Knowledge that Transforms

Certain extremum estimators have asymptotic distributions that are non‐Gaussian, yet characterizable as the distribution of the arg max of a Gaussian process. This paper presents high‐level sufficient conditions under which such asymptotic distributions admit a continuous distribution function. The plausibility of the sufficient conditions is demonstrated by verifying them in three examples, namely, maximum score estimation, empirical risk minimization, and threshold regression estimation. In turn, the continuity result buttresses several recently proposed inference procedures whose validity seems to require a result of the kind established herein. A notable feature of the high‐level assumptions is that one of them is designed to enable us to employ the Cameron–Martin theorem. In a leading special case, the assumption in question is demonstrably weak and appears to be close to minimal.

We investigate how formularies used by pharmacy benefit managers (PBMs) affect equilibrium manufacturer rebates for branded drugs through tiering and exclusion. We develop a theoretical model of multidimensional contracting in which a PBM negotiates with drug manufacturers over menus of formulary‐contingent rebates and chooses a formulary. We then estimate consumer demand responses to tier placement for statins using claims data from Princeton University, a large employer contracting with a single PBM to offer prescription drug coverage to its employees. Combining the theoretical model with demand estimates and observed list prices, we quantify how allowing for differential tier placement and exclusion affect equilibrium rebates. Our predictions are consistent with available aggregate rebate data, and we find that allowing a PBM to place branded drugs on preferred‐ and non‐preferred tiers can substantially increase negotiated rebate payments.

Apprenticeships play a key role in enabling successful school‐to‐work transitions in many countries, but in the presence of imperfect information, the specificity of this type of training may entail important costs for those working outside their training fields. I study this issue in one of the most prominent training settings, the German apprenticeship system. Using administrative data and a broad occupational classification, I find that 40% of individuals work in occupations different from their training. I estimate the cost of mismatch using vacancy instruments and extend methodological approaches in high‐dimensional selection settings. Lacking training in one's occupation entails an average wage penalty of 14%, the equivalent of two years of work experience. The penalty increases with the task distance between training and occupation. My findings suggest that retraining is crucial to mitigate the adverse consequences from imperfect information in specialized training settings.

Studies on agricultural technology adoption often focus on one input, practice, or package, which is analytically useful, but may overlook the complexities involved with multidimensional learning needed for a lot of agricultural decisions. In Kenya, we study farmers' dynamic learning (from oneself and others) and adoption decisions over six seasons after randomly inviting them to participate in agronomic research trials, comparing different combinations of inputs during three consecutive seasons. As a response to the trials, adoption increases steadily despite the absence of positive profits multiple seasons after exposure to the trials. Know‐how increases rapidly and faster for high skill farmers who experiment the most, at the cost of making new mistakes. The findings are consistent with a theoretical model with multidimensionality of input and practice decisions and differential learning from one's own experience by skills, where complementarities imply that adoption of an input requires finding how to re‐optimize other dimensions, which adds to the cost of adoption.