2026 Working Papers

Induced order statistics (IOS) arise when sample units are reordered according to the value of an auxiliary variable, and the associated responses are analyzed in that induced order. IOS play a central role in applications where the goal is to approximate the conditional distribution of an outcome at a fixed covariate value using observations whose covariates lie closest to that point, including regression discontinuity designs, k-nearest–neighbor methods, and distributionally robust optimization. Existing asymptotic results allow the dimension of the IOS vector to grow with the sample size only under smoothness conditions that are often too restrictive for practical data–generating processes. In particular, these conditions rule out boundary points, which are central to regression discontinuity designs. This paper develops general convergence rates for IOS under primitive and comparatively weak assumptions. We derive sharp marginal rates for the approximation of the target conditional distribution in Hellinger and total variation distances under quadratic mean differentiability and show how these marginal rates translate into joint convergence rates for the IOS vector. Our results are widely applicable: they rely on a standard smoothness condition and accommodate both interior and boundary conditioning points, as required in regression discontinuity and related settings. In the supplementary appendix, we provide complementary results under a Taylor/Hölder remainder condition. Our results reveal a clear trade–off between smoothness and speed of convergence, identify regimes in which Hellinger and total variation distances behave differently, and provide explicit growth conditions on the number of nearest neighbors.

We study how teachers' rule violations in grading affect students' ethical behavior. Using administrative data from high-stakes exams, combining teacher-assigned internal scores with externally graded national exam scores, we track teacher grading violations and subsequent student cheating. We explore three potential mechanisms: imitation (learning that rules can be broken), positive reciprocity (responding favorably to favorable treatment), and negative reciprocity (retaliating against unfavorable treatment). Exploiting within-student variation in exposure to different teachers, we find students are significantly more likely to cheat when teachers break the rules to their detriment (systematically undergrading), consistent with both imitation and negative reciprocity. However, when teachers systematically overgrade, responses vary by community structure. In heterogeneous communities, overgrading increases student cheating, suggesting imitation dominates. In homogeneous communities, students respond by cheating less, consistent with positive reciprocity dominating. This pattern holds across multiple homogeneity measures, including surname concentration and residential clustering. Survey measures of mutual respect and support between students and teachers confirm this pattern.

n the linear-in-means model, endogeneity arises naturally due to the reflection problem. A common solution is to use Instrumental Variables (IVs) based on highe rorder network links, such as using friends-of-friends’ characteristics. We first show that such instruments are unlikely to work well in many applied settings: in very sparse or very dense networks, friends-of-friends may be similar to the original links. This implies that the IVs may be weak or their first stage estimand may be undefined. For a class of random graphs, we use random graph theory and characterize regimes where such instruments perform well, and when they would not. We prove how weak-IV robust inference can be adapted to this environment, and how scaling the network can help. We provide extensive Monte Carlo simulations and revisit empirical applications, showing the prevalence of such issues in empirical practice, and how our results restore valid inference.

Most digital platforms are funded through advertising rather than direct payments. Why? We argue that three main factors could explain this prevalence: users are more sensitive to monetary prices than to ad loads, microtargeting improves the match quality between users and ads, and platforms can personalize ad loads and thus price discriminate. We conduct a field experiment on Facebook with 1,810 users who install a browser extension that (i) hides nearly all ads or (ii) replaces microtargeted ads with untargeted ones. We find that hiding 82% of ads increases time on the platform by only 6%, showing that users are highly insensitive to ad loads. Removing targeting sharply reduces ad clicks and long-run engagement, indicating that targeting increases the match quality between users and ads. Finally, two-thirds of ad-load variation occurs across users, consistent with ad-load discrimination. Counterfactual simulations indicate that an ad-funded model performs at least as well as a subscription model i terms of profits and delivers higher consumer surplus. The key mechanism is that users are much less sensitive to ad loads than to monetary prices, making advertising a relatively efficient revenue source.

We examine the causal effect of exposure to fast-food outlets on adolescent z-BMI using data from the UK Millennium Cohort Study. We develop a novel approach to modelling persistence in adolescent BMI by clustering early childhood BMI trajectories, capturing biologically and behaviourally persistent obesity risk profiles. Including these profiles in the model allows us to separate baseline susceptibility from contemporaneous environmental effects. For identification, we exploit the near-universal transition from primary to secondary school in Great Britain, which creates plausibly exogenous variation in exposure to fast-food outlets around schools. Using this variation, we find that adolescents with at least one major-brand outlet within 400 metres of their school have, on average, a 0.158 standard-deviation higher z-BMI. Effects decline at larger distances, are limited around the home, and do not extend to other food outlets.

