Papers

We study difference-in-differences (DID) estimation of average treatment effects on the treated when the researcher is uncertain about which pre-treatment periods satisfy the parallel trends assumption. Roth (2022) shows that pre-trend testing can induce bias, complementing the statistical literature on post-selection inference. We propose the model averaged difference-in-differences (MADID) estimator, a weighted average of the candidate $2\times2$ DID estimators. The weights are normalized exponential functions of the candidate residual sums of squares, so implementation requires no specification test. We derive MADID's asymptotic properties under conditions that make invalid comparisons detectably more variable than at least one valid comparison. Under these conditions, MADID is consistent when the parallel trends assumption holds for at least one pre-treatment period. Although its joint limiting distribution across post-treatment periods is generally non-Gaussian, inference can be conducted by subsampling or simulation.

When auction bidders misperceive their rivals, a seller can change not only the amount of information rent but also the states in which it is delivered. We ask when this payment-design opportunity survives a finite bidding interface, nonnegative-payment restrictions, and comparison with conventional auctions under the same beliefs. We formulate payment design as a linear program in nonnegative bidder rents. A single state can safely carry a report's rent when it is both cheapest for its intended type and least attractive to potential mimics. We characterize this alignment, bound the cost of a restricted payment architecture when alignment is imperfect, and use shadow prices when the rankings conflict. Own-type projection yields a unique optimal expected-payment rule for a fixed allocation. Contiguous bid bands implement truthful reporting at every value, but create pooling losses and within-band surplus. Under a separate smooth-belief model, feasible truthful rules and revenue upper bounds support comparisons with a first-price equilibrium under the same beliefs. A calibration using 1,000 eBay proof-set auction records illustrates a partition-dependent reversal: first price dominates all exactly truthful, no-subsidy rules on ten equal-probability bands in one scenario, while a richer contingent rule on ten equal-dollar bands outperforms it. An operator should choose interface resolution, payment dependence, and integrity tolerances jointly. Validate the probabilities that support incentives, compare formats under the same beliefs, and distinguish redesign from deployment. Any receipt advantage must cover implementation costs before it can imply a profit advantage.

Firms increasingly evaluate deployed AI agents using repeated human ratings, yet observed score changes may reflect the measurement process as much as changes in the agent itself. Firms also rarely know how long an evaluation remains informative as deployed systems evolve. We study reliability as a latent state using 2,611 rubric-scored production interviews from a deployed voice-and-video interviewing agent, overlapping human reviewers, the agent's change log, and support tickets. Reviewer severity varied substantially: two reviewers assessing the same batches differed by 0.79 standard deviations on the composite index, and a shift in reviewer composition flattened the raw trend. After adjusting for reviewer effects and smoothing batch-level estimates, evaluated reliability increased by 0.53 standard deviations from March to August 2026. Reliability also shifted materially across logged deploys. At the estimated rate of unaccounted movement, forecast uncertainty reaches the magnitude of the largest observed deploy contrast after about five weeks, although this horizon is imprecisely estimated and based on a small number of step-like changes. An independent operational outcome moved in the same direction: interview-related support tickets fell by about 10% per month relative to technical-issue tickets. These data do not identify a causal effect of evaluation on performance. Instead, they show that longitudinal AI evaluation requires three checks: comparability across raters, evidence currency under drift, and corroboration with consequential operational outcomes.

Finding which variables cause which others in multivariate time series, and at what lags, is central to science and policy, yet existing methods force a choice between flexible confounder adjustment, data-driven lag selection, and inference that controls the false discovery rate (FDR). ORACLE-VARX does all three in one pipeline. First, double/debiased machine learning (DML) removes nonlinear confounder effects from the outcomes and the lagged series. Second, adaptive causal lag estimation (ACLE) picks the lag order at each time step by sequential significance tests, tracking regime changes. Third, entry-wise $z$-tests with Benjamini–Hochberg correction select directed edges at a target FDR. We prove that in each rolling window, the debiased coefficients are asymptotically normal around a window-averaged target, so their $z$-tests are asymptotically valid. On a synthetic benchmark with time-varying structure and nonlinear confounding, ORACLE-VARX (LightGBM) tracks the true lag order best (RMSE $0.96$ vs $1.1$–$1.5$), has edge FDR $0.047$, close to PCMCI ($0.045$) and below VAR ($0.129$) and VAR-LiNGAM ($0.187$), and forecasts better than all three. On nine U.S. sector ETFs with macroeconomic confounders, it yields interpretable causal graphs whose lag order rises in high-volatility regimes.

