GARCH-LSTM hybrid models have become increasingly popular in the volatility forecasting literature. To date, these models have almost exclusively focused on point estimation and accuracy measures such as RMSE and MAPE, while failing to assess the uncertainty of these point estimates. We approach this gap in two ways. First, we extend the existing GARCH-LSTM model by modeling the conditional mean with an ARMA structure. Second, we develop a forward bootstrap algorithm to build multi-step-ahead prediction intervals for future realized volatility. In order to address the notorious undercoverage of prediction intervals, we implement two solutions. One draws bootstrap innovations from the empirical residual distribution to mimic the fat-tail property of financial series. The other utilizes principles from conformal prediction to systematically extend the prediction interval length. We evaluate the methods in simulations with heavy-tailed and time-varying GARCH processes and in rolling-window forecasts of four U.S. stock indices from 2020 to 2025. The ARMA-based hybrid gives the lowest point-forecast errors in most settings, and combining both remedies with the mean-based hybrid yields intervals closest to the nominal level among competing methods.
Current evaluations of agentic AI systems primarily rely on benchmarks. However, the static nature of benchmarks makes them uninformative about the value of auditing interventions at deployment time. For reliable deployment, we often need to determine within an agent's trajectory whether or not to intervene, e.g., by verifying the agent's current output or steering it toward different behavior, using either human review or more costly models or time-consuming agentic supervision. The value of such interventions is fundamentally a causal inference question, asking whether the causal effect of such an intervention is beneficial for the final outcome. Estimating this causal effect is particularly challenging due to the dynamic nature of the problem: the impact of an intervention depends on the current context and environment, as well as the agent's future behavior. We develop the first methodologically grounded framework for evaluating the causal effects of randomized interventions within agent trajectories. By leveraging a relatively small dataset of agent trajectories collected under randomized interventions such as verification or screening, we can estimate the dynamic causal effect of a rich class of (contextual) intervention policies. The structural parameters that we estimate can also be used to construct safety leaderboards that capture the risk of various agentic actions, which can be used by practitioners for audit prioritization. Applying our methods to tau-bench Retail, ToolSandbox, and ImpossibleBench, we find that intervention effects vary substantially across agent actions and states. More broadly, our work suggests a rich research agenda at the intersection of causal inference and agentic evaluation, which we believe is essential for the reliable and safe deployment of AI agents in the real world.
Social media sentiment is increasingly used to predict stock returns. Its reliability depends on the composition of the crowd generating the feed. For foreign firms, the home-market investors who know them best are underrepresented on English-language Twitter. Hence, their feed should carry a less precise signal. Reduced-form analysis of firm-level Twitter sentiment shows that sentiment noise predicts next-quarter losses less consistently for foreign stocks than for domestic stocks. The loss-on-noise slope is negative in 78 percent of quarters for domestic stocks but in only 48 percent for foreign stocks, with a significant mean difference in slopes. We interpret this difference through a noisy rational expectations model in which the feed carries a public signal produced by three types of contributors. Specialists post repeatedly with information about the firm, attention-driven contributors post once with information about the firm and a transient fad, and automated accounts post repeatedly with no information about the firm. The share of each contributor type and the size of the fad determine the precision of the signal, which, in turn, determines how much of the feed's noise reaches prices. The model gives repeat-author share, fad and precision from the vendor's aggregate metrics, without account-level data, whereas prior work backs out precision from price informativeness. The structural estimates show that feeds for foreign stocks have a higher share of repeat authors, a larger fad component and a less precise signal. Thus, the foreign signal did not disappear but got diluted.
