Papers

Many structural and dynamic economic models imply that key parameters are identified by conditional quantile restrictions. Building on the exponential-weighting approach of Bierens (1990) and recent advances in penalized maximum statistics for conditional moment restrictions (Chen et al., 2025), we develop a unified inference framework for such parameters. We propose an adaptive $\ell_1$-penalized supremum statistic that transforms the conditional restriction into a continuum of unconditional moment conditions and aggregates evidence across quantile indices. The penalty regularizes the maximization over the weighting direction. Under the stated uniformity conditions, the known-parameter adaptive selector has no lower maximin local power than the unpenalized test and yields a strict maximin local-power gain whenever some positive candidate penalty has a strictly larger population maximin criterion than the zero penalty. We extend the theory to settings with pre-estimated nuisance parameters, characterizing the additional terms induced by the plug-in step in the limiting process. We derive an analytically corrected variance estimator that accounts for plug-in estimation uncertainty and establish the validity of a Gaussian multiplier bootstrap under the null and sequences of local alternatives. Monte Carlo simulations show that the proposed CvM-KS aggregation scheme has rejection rates relatively close to nominal size under pre-estimation in the linear design and approaches the nominal level in nonlinear designs as the sample size grows. The reported power comparisons between the adaptive and unpenalized procedures are design-specific, with higher empirical rejection frequencies for the adaptive procedure along some directions.

While corporate sustainability mandates are expanding, the systemic reliance on self-reported emissions data exposes financial markets to pervasive greenwashing. Current literature relies heavily on subjective ESG ratings or textual sentiment analysis, leaving a critical econometric gap in objectively quantifying physical climate realities. To resolve this information asymmetry, we fuse U.S. SEC financial fundamentals with facility-level EPA greenhouse gas registries to establish a mathematically guaranteed baseline of physical corporate emissions. Leveraging a gradient boosting architecture and Mondrian Conformal Prediction, we quantify the shortfall between self-reported data and this algorithmic baseline into a novel Conformal-Weighted Continuous Divergence (CWCD) metric. Evaluating this divergence via a cross-sectional lead-lag econometric design, we uncover a robust mechanism of market discipline: algorithmic emissions divergence exhibits a severe, statistically significant negative relationship with subsequent market valuation (Tobin's Q) and operational profitability (ROA). Providing definitive evidence against the market blindness hypothesis, this study proves that institutional capital actively prices environmental deception not merely as an ethical lapse, but as a leading indicator of fundamental corporate mismanagement. Ultimately, these findings provide the quantitative justification necessary for asset managers and regulators to deploy algorithmic auditing infrastructure at scale.

Day-ahead electricity price forecasts support trading and storage decisions, but for battery arbitrage predicting intraday price spreads is more relevant than predicting individual hourly prices. Here we show that a temporal hierarchy forecasting (THieF) framework that jointly reconciles forecasts of hourly electricity prices and all intraday price spreads consistently improves performance across two major European electricity markets and three different forecasting architectures. Using five years of out-of-sample data from Germany and Spain, we obtain accuracy improvements of up to 19.7% and profit gains of up to 10.4% relative to unreconciled hourly price forecasts. The gains persist even for a highly accurate pretrained TabPFN foundation model. Our results demonstrate that exploiting coherent relationships between economically relevant forecasting targets can improve both predictive accuracy and decision value, and that better statistical forecasts do not necessarily imply better economic decisions.

We provide sufficient conditions for the consistency of penalized least squares procedures that select the order (dimension) of a regression model from a sequence of nested classes, allowing for dependent, martingale-difference errors. The main contribution is to relax the classical identifiability requirement: parameters indexing classes larger than the true order need not be identified, provided the additional, excess directions admit a linear approximation to the truth in a neighbourhood of the true parameter. This relaxation lets the number of candidate models grow with the sample size, removing the usual need for a fixed upper bound. We verify the resulting high-level conditions for two classes of nonlinear regression models used in applied work: multiple-regime smooth transition regression and mixture-of-experts models with generic generalized linear model experts. Under a BIC-type penalty, the resulting order selection rule is consistent in either model class whenever the number of candidate models grows slower than the logarithm of the sample size.