Synthetic control relies on pre-treatment fit to balance the unobserved factors that drive untreated outcomes. The argument fails when an observed common shock, such as a commodity price, moves with the latent factors before treatment and departs from them afterwards and pre-treatment fit then says almost nothing about the synthetic unit's exposure to the shock. I derive a period-specific bound that links the post-treatment bias of any weighting estimator to pre-treatment misfit through a leverage statistic, and show that the statistic decomposes exactly into a latent-factor component and the partial leverage of the observed shock. The decomposition yields a diagnostic for such silent factor that uses no post-treatment outcome of the treated unit. Balancing an observed exposure index removes the problem, but simplex weights can achieve balance only when the treated unit lies inside the donors' exposure hull. Outside the hull, every simplex-weighted estimator carries an exposure imbalance with an explicit lower bound. For that case I propose an exposure-augmented synthetic control that corrects the remaining imbalance with a cross-donor regression, characterise what the correction identifies and when it fails, and derive bias-aware confidence intervals with a breakdown value. Monte Carlo experiments, including designs in which the correction's assumptions fail, map its gains and its limits. Iran's 2012 sanctions coincided with the 2014 oil collapse, and Iran is more oil-dependent than any of its natural comparators. The 2012-15 output loss of about 12 percent survives every adjustment. The persistent loss that standard synthetic control reports after 2016 does not. Any adjustment for oil exposure removes it, and the long-run effect is not identified with these comparators.

Clinical trials with survival endpoints lose information when participants are censored before death is observed. We develop target-preserving estimators that use posttreatment disease history, such as recurrence or progression, to recover information lost to censoring for marginal survival and restricted mean survival effects. The difficulty is that the intermediate event is downstream of treatment: naive adjustment can change the causal estimand, and the useful information enters only through the observed coarsening. We derive observed-data influence functions with and without recurrence history and obtain an exact gain identity. The identity shows that efficiency improvement is driven by the censoring hazard, the split of the alive risk set into recurrence states, and the residual-survival separation between those states. In the no-covariate illness-death model, the Aalen-Johansen estimator realizes the recurrence-augmented efficient score after standardization to the marginal target. With covariates, correctly specified Cox-Breslow transition hazards provide a root-N plug-in benchmark, while a hazard-induced one-step estimator gives rate robustness and, under primitive transition and censoring learner rates, canonical inference with a second-order product remainder. A semi-synthetic metastatic breast cancer study calibrated from digitized progression-free survival and overall survival curves illustrates the gain identity. The framework applies broadly to censored time-to-event studies with informative intermediate histories.

Matrix autoregressive (MAR) models offer a parsimonious framework for modeling matrix-valued time series, yet tools for estimation and inference for their impulse response functions are lacking. We develop asymptotic and bootstrap-based inference for impulse responses of stable MAR($p$) models. We derive the joint asymptotic distribution of the coefficient and covariance estimators, which permits closed-form delta-method standard errors. To address finite-sample bias, we propose ProBAB-MAR, a bias-corrected bootstrap that projects the corrected coefficients back onto the Kronecker parameter space, and prove that it attains asymptotically correct coverage. Monte Carlo simulations show that delta-method intervals undercover in small samples, while ProBAB-MAR achieves near-nominal coverage with intervals considerably narrower than those from an unrestricted VAR. An application to euro area inflation illustrates how shocks transmit across countries and inflation categories.

We study the existence of real-valued social welfare functions (SWFs) on the set of infinite utility streams that satisfy anonymity and almost weak Pareto. We characterize the domains of one-period utilities, \(Y\subset \mathbb{R}\), for which such SWFs exist. We show that an SWF satisfying these two axioms exists if and only if \(Y\) contains no subset that is order-isomorphic to the set of negative and positive integers. Thus, the restrictions on \(Y\) required for the existence of an SWF satisfying anonymity and almost weak Pareto coincide with those required for an SWF satisfying anonymity and weak Pareto. Moreover, the SWF we construct is invariant under arbitrary permutations and satisfies stationarity, two properties that are particularly useful for decision making in infinite-horizon economies.