Collective action problems emerge when individual incentives and group interests are misaligned, as in the case of climate change Individuals involved in collective action problems are often considered to have two options : contribute towards a public solution or free-ride. But they might also choose a third option of investing in a private solution such as local climate change adaptation. Here we introduce a collective action game featuring wealth inequality caused by luck or merit and both public and private solutions with participants from 34 countries. We show that the joint existence of wealth inequality and private solutions has a consistent effect across countries: participants endowed with higher income choose the private solution almost twice as often as those endowed with lower income ; and this finding cannot be explained by different sources of wealth (luck vs. merit) or by cultural or economic factors. We also show that preferences for private solutions undermine support for public solutions, resulting in wealth inequality increasing in every country. In contrast, we identify two universal pathways to successful public solution provision: early contributions to public solutions and conditional cooperation. Our findings highlight the ubiquity of the ‘private solution problem’ and its potential consequences for global collective action problems.

Is deplatforming effective in reducing toxicity on social media? To answer this question we study a policy change on Reddit in June 2020 which led to a simultaneous ban of thousands of forums containing hateful content, but not the users of these forums. We use data on the near universe of comments left on Reddit to examine the impact of the ban on user behaviour in a differences-in-differences design. We find that the most active users of banned subreddits comment more after the policy change and substitute to new forums in the weeks after the ban. The increase in activity persists in the long run, but is not associated with higher toxicity: instead, the comments left by affected users outside banned subreddits contain 20% fewer instances of hate speech. We do not find evidence that the policy leads to lower quality of engagement, negative spillover effects or recreation of banned subreddits elsewhere on the platform. Overall, the results suggest that moderation targeting toxic digital spaces can be effective in combating hate speech without lowering user engagement, and thus can be aligned with platforms incentives.

This chapter explores the intersection of religion and economics on the 250th anniversary of Adam Smith’s The Wealth of Nations, first published in 1776. While Smith is often viewed as a secular figure in economics, his work was deeply influenced by the moral philosophy of his time, which was shaped by Christian thought. I discuss how economists think about the religious themes in Smith’s work in the 21st century and review what we know today about the connection between religion and economic outcomes.

Public support for raising minimum wages as a policy response to economic inequality is increasing; however, empirical evidence from highly informal and weakly regulated labour markets remains limited. This study estimates the impact of minimum wage increases on earnings and hours worked in Pakistan, drawing on 21 waves of nationally representative Labour Force Survey data between 1992 and 2021. By leveraging national time variation in statutory minimum wages and pre-policy district exposure, proxied by the proportion of workers earning below the minimum wage prior to policy changes, we find that increases in the minimum wage are associated with statistically significant but modest gains in real hourly earnings, with stronger wage pass-through observed in local labour markets with higher initial exposure. The benefits are disproportionately greater for male workers; however, the policy has achieved only limited and uneven progress in reducing gender pay disparities. On the intensive margin, minimum wage increases are associated with reductions in hours worked, particularly among women. This pattern is consistent with adjustment through hours in segments characterised by part-time work and weaker compliance. Overall, the findings indicate that minimum wage policy can increase earnings in low-wage areas under conditions of partial compliance, yet has limited capacity to address persistent structural gender inequality in highly informal contexts. These results underscore the need for stronger enforcement and complementary, gender-sensitive labour market interventions

Abstract: We study how structural disadvantage (e.g., race, class, gender) shapes the formation of subjective beliefs about the returns to effort. We formalize and distinguish between psychological constructs such as locus of control, self-efficacy, and grit, and study their response to structural disadvantage and policy interventions. In our model, individuals share the same (unknown) innate ability, and beliefs about success can only be updated through effort. However, agents fail to internalize the dynamic feedback between effort and belief formation. Structural disadvantage raises the threshold of belief required to justify effort, increasing the likelihood of falling into a pessimistic low-effort trap. We characterize the conditions under which psychological interventions that bound beliefs from below and enhance grit improve welfare, and when such interventions must be complemented by policies that relax external constraints to be effective.

Using Danish administrative data linked to two independent, validated postpartum depression screenings, we study how postpartum mental health shocks shape women’s labor market trajectories. Event-study estimates show no pre-birth differences in trends between depressed and non-depressed mothers, but persistent employment gaps that widen immediately after birth. Health-care utilization patterns indicate that these differences reflect acute mental health shocks rather than pre-existing trends. The penalties are concentrated among less educated mothers and those in less family-friendly jobs. Our results highlight postpartum depression as a meaningful and unequal contributor to the motherhood penalty.

We leverage the introduction of the first antibiotic therapies in 1937 to examine the long run effects of early childhood pneumonia on adult educational attainment, employment, income, and work-related disability. Using census data, we document large average gains on all outcomes, alongside substantial heterogeneity by race and gender. On average, Black men exhibit smaller schooling gains than white men but larger employment and earnings gains. Among Black men (and women), we identify a pronounced gradient in gains linked to systemic racial discrimination in the pre–Civil Rights era: individuals born in more discriminatory Jim Crow states realized much smaller gains than those born in less discriminatory states. There is no similar gradient among white Americans. Women of both races exhibit smaller education and earnings gains than men on average, consistent with cultural and institutional barriers to women’s work. Our findings highlight the role of opportunities in shaping the extent to which investments in early-life health translate into longer run economic gains.