of zero. Finally, we derive a lower bound on the mean deviation, using only variance and skewness, that is robust to a scarcity of data near the frontier. We apply our methods to estimate a firm-level frontier production function and mean inefficiency.