Operational research evaluates tournament formats by their efficacy, the probability that the design identifies the stronger competitor, and computes it almost exclusively by simulation under an assumption that is rarely tested: that the trials a format aggregates are independent. We show that this assumption systematically overstates what shortening a contest costs, and we measure the correction. When trials are serially correlated, a format carries N/[1+(N-1)rho] effective trials rather than N, so a twenty per cent cut costs 10.6 per cent of discriminating power under independence but only 5.2 per cent at the correlation we estimate, and only 2.1 per cent in a sixty-trial format. The ingredients are estimated from an implemented change in contest length: in September 2023 professional Counter-Strike cut regulation rounds per map from thirty to twenty-four. Across 72,576 maps we measure discrimination as the ability slope over the residual standard deviation on the first half of a map, a contest fixed by rule at fifteen or twelve rounds with no stopping rule. Against the calibrated benchmark of 0.948, discrimination falls by thirty-five per cent in a ten-month window around the switch (0.648, 95 per cent interval 0.52-0.81) and by ten per cent over the full panel (0.901, 0.82-0.99). The correction needs one number, recoverable from ordinary results data as a design effect.

A new form of the fiscal multiplier suited to data expressed as gross growth rates is proposed, together with a structural vector autoregression that isolates discretionary fiscal policy from regime and rule-based components. Because cumulating growth-rate responses over a horizon involves a product rather than a sum, the proposed multiplier is multiplicative: both numerator and denominator are polynomials in the size of the fiscal impulse, so the multiplier varies nonlinearly with the scale of the expenditure programme – a dependence that conventional additive multipliers cannot capture. The multiplier requires no discounting, no ad hoc conversion factor, and no auxiliary estimate of an average expenditure share of output, and it is computed directly from nominal, non-seasonally-adjusted data, matching the terms in which government budgets are actually expressed. A version of the narrative approach to identification is introduced. It allows to decompose the influence of fiscal policy into a regime component, a standard policy rule embedded in the autoregressive structure, and discretionary shocks. Removing the regime component from the residuals yields smaller identified shocks, and correspondingly smaller multipliers, than found in SVARs not controlling for regime changes – closing much of the gap between SVAR- and DSGE-based estimates. It is found that multipliers vary substantially across countries and across types of government expenditure, that regime volatility itself differs markedly by country, and that several economies display cyclicality in spending and multiplier paths that only becomes visible once responses are cumulated multiplicatively. The resulting multipliers are nominal, derived from unadjusted data, proven to be sensitive to shock size, and purged of regime effects, offering a tool with direct relevance for budget planning and fiscal policy evaluation.

Generative AI enables authors to produce more manuscripts at lower cost. But evaluation by expert editors and referees remains scarce. In this context, the value of an author's submission rights can increase. Major research repositories and journals are reacting by imposing submission quotas and fees. Non-transferable author quotas transform submission of a manuscript into a revealed-priority decision. Authors construct a portfolio of their highest-ranked manuscripts. A common fee charges an author for every manuscript they submit. Both methods have drawbacks. Quotas do not flexibly allocate scarce review capacity to authors who have unusually strong realised candidate sets. Fees can generate heterogeneous submission thresholds that may be systematically related to an author's baseline portfolio quality. In this paper, I derive the condition under which submission quotas yield weakly higher mean reviewed value than fees, given an equal expected review load. When project values are bounded and candidate manuscripts are abundant, an equal-load fee can price out some authors entirely.

The study of the factors driving the dynamics of healthcare spending is of paramount importance to guide policymakers in the allocation of resources and to measure the effectiveness of the healthcare system under consideration. The identification of these drivers can be supported by the use of machine learning techniques, which enable the discovery of hidden patterns within vast amounts of data. In contrast to black-box methods, Symbolic Regression (SR) is an approach that allows for the identification of analytical models that explicitly capture the functional relationships within the data, thus enhancing interpretability. In this paper, we present the use of SR for identifying the drivers of healthcare expenditure in Italian regions. Given the dynamic and complex nature of this phenomenon, we generated several models based on distinct temporal windows, enabling us to analyze the drivers across different time horizons. In addition to identifying the main drivers based on variable frequency in the generated models, we also conducted a study on recurring substructures. The results show that SR was able to generate models with a good level of predictive accuracy for private healthcare expenditure, enabling a reliable analysis of its driving factors, but failed to do so for public healthcare expenditure.

