アブストラクト：While average treatment effects are a common focus in causal inference, such measures often mask important distributional characteristics of counterfactual outcomes. This work considers the estimation of counterfactual densities under continuous treatments, thereby allowing richer and more detailed insights into the effects of interventions. We propose a Neyman-orthogonal moment condition that treats the conditional outcome density and the generalized propensity score as nuisance parameters. Leveraging this orthogonality within a debiased machine learning (DML) framework ensures the asymptotic normality of the parameter of interest, even when employing flexible machine learning methods for nuisance estimation. However, two challenges arise in finite samples due to the structure of the proposed moment conditions. First, the double summation within the moment conditions makes standard cross-fitting approaches susceptible to poor estimation performance, especially in small- or medium-sized datasets. To address this, we derive theoretical conditions under which DML can be implemented without sample splitting, thus mitigating performance degradation. Second, the proposed moment conditions involve integral over the nuisance estimates, meaning numerical integration errors can negatively affect estimation accuracy. Hence, it is desirable to use nuisance estimators that allow for easy analytical integration. As an illustrative example, we employ random forests as the nuisance estimator to satisfy these two requirements. We demonstrate the effectiveness of the proposed method through simulation studies.

Abstract：(Tentative) We draw on new granular data from cities around the world to study how the spatial distribution of income within cities varies with development. We document that in less-developed countries, average incomes of urban residents decline monotonically in distance to the city center, whereas income-distance gradients are flat or increasing in developed economies. We also show that urban neighborhoods with natural amenities – in hills and near rivers – are poorer than average in lessdeveloped countries and richer than average in developed ones. We hypothesize that these patterns arise due to the differences in the provision of residential and transportation infrastructure within cites. Using a quantitative urban model, we show that observed differences in residential and transportation infrastructure help explain a significant fraction of how the spatial income distribution within cities varies with income per capita.

Abstract: In one-shot delegation the principal optimally imposes an upper bound on actions when the agent is possibly upward-biased. In repeated delegation a biased type has greater incentive to signal her honesty in the early period than an honest type to induce the principal to relax the upper bound in the later period. Both types are locked in a signaling race and pool at downward-distorted actions in equilibrium. The optimal delegation set in the early period imposes a binding action lower bound despite the agent’s upward bias. The optimal action upper bound is less restrictive than that in a one-shot game.