We study randomized experiments involving two interacting populations, such as buyers and sellers in a marketplace. In the two-sided experiments we consider, we randomize the two populations separately and independently. For a pair consisting of one member from each population, the two assignments jointly determine one of four exposure conditions. Under a local interference assumption, we consider a broad class of linear estimands, including total, interaction, and buyer- and seller-side spillover effects. Our first main result establishes that researchers can estimate these effects using ordinary least squares and conduct asymptotically valid design-based inference using the conventional two-way cluster-robust variance estimator, clustered at the buyers' and sellers' levels. Our second main result develops a sharper variance estimator for a single linear estimand that better preserves dependence within the buyer and seller dimensions and is asymptotically less conservative than the two-way clustered estimator and existing alternatives. Our third main result establishes the theory for covariate adjustment and recommends a two-way analysis-of-variance-type covariate representation to ensure efficiency gains.

The pre-season strengths of European football clubs are usually measured by two proxies in the literature. Football Club Elo Ratings provide strictly performance-based Elo ratings from the early days of the European Cups, while Transfermarkt valuations are crowd-based estimates of squad market values. This paper compares them by evaluating their ability to forecast the results of matches played in the UEFA Champions League and the UEFA Europa League between the seasons 2020/21 and 2024/25. The two indicators yield almost identical out-of-sample accuracy when used separately. Combining the two measures leads to a modest improvement, but the best aggregation procedure is sensitive to the forecast target. Our results suggest that seeding based on Elo ratings would be (closely) optimal.

Comparing two populations at the same physical covariate value requires more than conditional means or isolated target-point decisions: researchers may need evidence about an entire conditional-distribution ordering over a continuum, even when covariate margins differ. This paper makes that common-value comparison estimable under an explicit structure–flexibility tradeoff and turns the resulting surface into simultaneous evidence for first-order stochastic dominance. Population-specific margins map the common covariate value into each group, while a fitted copula-derivative representation links conditional distributions across the region. Uniform inference propagates uncertainty from both the margins and dependence model through a one-sided statistic with unknown binding locations. Under correct specification within a finite copula class, smoothness and trimming conditions, and a uniquely best candidate family, the procedure admits uniform control and consistent calibration. Simulations show increasing rejection as alternatives become more distinguishable, alongside model-selection sensitivity and small-sample size distortion. In a descriptive PSID application, the high–low parental-education comparison satisfies the two-direction criterion after multiplicity adjustment, whereas adjacent education-group comparisons remain inconclusive. The framework therefore supports region-wide distributional comparison while making its structural and inferential boundaries explicit.

This paper proposes discrete-time approximations to rough continuous-time models of realized variance (RV). The leading rough models can be viewed as autoregressive processes driven by fractional Gaussian noise. We show that the Wold representation of this noise concentrates its dependence at the first lag when the Hurst parameter is below one half. Augmenting the autoregressive (AR) and heterogeneous autoregressive (HAR) models with a first-order moving-average (MA(1)) component therefore approximates the roughness, and the MA coefficient maps almost linearly into the Hurst parameter. We refer to these extensions as the "rough" AR and "rough" HAR models. Estimating them on the log RV of ten ETFs, we find negative MA coefficients for every asset, and the implied Hurst parameters align closely with the estimates from the continuous-time models. In the HAR literature, the negative MA(1) component is a significant feature that has been largely overlooked. In out-of-sample comparisons, the "rough" models outperform their classical counterparts for nearly every asset and horizon, with the largest gains at short horizons, and their accuracy is comparable to that of the rough continuous-time models but much easier to estimate by standard off-the-shelf software.

This paper studies the practice of combining multiple outcomes into a summary index to estimate a causal effect. For common estimators and index constructions, the estimate equals a weighted sum of the estimated effects on the components, with weights that are implicit and rarely reported. The paper derives the weights and shows that, for inverse-covariance-weighted indices, they can be negative and unrestricted in magnitude, so the index effect can have the opposite sign to every component effect. The paper proposes two procedures for valid inference on the index effect: a variance estimator that accounts for the data-dependent weights, and a shifted t-test that requires no such correction. Conventional t-tests of the null of no effect remain valid. Contrary to common claims, summary indices do not generally improve power. Three published studies illustrate the results.