A common empirical strategy in triple-differences (DDD) is to include either covariate trends or covariate levels to a three-way fixed effects (3WFE) regression. This strategy is typically motivated by the conditional parallel gaps assumption which assumes that deviations from parallel trends are similar among units with comparable observed covariates. We formally study both 3WFE specifications and show that, in general, neither consistently estimates the average treatment effect on the treated (ATT). Our diagnosis has two parts. First, we show that the OLS estimands of both specifications fail to satisfy the covariate balancing condition that is sufficient for them to equal the ATT. Second, we characterize the additional assumptions for each OLS estimand to equal the ATT. To address these limitations, we propose a matched triple-differences framework that accommodates a general class of matching procedures, including nearest-neighbor matching and kernel matching. Within this framework, we construct a class of consistent matching estimators, establish their asymptotic normality, and provide a consistent variance estimator that accounts for the variability introduced by the matching step. Our empirical application shows that the proposed matched triple-differences estimator can yield estimates and qualitative conclusions that differ from those obtained using the 3WFE regression.

Random assignment can make ordinary least squares (OLS) inference insensitive to outcome dependence, yet iid resampling can still fail because assignment and resampling need not remove the same covariance terms. With binary treatment, the iid variance target differs from the sampling variance by exactly minus aggregate cross-unit covariance of treatment effects. Under a Gaussian first-order limit, positive covariance leads to over-rejection and negative covariance to under-rejection. Two designs can generate the same distribution for each observation but require variance corrections of opposite signs, so no marginal-only variance correction is first-order exact for both. For first-order Gaussian inference, only one covariance component—the covariance carried by the randomized-regressor score—must be recovered. Standard dependence estimators on that score restore validity under suitable ordered or grouped dependence conditions. In simulations with ordered data, fixed-bandwidth calibration reduces several large-bandwidth size distortions.

What role do journalists play in determining the political slant of the news they produce? We develop and estimate a model where journalists and newspaper outlets contract over both slant and wages. The model implies a set of conditions under which we can consistently estimate the role of journalist preferences in driving the observed variation in slant across outlets by leveraging journalist transitions between outlets. To measure slant, we train a transformer-based, machine learning model using articles tweeted by politicians and apply it to a full-text database of 9+ million newspaper articles published in the US between 2013 and 2018. Under our model and identifying assumptions, our estimates (a) reject the hypothesis that journalists have zero ideological preferences over the content they produce and (b) imply that 10% of the variation in slant can be explained by journalists. When journalists move across outlets, their average slant shifts by 77% of the gap between their destination and origin outlet averages. In counterfactuals that reweight journalist party composition, moving from the observed left-leaning distribution to an even distribution of Democrats, Republicans, and non-partisans shifts average article slant in a more conservative direction, but only by 0.04 standard deviations in slant.

Imperfect measurements allow different underlying distributions to generate the same observations. Restrictions linking unobserved components can make it difficult to characterize all compatible distributions and compute the resulting range of parameter values. We ask how to construct computable parameter bounds and tighten them without additional data or stronger assumptions. We develop a general framework for partial identification under imperfect measurement based on distributionally robust optimization. We use measurement restrictions to construct a neighborhood around a benchmark distribution, ensuring that every compatible distribution lies within it. Optimizing over this neighborhood gives parameter bounds through a common procedure that can be adapted to different measurement problems. The choice of benchmark affects how informative these bounds are. We establish conditions under which changing it can tighten the bounds while preserving every parameter value consistent with the same data and assumptions. Applications to networks, missing outcomes, and regression illustrate how the framework tightens bounds and can recover the full range of compatible values.