Causal inference under treatment interference requires assumptions to reduce the exponentially large number of potential outcomes. We develop hierarchical finite-population models for potential outcomes by separating these assumptions into three layers: interference neighborhoods, exposure mappings, and outcome models. Given an interference neighborhood and exposure mapping, we show that potential outcomes admit a nonparametric decomposition into direct, spillover, and treatment--spillover interaction components. This decomposition provides a common coordinate system for nonparametric, semiparametric, and parametric models of finite-population potential outcomes. For a given potential-outcome model, we define two classes of estimands. Average causal effects (ACEs), such as direct and total treatment effects, contrast potential outcomes under fixed treatment--exposure conditions. Expected causal effects (ECEs), such as expected direct and indirect effects, contrast expectations of potential outcomes under a specified allocation policy. For both classes, we bound the variance of the canonical Horvitz--Thompson estimator by three terms: a diagonal weight term, a term from overlapping interference neighborhoods, and a residual design-dependence term. Our upper and lower bounds show that under graph-unaware designs, nonparametric design-unbiased estimation of ACEs can require sample sizes growing exponentially in the interference degree, whereas polynomial growth can suffice for design-aligned ECEs. Under neighborhood interference, we construct graph-aware independent-set designs with variance bounds polynomial in the maximum degree for direct and total treatment effects on every graph, whereas graph cluster randomization can incur worst-case variance exponential in the degree.
For a stochastic program $\min_{z\in Z} c^T z + h(z)$ with a random cost vector $c$, regret, or the expected value of perfect information, is usually computed by re-solving the problem scenario by scenario. We show that regret is determined by the covariance between the cost vector and the optimal decision $\pi^s(c)$. The identity is exact and holds without regularity assumptions once the covariance is integrated along the path from the mean to each scenario. With $c_s=E[c]+s(c-E[c])$, $Reg=-\int_0^1 s^{-1} Cov(c_s,\pi^s(c_s))\,ds$. Three consequences follow. First, the plain covariance is always an upper bound: $0\le Reg\le-Cov(c,\pi^s(c))$. Second, when the optimal decision rule is affine, as in unconstrained and equality-constrained (e.g., budget-constrained Markowitz) quadratic programs, regret equals exactly one half of the covariance. For purely linear objectives, an affine rule forces zero regret. Third, the ratio of regret to covariance lies in $[0,1]$ and is governed by the shape of the covariance profile along the path. The noise level at which this profile first becomes positive is a decision-stability margin that tightens the bound. The identity also holds scenario by scenario for any predicted cost, so it applies to linear and combinatorial programs with discontinuous decision rules. We give dimension-free concentration bounds and a central limit theorem for the plug-in estimator. Across 7 problem classes, 8 cost distributions, and the shortest-path benchmark of Elmachtoub and Grigas (2022) with $10^6$ draws per configuration, the identities hold to within quadrature error, and the covariance bound holds in every scenario.
We study multi-task demand learning for retail pricing, where firms seek to estimate heterogeneous linear price-response functions across many decision contexts. Each context has rich covariates but limited price variation, motivating transfer across tasks. However, such transfer is complicated by endogeneity: prices may be arbitrarily correlated with unobserved task-level demand determinants. We propose a meta-learning framework that identifies the conditional mean of task-specific causal demand parameters despite this confounding, provided each task contains at least two distinct, locally exogenous price points. The key is to carefully design the information available to the learner. We condition on the full price history to address cross-task confounding, but withhold the demand outcomes associated with two locally exogenous price points to preserve identifiability. These withheld outcomes provide supervision through a randomized query. We establish that this information design is maximally uniformly valid for the squared-loss objective: any refinement of the conditioning set that reveals information about the withheld outcomes or the query index admits a data-generating process under which the objective no longer identifies the conditional mean causal target. Synthetic experiments demonstrate improved estimation of causal demand parameters under strong confounding, while retail experiments show improved held-out demand prediction relative to standard transfer-learning baselines.
This paper studies the task of estimating heterogeneous treatment effects in causal panel data models with covariate effects. We propose a novel Covariate-Adjusted Deep Causal Learning (CoDEAL) framework that cohesively deals with the underlying heterogeneity and nonlinearity of both panel units and covariate effects. CoDEAL integrates nonlinear covariate effect components (parameterized by a feed-forward neural network) with nonlinear factor structures (modeled by a multi-output autoencoder) to form a heterogeneous causal panel model. The nonlinear covariate component flexibly captures complex covariate influences on outcomes, and the nonlinear factor decomposition enables CoDEAL to effectively capture both cross-sectional and temporal dependencies inherent in the data panel. This latent structural information is subsequently integrated into a customized matrix completion algorithm, thereby facilitating more accurate counterfactual imputation. Moreover, the use of a multi-output autoencoder explicitly accounts for heterogeneity across units and enhances the model interpretability of the latent factors. We establish theoretical guarantees on the convergence of the estimated counterfactuals and demonstrate the compelling performance of the proposed method using extensive simulation studies and real-world data applications.