This paper develops a Poisson regression model with multivariate sample selection, in which the outcome is observed only when several potentially correlated selection conditions are satisfied. To the best of our knowledge, this is the first Poisson sample selection model that allows for an arbitrary number of selection equations. We derive the conditional mean of the observed outcome under joint normality of the outcome and selection errors and obtain a multivariate selection-correction term. We prove identification of the model parameters and show that, under suitable support and rank conditions, the outcome parameters can be identified without an exclusion restriction. We also propose two-step estimation procedures based on nonlinear least squares and Poisson pseudo-maximum likelihood. To the best of our knowledge, this paper is the first to apply PPML to a Poisson regression model with sample selection. The consistency of both estimators is established, and a robust two-step sandwich covariance matrix is proposed to account for the estimation error from the first-step selection model. In addition, factorial moments are used to recover the variance of the latent outcome error and the correlations between the outcome and selection errors. Monte Carlo simulations show that ignoring sample selection leads to persistent bias when the outcome and selection errors are correlated, while the proposed PPML estimator substantially reduces this bias and is more stable than nonlinear least squares, especially under moderate and strong selection dependence.

Machine learning (ML) techniques are increasingly drawing interest in the choice modelling (CM) field. The focus has primarily been on comparing the performance of these contrasting approaches or on improving behavioural insights for ML techniques, rather than translating ideas from one field into the other. In the present paper, we specifically focus on knowledge transfer from ML into CM in the context of model performance evaluation. In CM, model performance is typically evaluated using log-likelihood and related indicators, which are aggregate fit metrics that focus on overall fit. Conversely, in ML, the focus is on alternative-level misclassifications and correct classifications, which provide a more nuanced view of the results. To bridge these approaches, we explore the use of a probabilistic version of the confusion matrix, which reports the average probability of the model predicting each alternative, conditional on which alternative was observed to be chosen, across all choice tasks. This enables the computation of probabilistic ML metrics for both classic choice models and ML algorithms. We analyse model performance jointly in terms of overall fit and alternative-level predictions. Our findings demonstrate that models with similar log-likelihood can exhibit substantially different confusion matrices, revealing different probability patterns that aggregate metrics cannot capture. This framework identifies where models systematically `confuse' alternatives, highlighting trade-offs between alternatives, and potentially guiding model specification. Furthermore, evaluating these matrices and metrics out-of-sample reveals alternative-level prediction shifts that significantly impact forecasting performance.

We examine semiparametric solutions to contamination bias for nonbinary treatments. Deepening the discussion by Goldsmith-Pinkham et al. (2024), we detail how spline functions approximate conditional expectation and propensity score functions under weak functional-form assumptions. Reanalyzing 18 regressions across 11 studies, we compare standard linear regressions against parametric and semiparametric versions of three contamination-robust estimators. We document large point-estimate discrepancies between parametric and semiparametric approaches and find that adopting flexible semiparametric solutions may not increase statistical uncertainty substantially. We recommend that researchers verify the robustness of their conclusions to the use of semiparametric tools that address contamination bias.

Structural similarity, the extent to which economic mechanisms respond alike to common shocks, matters for economic forecasting, policy transfer, and other decisions that rely on evidence from comparable settings. For mechanisms approximated by linear models, this idea has a simple geometric representation: The smaller the angle between their coefficient vectors, the more similarly they respond to the same shock. This paper introduces the cosine of this angle as a measure of structural similarity and develops the corresponding inference procedure. Applied to New Jersey's wage mechanism around the 1992 minimum-wage reform, the approach yields interesting evidence: Although the reform clearly affected the low-wage part of the wage distribution, the underlying mechanism linking worker characteristics to wages was no different from what would have prevailed without it.

This paper develops inference for a Gaussian-nested hypergeometric family of distribution functions. The family \[ G_c(z) = \frac12 + z\,\frac{\Gamma(c-1/2)}{2\sqrt2\,\Gamma(c)}\,{}_1F_1\!\left(\frac12;c;-\frac{z^2}{2}\right),\quad c\ge\frac32, \] contains the standard normal distribution at the boundary $c=3/2$. Away from the boundary, the density has algebraic tail behaviour $g_c(z)\sim(c-3/2)|z|^{-3}$, so the parameter $c$ indexes a directed heavy-tailed deformation of the Gaussian law. The distribution also admits an equivalent beta-precision normal scale-mixture representation, in which the Gaussian boundary corresponds to degenerate unit precision. I establish the admissibility of the hypergeometric family, derive minimum-distance estimators, and obtain both regular interior asymptotics and nonstandard boundary asymptotics under the normal null. The standardised fit-improvement statistic converges to the mixture distribution $\tfrac12\delta_0+\tfrac12\chi_1^2$. I extend the theory to plug-in location-scale procedures, including a robust median/IQR version. Simulations document accurate null size and directed power against heavy-tailed alternatives. An application to daily S&P 500 returns, both unconditionally and after GARCH(1,1) filtering, illustrates the empirical implications for tail fitting and risk quantiles, and documents that GARCH filtering substantially reduces but does not eliminate the symmetric heavy-tailed departure detected by the test.