A growing number of countries are adopting policies to reduce greenhouse-gas emissions, a process expected to reshape production systems and labour markets by altering relative prices and changing production and demand patterns across sectors. We develop an agent-based monetary input–output network model to analyse the medium-term effects of climate policies on production networks and sectoral labour allocation. The model features heterogeneous industries and households, incorporates greenhouse-gas emissions accounting, and is calibrated to a multi-regional input–output database covering the world economy. We simulate two alternative climate-policy scenarios over 2026–2035: one based on countries' nationally determined contributions and another targeting larger reductions in global emissions. The results reveal significant economy-wide reallocation: fossil-energy sectors contract, low-carbon energy sectors expand, and their positions within production networks change asymmetrically. Fossil energy loses relevance as a direct supplier and exerts less forward influence, while its remaining production becomes more dependent on intermediate inputs and upstream suppliers; low-carbon energy primarily strengthens its role as a supplier. Carbon pricing also generates sectoral labour gaps that, under mobility frictions, imply greater potential transitional unemployment. Because the model allows only limited technological change and adjustment within existing production links, the simulations provide a counterfactual assessment of how existing production structures respond to a relatively rapid climate transition, revealing the adjustment pressures that emerge when policy ambitions advance faster than the productive system's capacity to adapt.

We study Balbuzanov and Kotowski's (2019) exclusion core for object allocation problems with complex endowments. We focus on mixed-ownership object allocation problems in which each object is either individually owned by one agent or publicly owned by all agents. Given a priority order over agents, an endowment structure, an initial allocation, and agents' preferences, we construct serial top trading cycles (STTC) graphs and define the associated STTC algorithm. The algorithm is the central tool for our analysis of exclusion-core-stable allocations. First, we show that, among domains defined by restrictions on endowment structures, the mixed-ownership domain is the largest on which the STTC algorithms characterize the exclusion core. Second, we provide a simple test for exclusion core stability: an allocation is exclusion core stable if and only if it is a fixed point of the STTC algorithm initialized at that allocation, independently of the priority order. Third, we show that the exclusion core is a von Neumann–Morgenstern (1944) stable set with respect to exclusion-domination. Finally, we study a dynamic exclusion-blocking process and show that, from any allocation, there exists a self-contained exclusion-blocking path to the exclusion core.

This study develops a survey-calibrated, path-dependent agent-based microsimulation for evaluating crime and violence reduction policies in Bolivia. A synthetic population is constructed from the 2025 Household Survey and the fourth-quarter 2025 Continuous Employment Survey using official expansion weights and cross-survey donor matching. The model combines heterogeneous agents, place-based risk, criminal adaptation, institutional capacity, community organization, and multiple policy channels. Uncertainty is propagated through adaptive Gaussian-process and tree-ensemble emulators, global sensitivity analysis, tempered ensemble MCMC, and predictive meta-calibration. Simulated portfolios include focused policing, community control, Guardabosques-type incentives, social prevention, environmental interventions, justice reforms, investigation, rehabilitation, and financial-network disruption. Results are interpreted as structural policy scenarios rather than causal forecasts, with explicit attention to displacement, distributional effects, implementation capacity, and multi-objective trade-offs.

Implied stock-market volatility is widely used to identify uncertainty shocks and discipline models of their macroeconomic effects. This paper gives distributional content to the option-price variation underlying these exercises. A normalized at-the-money forward put price, the Downside Protection Premium (DPP), closely tracks the VIX and preserves the principal macroeconomic responses in the specifications examined. Its price and local strike slope separate risk-neutral shortfall probability from conditional severity. Severity accounts for almost all monthly premium comovement and 91.5 percent of the recession log premium. Conditional gains also rise substantially, while conditional losses remain larger in phase averages. Despite this severity dominance, quotations across strikes frequently exclude a common-scale explanation, conditional on the maintained forward and premium construction. Realized-volatility and future-volatility-news innovations both raise protection costs primarily through severity, yet produce different estimated employment responses in common impact-severity units; simultaneous significance depends on the inference formulation. Together, these findings supply joint empirical restrictions on the connection between uncertainty, option prices, and economic activity.