Digital traces may complement population surveys in the measurement of well-being, but the two sources do not necessarily capture identical constructs. We compare the Human Flourishing Geographic Index (HFGI) with World Values Survey and European Social Survey measures across three domains using structural equation models. The results show strong within-source coherence and construct-specific cross-source correspondence. Cross-source convergence varies across domains; the observed pattern is consistent with stronger correspondence where the underlying indicators are more closely aligned conceptually. Near-perfect agreement would be unexpected, since surveys elicit standardized self-reports, whereas HFGI captures naturally occurring public expressions. HFGI should therefore be viewed as a distinct but empirically connected measurement system that complements survey-based evidence on well-being.
Forecasting local elections is difficult because electoral competition is geographically heterogeneous, highly overdispersed, and characterized by many party-municipality combinations with little or no electoral activity. This article develops a bounded neural zero-inflated negative binomial (ZINB) model for forecasting party-level vote counts in Swedish municipal elections. The model combines a count component for expected votes, a zero-inflation component for structural electoral absence, municipality-specific dispersion, and learned embeddings capturing latent party, municipality, and party-municipality structure. Model training relies on data from the 2014, 2018, and 2022 Swedish municipal elections and performance is evaluated on a held-out set of municipalities. Testing results show strong predictive accuracy in both vote counts and vote shares. Explanatory diagnostics indicate that latent party structure, local party-municipality relationships, electoral persistence, organizational capacity, and candidate-electorate representational similarity are among the most important determinants of predictive performance. Following model evaluation and diagnostics, the selected specification is refit on all historical data and applied to generate forecasts for the 2026 municipal elections. The proposed framework provides a flexible approach for studying local electoral competition in multiparty systems characterized by heterogeneous party presence and substantial geographic variation.
Attention-based time series forecasters commonly group consecutive observations into patches. Patch length is known to have a substantial impact on forecasting performance, yet its selection remains largely empirical. We develop a theoretical framework that explains why patch length matters for forecasting accuracy and provides a principled basis for its selection. For linear state-space models, we derive an exact factorization of the forecasting coefficient matrix that separates forecast dynamics from filtering dynamics. We show that dominance of a single forecast component at long horizons does not generally imply a rank-one representation. This explains why longer horizons alone need not reduce representational complexity. We combine the factorization with estimation bounds to develop a patch-selection criterion that balances representation and estimation errors under temporal dependence. Our analysis establishes conditions under which balanced patching is justified and clarifies how predictive structure, head count, training sample size, and long-run dependence enter the selection criterion.
This paper examines robustness and causal interpretation of panel estimates using double/debiased machine learning. The prototype treatment setup is the 2018 US banking regulation. Under common treatment timing, we characterize identification and nuisance restrictions and derive exact bank-level representations of specified "within" and "first-difference" scores. Simulations assess panel model transformation-induced specification bias and inference under weaker overlap. Empirically, estimates vary with asset-size classification and prior growth. Prior-growth adjustment reduces overlap-weighted estimates across four Machine Learners (ML); doubly robust estimates depend on the fitted propensity and concentrated weights. Both bank groups received relief, so we estimate differential exposure effects. The findings highlight specification issues and the relation between functional-form robustness and causal identification.
We comment on the optimally-transported generalized method of moments (OTGMM) of Schennach and Starck (2026) and give counterexamples to Theorems 2-6. In the small-error analysis, the assumptions do not ensure consistency or asymptotic normality (Theorems 2-3). In the large-error analysis, Theorem 4 states the equivalence between the OTGMM estimator and the GMM estimator with modified moments, and Theorems 5-6 assert consistency and asymptotic normality. Through counterexamples, we show that the equivalence in Theorem 4 fails and a component of the Lagrange multiplier has different limits under the two estimators (Theorems 5-6). We also point out that, under misspecification, the population value selected by OTGMM depends on the metric. We give sufficient conditions and establish results following the original motivation for OTGMM. Our corrections to Theorems 4-6 change their assumptions or interpretation more substantially than those to Theorems 2 and 3. Furthermore, we correct Assumption 16 and the test's null distribution.