Economic networks are often estimated separately over two periods, and changes in their edge sets are interpreted as structural rewiring. Since both networks are estimated, observed turnover also reflects graph-selection error. We study the two-snapshot Hamming-turnover functional under a homogeneous edge-misclassification model. With known sensitivity and specificity and conditional independence of the estimated edge indicators across periods, latent turnover admits a closed-form unbiased adjustment based only on observed turnover and the two estimated graph sizes. We then examine the effects of sparsity, calibration error and dependence across periods. When the number of true links is proportional to $p$, a false-positive probability of order $p^{-1}$ generates expected spurious turnover of order $p$. An error of the same order in calibrating the false-positive probability can likewise leave order-$p$ bias after adjustment. We also derive the bias induced by cross-period dependence and give sufficient conditions for consistency relative to network size. Numerical results illustrate the finite-sample implications.

Local asymptotic minimax (LAM) risk is a foundational efficiency criterion in statistics and econometrics. The literature uses two definitions of LAM risk: one which appears in classical lower bounds and another which appears in arguments establishing attainment of those bounds. Conventional efficiency arguments are consistent with any estimator-dependent weighted average of the two, and consequently do not reveal which of these generalized $\alpha$-LAM risk indices describes researchers' actual preferences. We take a decision-theoretic approach to systematically resolve this ambiguity. We axiomatically characterize the set of preferences with a generalized $\alpha$-LAM representation, and we document that such preferences may violate basic rationality requirements such as Monotonicity. Motivated by this, we axiomatically characterize the subset with constant-weight representations. Within this class, a Sample Uncertainty Aversion axiom uniquely selects attainment LAM. We argue that Sample Uncertainty Aversion is normatively appealing, and we therefore recommend attainment LAM as the functional form for LAM risk.

Multivariate longitudinal data may exhibit non-Gaussian margins, nonlinear dynamics, and response vectors with composition that varies across waves. To account for these features, we introduce a vector drawable vine (VD-vine) copula that extends conventional drawable vine copulas from scalar to vector-valued nodes. Here, the response vector at each wave forms a multivariate marginal, and serial dependence is captured through a sequence of linking vector copulas. We establish that the VD-vine is itself a vector copula and reduces to a conventional drawable vine for scalar nodes. Recursive forward and backward conditional transports are derived that enable efficient likelihood evaluation and predictive simulation, with parsimonious reductions under finite-order Markov and stationary restrictions. Unconstrained parameterizations for Gaussian and FGM linking vector copulas, flexible multivariate marginals, and Bayesian variational inference provide a practical implementation. Simulations show improved predictive accuracy when the marginals are asymmetric and serial dependence is multivariate, with little loss under a correctly specified Gaussian panel vector autoregression. In an eight-wave Australian panel of 1,093 individuals with varying response vectors, the full VD-vine delivers the best cross-validated distributional forecasts among the models considered, establishing the benefit of capturing asymmetry and nonlinear dependence.

In evidence-based policymaking, typically one experimental sample is observed, then a learned policy recommendation is implemented at scale. Policies learned from the experimental data can perform well in expected welfare, yet random sampling in the experiment can produce recommendations with poor welfare outcomes. In this paper, we ask: how should policy learning algorithms balance expected welfare against sampling risk? Our main contribution is to show that algorithmic stability plays a central role in characterizing and navigating the tradeoff. Intuitively, if a policy learning algorithm's recommendation remains stable when one experimental unit is replaced, then that algorithm has limited sampling risk. We propose a method for policy learning called policy-vote bagging, which learns treatment decisions on many subsamples then averages their votes into treatment probabilities. Relative to using one subsample, averaging across subsamples preserves expected welfare and improves expected utility for a risk-averse researcher. We derive sharp bounds linking estimation accuracy, subsample size, and welfare variation, including an exact guarantee under CARA utility.