The role of women on corporate boards and its impact on innovation remains heavily debated. While existing literature offers conflicting perspectives on whether female board representation spurs or hinders innovaiton, most research has focused exlusively on large firms. This studt adresses this gap by investigating the causal impact of women on boards on both "hard" (technological) and "soft" (non-technological) innovation within SMEs. Using a longitudinal dataset f 2762 Italian innovative SMEs from 2015 to 2024 and a 2SLS identification strategy, we find highly nuanced patterns. For hard innovation, the instrumented model reveals a robust positive causal impacr, reversing a negative baseline coefficient and exposing a severe negative selection bias. For soft innovation, a stable positive impact is confirmed. Our findings provide crucial causal evidence for SMEs' governance, while offering tailored insights for policymakers and practitioners to incentivize women' management roles in SMES.

We derive sufficient conditions for the existence, non-existence, uniqueness and global attractivity of stochastic recursive utilities. Our approach uses abstract order-theoretic fixed-point arguments, allowing us to accommodate a large class of Markov consumption-growth models with unbounded per-period consumption and a broad range of recursive preferences, including non-expected utility specifications. We illustrate the usefulness of the framework through macro-finance applications to widely used consumption-growth models under several preference specifications, including Epstein–Zin preferences, multiple priors, smooth ambiguity, and Hansen–Sargent robust-control preferences. In each application, our general conditions reduce to a simple, economically interpretable parameter test that shows how fundamental parameters influence existence and uniqueness. While uniqueness of recursive utility can be difficult to guarantee in unbounded environments, in a Markov setting with non-unit elasticity we establish uniqueness within the class of recursive utilities having a so-called separability property.

We study mechanisms in which agents' preferences depend both on an object allocation and on a third party's beliefs about private information. A mechanism determines an allocation, transfers, and sends signals to the aftermarket. We compare two informational environments: (i) private transfers, where the aftermarket observes allocations and signals, and (ii) public transfers, where transfers are publicly observable. Our main result shows that any equilibrium outcome implementable with hidden transfers can be approximated arbitrarily well with public transfers. Thus, transfer observability generates at most a vanishing informational distortion and helps rationalize the common assumption that transfers remain hidden from aftermarkets.

Cross-fitting is routine in much of applied research. While conventional confidence intervals that ignore cross-fold dependence are asymptotically valid in several settings, they undercover in many applications that share a common form of nonregularity: from the classic cross-validation problem of testing whether a fitted model outperforms another, to testing for heterogeneous treatment effects with machine learning, to estimating the value of a potentially non-unique optimal treatment regime. Exploiting a new locality condition, I show that a large class of cross-fitting estimators still satisfies a central limit theorem despite the nonregularity, but with an asymptotic variance that must be adjusted for the cross-fold correlation. Then, I propose a method for estimating this correlation and construct new confidence intervals that attain asymptotically nominal coverage. Finally, I show that the proposed confidence intervals attain approximately nominal coverage in a simulation study with random forests and neural networks.

Many policies assign treatment according to whether a score crosses a cutoff. Regression discontinuity (RD) designs identify the effect of treatment assignment, but the threshold policy itself may also reshape individuals' incentives, inducing behavioral responses that affect outcomes even holding treatment status fixed—a channel that conventional RD designs cannot capture. We develop a framework that exploits randomized variation in policy thresholds, together with rank-invariance-type restrictions, to identify the treatment-assignment and incentive-response effects. We illustrate the empirical relevance of this distinction using data from a merit-based scholarship experiment in Malawi. In this illustration, the conventional RD estimate is positive, while the incentive-response effect is negative and more than twice as large in magnitude. These findings suggest that threshold policies may generate unintended adverse behavioral responses, potentially reflecting discouragement induced by demanding thresholds, and caution against relying solely on conventional RD estimates when evaluating threshold policies.

Professionals are increasingly turning to artificial intelligence (AI) for career advice. We tested whether large language models (LLMs) provide different advice to female and male job candidates in the context of career negotiations. Across two large-scale audit experiments, including four leading models, the LLMs not only recommended lower salary targets for female than male personas, but also constructed distinct negotiation agendas, prioritizing flexibility, caregiving, and well-being more highly for female candidates, and financial rewards, mobility, and advancement more highly for male candidates. These differences closely tracked independent human ratings of employment terms associated with primary caregiver and breadwinner roles, respectively. Our findings identify a subtle pathway through which personalized AI advice may reinforce gender inequality at work: steering otherwise equivalent users toward traditional gender roles.