Consider an increasing number of consistent estimators to be averaged when only estimated weights are available. The underlying parameter of interest can be identical across estimators (homogeneity) or not (heterogeneity). The contribution of the paper is threefold. First, it is shown that the interaction of the estimated weights with the estimators can generate specific bias terms. This constrains the number of estimators that can be aggregated when weight estimation is ignored in inference. Second, the paper proposes estimated adaptive weights, which allow for standard Gaussian inference in a uniform manner and are asymptotically optimal both under homogeneity and heterogeneity. Third, conditions ensuring the validity of the Cochran (1937) Q test of homogeneity with estimated variance are given.

We study the omitted variable bias (OVB) problem in canonical difference-in-differences (DiD) designs when unobserved confounding induces departures from the parallel trends assumption. Our results provide a novel characterization of the OVB formula for the average treatment effect on the treated (ATT), which is of independent interest. We show how the ATT bias is mainly governed by the strength of confounding in the treatment assignment mechanism and provide alternative ways of quantifying this strength, such as (i) changes in the average odds of treatment among the treated, (ii) confounding imbalance between treated and control units, or (iii) variation explained in treatment odds among the untreated. Building on these results, we offer sensitivity statistics for routine reporting, describing the minimum strength of confounding required to overturn the conclusions of a DiD study, as well as formal bounds on the strength of confounders based on comparisons to observed covariates or pre-trends. Finally, we provide flexible and efficient statistical inference methods for the bounds on ATT, which can leverage modern machine learning algorithms for estimation. We demonstrate the utility of our approach in an empirical example that estimates the effects of minimum wage on teen employment.

In the canonical Difference-in-Differences design, the control group's post-treatment change serves as an imputation of the treated group's counterfactual change in the same period, an imputation justified by parallel trends. However, differences in group composition can produce between-group differences in how outcomes would evolve over time, rendering this imputation vulnerable to confounding. An alternative imputation – such as one based on the treated group's pre-treatment change – avoids such between-group confounding but introduces the risk of confounding from within-group temporal shifts. Ideally, both imputations, each vulnerable to different sources of confounding, would have concordant values, thereby yielding the same causal conclusions. When the imputations are discordant, conclusions under parallel trends hinge more critically on that assumption since alternative imputations would point to different results. Yet in these scenarios, existing pretrends-based sensitivity analyses can show low sensitivity because they ignore post-treatment deviations from pretrends in the control group. This paper therefore proposes a discordance-based sensitivity model in which parallel pretrends are necessary but not sufficient for low sensitivity. I formally justify this model in terms of the expected distance between the ATT under parallel trends and under alternative assumptions, weighted by the joint plausibility of those assumptions. I then provide a decision-theoretic rationale for benchmarking violations of parallel trends using the worst-case discordance between the parallel trends imputation and alternative imputations. Finally, I apply both pretrends- and discordance-based sensitivity models to assess how a labor supply shock influenced electoral support for apartheid-era policies in South Africa, showing how the two approaches yield different results.

Discovering interpretable subgroups whose complier effects deviate from the average is a central goal of instrumental variable analysis under imperfect compliance, yet existing tree-based methods degrade when most covariates are irrelevant to the effect. We propose Shrinkage Bayesian Causal Forest with Instrumental Variable (SBCF-IV) for discovering and estimating subgroups with heterogeneous Complier Average Causal Effects (CACE) in sparse high-dimensional settings. SBCF-IV places a sparsity-inducing Dirichlet prior on the splitting probabilities of the Bayesian Additive Regression Trees that estimate the conditional intention-to-treat and the complier share, concentrating posterior mass on the few covariates that moderate the complier effect and thereby regularizing effect estimation. The posterior split frequencies additionally enter a downstream CART as variable-level costs that steer the partition toward relevant moderators, providing an interpretable division of the covariate space. Monte Carlo experiments show that, as the share of irrelevant covariates grows, SBCF-IV recovers the true partition more reliably than its non-sparse predecessor BCF-IV at the tree and unit level, and retains nominal coverage where BCF-IV's intervals deteriorate. We apply the method to the Oregon Health Insurance Experiment and the 401(k) eligibility data.

We develop a class of linear state space models for matrix-valued time series data where the state is a latent matrix normal process. We derive matrix versions of the Kalman filter, log-likelihood, and smoother enabling estimation of the latent state matrix as well as the model's parameters. To conduct Bayesian inference, we provide algorithms that draw from the joint posterior distribution of the latent state matrices conditional on the observed data and parameters. We apply these methods to a large panel of U.S. macroeconomic time series across the 50 U.S. states. The proposed framework accommodates mixed-frequency data, heteroskedasticity, and outliers within a unified matrix-valued structure. Empirically, we find that a small number of latent factors captures the joint dynamics across states and variables, providing a parsimonious and scalable approach to modeling high-dimensional macroeconomic systems.

This paper develops a framework for individualized treatment allocation when interventions shift equilibrium prices and generate spillovers across treated and untreated units. The planner chooses which units receive a subsidy while allowing equilibrium prices to adjust endogenously. We show that the resulting welfare function is supermodular under broad and interpretable conditions, implying complementarity across treatment assignments and enabling exact polynomial-time optimization. This structure clarifies how equilibrium spillovers shape the trade-off between universal and targeted distribution and makes the planner's problem computationally tractable despite interactions across units. We characterize when universal or targeted subsidies are optimal and show how market conditions and heterogeneity shape the optimal allocation. We further establish statistical guarantees for plug-in allocation under estimation uncertainty in demand and supply. Finally, we illustrate the framework in a coupon allocation problem calibrated with household expenditure data from the Philippines.

Applied instrumental variables (IV) practice reports a first-stage F, now often the conditional F of Sanderson and Windmeijer (2016), and reads a large value as license to interpret the second stage. We show that no first-stage diagnostic can provide it. With a scalar instrument, a scalar treatment, and covariates entered linearly, the 2SLS estimand splits into a signal that a saturated specification would target and a contamination, the covariance between curvature in the instrument propensity and a covariate level function. The same nuisance sits in both terms, so it biases the estimand and inflates the reported strength at once. When the instrument is nearly collinear with the covariates the signal vanishes and the strength is manufactured. When the strength is honest the curvature still biases the estimand through the outcome, where no first-stage number reaches it. We prove that no functional of the joint distribution of instrument, treatment, and covariates can detect this second bias, and we give a directed test built from reduced-form regressions, a corrected estimator, and a reportable contamination share. Two applications show the modes. A husband's insurance instrument (Olson, 1998) has a conditional F above 36,000, yet correcting a linear income control more than doubles the estimate. The instrument of Nunn and Wantchekon (2011) loses most of its first stage once geography enters flexibly.

Modern economic and financial data are increasingly organized as multiway arrays, with observations indexed simultaneously by geographic regions, industrial sectors, asset categories, and other economic characteristics. Representing such data as tensor-valued time series preserves their intrinsic multiway structure. Although substantial effort has been devoted to modeling the conditional mean of tensor-valued time series, comparatively less attention has been paid to their conditional covariance dynamics. The latter remains challenging because unrestricted multivariate covariance models involve many parameters and substantial computational cost. To address these challenges, we propose the Tensor-BEKK (T-BEKK) model, a tensor-structured BEKK specification that retains the positive definite covariance recursion for the vectorized process while imposing Kronecker structures on the intercept and the ARCH and GARCH coefficient matrices. The model reduces the parameter dimension and provides mode-specific interpretations of the covariance intercept, ARCH effects, and GARCH persistence. We establish stationarity, identification, and the asymptotic properties of the Gaussian quasi-maximum likelihood estimator. We further develop mode-specific restricted score tests tailored to the tensor structure, inference procedures for nonzero spillover intensities within each mode, and a portmanteau diagnostic test based on quadratic form residuals. For higher-dimensional settings, we also introduce the Tensor-Factor-BEKK (TF-BEKK) model. Under a first-step negligibility condition, its feasible second-step QMLE is asymptotically equivalent to the oracle QMLE based on the latent factors. Simulations and two empirical applications, covering currency futures and Chinese equity tensor portfolio allocation, illustrate the finite-sample behavior and empirical usefulness of the proposed methods.