For some integer $\ell\geq 2$, consider $m=2\ell-1$ voters, each of which has a preference ranking over $n$ alternatives. The Smith set is the smallest nonempty set of alternatives each of which defeats every alternative outside the set in a pairwise majority comparison. Under the standard benchmark impartial culture where each voter uniformly and independently chooses a random preference ranking over all alternatives, we study the asymptotic probability that the Smith set has cardinality $s$ for fixed $\ell$ and $n\to \infty$. First, we prove that the probability that the Smith set has constant cardinality $s$ goes to zero at a rate of $\Theta_{\ell,s} (n^{-s(\ell-1)/\ell})$. Next, if $\min\{s,n-s\}\to\infty$, we show that this probability decays superpolynomially. Further, when $s$ and $n-s$ are both $\Theta(n)$, we prove that the probability decays exponentially and determine the exact exponential base. Finally, we prove that the probability that the Smith set contains all alternatives approaches one at a rate of $\Theta_\ell(n^{-(\ell-1)/\ell})$. These theorems resolve the conjectures in [Liu et al, Ann Stat 2026] under impartial culture.

LLM judges are increasingly used to assess whether AI outputs meet workplace requirements, but agreement on response rankings does not establish agreement on acceptance rates or occupational aggregates. We introduce O*NET-BENCH, an audit suite derived from an existing survey of 45,796 worker ratings, and evaluate 33 pre-existing judge configurations across six model families on 4,501 test ratings. Twenty-five configurations achieve tie-aware pair accuracy of at least 0.60, although a train-fitted response-only TF-IDF baseline nearly matches the strongest judge. Despite this ordering agreement, judges estimate that 3.0%-97.9% of responses are acceptable, compared with 61.1% for occupation-matched workers. In one fine-tuned lineage, changing from pointwise scoring to a bundled few-shot/listwise protocol improves response ordering while reducing agreement with worker means at the task and occupation levels; this reversal replicates on a task- and worker-disjoint validation split under prespecified criteria. Cross-validated calibration largely removes mean bias, but calibrated scores explain at most 8.5% of individual worker-rating variance. Prediction-assisted estimation yields at most small precision gains at the studied label budgets. These results show that ranking agreement alone is insufficient for occupational measurement. Judges should be validated against the acceptance rates and aggregates their scores will be used to estimate.

This study considers the problem of portfolio choice, where we recommend a portfolio to an investor to maximize the expected utility of their wealth. Our goal is to construct an asymptotically optimal portfolio choice rule in terms of expected utility regret, the difference between the expected utility of an oracle investor and that achieved by a portfolio chosen from data. We propose the Expected Utility Regret (EUR) rule, which jointly selects a portfolio class and estimates its weights. In a regular parametric return model, a single EUR rule attains both the minimax and the Bayes lower bounds, including their leading constants, without using the prior distribution that defines the Bayes criterion. We then derive the mean–variance and risk-parity portfolios as special cases of this framework. Under smooth increasing and concave utility, the EUR rule and the sample mean–variance portfolio attain the same leading expected regret when expected excess returns approach zero sufficiently fast. When the returns divided by their volatilities have a joint distribution that does not depend on the order of the assets, the EUR rule and the risk-parity portfolio coincide.

Text, images, and other rich covariates are increasingly compressed into AI-learned representations and then used as controls in causal analysis. We study when this approach is valid and develop a practical framework for causal inference with learned representations. For a broad class of estimands, an imperfect representation distorts the target causal parameter by the product of two representation errors: one in the outcome regression and one in the balancing weight (or Riesz representer). This yields three constructive results. First, cross-fitted double machine learning (DML) provides valid Wald inference for the representation-dependent target. When representation errors are small, the same interval covers the causal parameter, and it can even attain the semiparametric efficiency bound. Second, fold-wise representation learning (or fine-tuning) is compatible with DML inference for the causal parameter. To this end, we develop convex- and star-aggregation pipelines for learning and combining representations. Third, when representation errors are substantial, we can provide interpretable sensitivity regions and root-$n$ inference for their endpoints. In a multi-modal demand application, seven representation-specific estimates and their star aggregate all imply a negative near-unit elasticity for rank-based price response, and the result remains robust over the reported sensitivity grid.