Instrumental variable analyses often rely on the assumption that instruments affect the outcome only through the endogenous regressor. In many applications, researchers can defend only a plausible range for direct effects of instruments, while conventional sensitivity analyses may be unreliable when instruments are weak. This paper proposes the profiled Anderson–Rubin (pAR) test, which considers all direct effects within a prespecified range and retains a candidate effect whenever at least one admissible direct effect is consistent with the data. Under the maintained sampling assumptions, the procedure controls false rejection for each compatible candidate without requiring strong instruments. The paper provides practical methods for constructing confidence sets and distinguishes substantive bounds from bounds tied to the realized instrument design. Simulations and applications to retirement saving and returns to schooling show that the procedure resembles conventional sensitivity analysis when instruments are strong but preserves substantially more uncertainty when identification is weak.

Exploiting the equality restrictions that nested Markov models encode requires their tangent-space geometry. For strictly positive finite-state models on arbitrary acyclic directed mixed graphs (ADMGs), we differentiate the recursive-head chart and prove that the range of its score map is the full tangent space. Intrinsic-set coordinate blocks form an algebraic direct sum, blocks of distinct districts are orthogonal, and the resulting Gram projection needs neither mb-shieldedness nor a district order. Exact counterexamples show that observational centering and kernel normalization alone do not certify tangency. For the node, complete-source edge, and compatible path-specific intervention targets considered here, boundary substitution yields a normalized configured active law and an order-free canonical-gradient formula, extended to finite mixtures by independent source redraws. A coherent one-step estimator with an exact remainder identity and a model-valid chart-flow targeted maximum likelihood estimator are efficient under stated local conditions. On a narrower source-isolated fixed-node subclass, a sequential estimator has an exact transition-factorized drift and up to $2^m$ nuisance-correctness regimes over $m$ active-district transitions; its all-correct influence function projects onto the canonical gradient, and an exact rational law exhibits a strict variance gap. These results separate order-free efficiency on arbitrary finite-state ADMGs from multiple robustness of a narrower construction.

We study exact-cardinality, equally weighted minimum-variance portfolio selection under a one-factor covariance model supplied in factor form. In the nonnegative homoskedastic regime, selecting the K smallest loadings is optimal. Allowing strictly positive asset-specific idiosyncratic variances makes the decision problem NP-complete even with positive integer loadings and a strictly positive-definite covariance matrix; with identity residual covariance, exactly one negative loading also suffices. We give exact pseudo-polynomial dynamic programs for one factor and fixed factor dimension and prove W[1]-hardness parameterized by K, including the positive-data family. Consequently, a general exact polynomial-time algorithm for Monge's (2017) equally weighted single-factor variance-input formulation would imply P=NP. For a normalized binary factor encoding, we construct a depth-zero projection from modular k-SUM that preserves exact cardinality and positive definiteness. The projection transfers Lin's (2026) fixed-k circuit lower bound under its stated width and quantifier conditions and, independently, yields a parity-based proof that the portfolio language is not in nonuniform AC^0 even with identity residual covariance and polynomially bounded integer coefficients. These are restricted-circuit results: no unrestricted P/poly lower bound and no separation of P from NP is claimed.

Policy learning aims to determine who should be treated based on individual characteristics. In high-stakes settings such as medicine and public policy where safety is a central concern, improving the average outcomes alone may not be sufficient: decision makers may also seek to protect individuals from harm, in line with the Hippocratic principle of “do no harm.” In this paper, we propose conformal policy learning (CPL), a policy learning procedure with a new distribution-free safety guarantee that controls the probability of assigning treatment to an individual who would be harmed relative to control. CPL views each treatment decision as testing a hypothesis of counterfactual harm and assigns treatment by thresholding conformal p-values. These p-values use observable proxies and selective calibration to address the challenge that the potential outcomes under comparison are never simultaneously observed. For randomized experiments, under standard exchangeability conditions, CPL provides finite-sample safety guarantee at a user-specified level, without imposing any outcome modeling assumptions. Moreover, when the outcome model is consistently estimated, CPL achieves asymptotically optimal welfare subject to the safety constraint. In observational studies, CPL with learn-then-balance weights achieves doubly robust safety guarantees. We evaluate CPL through extensive simulations and apply it to an empirical study of AI-powered interventions designed to reduce conspiracy beliefs.