We consider linear regression when some covariates are endogenous, a variable sufficient to control for endogeneity exists but is latent, and the practitioner can generate many proxies for the latent control. For instance, one could cheaply but imperfectly impute the latent control using different AI/ML methods, each generating a different proxy. There are several options the practitioner might consider in this scenario: including all proxies or factors computed from the proxies as controls, instrumental variables estimation, and some nonlinear variants thereof. We use a transparent model to derive the implicit estimand each approach targets, determine when these estimands recover the parameter of interest, and provide some practical recommendations. We show that with a sufficiently rich set of generated controls, practitioners can consistently estimate the parameter of interest without relying on validation data.
We study how a seller can learn a reserve price for a second-price auction from censored historical bid data. Because values below past reserves are unobserved, even unlimited data may not reveal which reserve earns the most revenue. Under a monotone hazard rate (MHR) assumption, we characterize exactly when an optimal reserve is identified, even though the value distribution may not be: either revenue has not yet begun to fall at the lowest historical reserve, so the best reserve lies among the observed bids, or the censored fraction is as large as MHR allows, which pins down the entire distribution. When identification fails, we derive the reserve that minimizes worst-case revenue regret over distributions consistent with the data, and prove that this unique minimax reserve and its regret depend on just two observable quantities: the censored fraction and the hazard just above the lowest historical reserve. With $N$ historical bidders, the optimal reserve is learned with expected worst-case regret $O(N^{-1/2})$ when it lies among the observed bids, and $O(N^{-1})$ from the censored count alone when the censored fraction is as large as MHR allows and the optimum lies at or below the lowest reserve. In the nonidentifiable regime, plugging estimates of the two quantities into the minimax reserve gives expected worst-case regret within $O(N^{-2/5})$ of the population minimum under local density smoothness, with a matching minimax lower bound on a smooth class. Beyond MHR, we solve the minimax problem for regular and unrestricted distributions and prove an $O(N^{-2/5})$ learning bound in the regular nonidentifiable regime, given known bounds on the boundary parameters. Experiments on synthetic and eBay data show gains from exploiting MHR and the benefits of the robust rule when parametric models are misspecified.
The persistent volatility of construction material prices poses significant risks to cost estimation, budgeting, and project delivery, underscoring the urgent need for granular and scalable forecasting methods. This study develops a forecasting framework that leverages the Construction Specifications Institute (CSI) MasterFormat as the target data structure, enabling predictions at the six-digit Section level and supporting detailed cost projections across a wide spectrum of building materials. To enhance predictive accuracy, the framework integrates explanatory variables such as raw material prices, commodity indexes, and macroeconomic indicators. Five time-series models, long short-term memory (LSTM), autoregressive integrated moving average (ARIMA), vector error correction model (VECM), vanilla Transformer, and Chronos-Bolt, were evaluated under both baseline configurations (using CSI data only) and extended versions with explanatory variables. Results demonstrate that incorporating explanatory variables significantly improves predictive performance across all models. Among the tested approaches, the LSTM model consistently achieved the highest accuracy, with root mean squared error (RMSE) values as low as 4.19 and mean absolute percentage error (MAPE) values of 1.60 based on Division 06 as a representative demonstration, representing improvements of up to 59% over the traditional statistical time-series model, ARIMA. Validation across multiple CSI Divisions within the residential-material confirmed the framework's scalability, while Division 06 (Wood, Plastics, and Composites) is presented in detail as a demonstration case. This research offers a robust methodology that enables owners and contractors to improve budgeting practices and achieve more reliable cost estimation at the definitive level.
We develop PanelGP, a conditional factor model for large panels of stock returns in which firm characteristics determine the factor loadings and economic conditions determine the expected factors. Both relations are estimated nonparametrically with Gaussian-process priors. The kernel choice nests instrumented principal component analysis (IPCA) as well as smooth nonlinear maps. Asset-specific loading innovations and common factor innovations separate idiosyncratic from systematic risk, so one estimated model forecasts both expected returns and their covariance matrix. Estimation uses sparse variational inference with a joint posterior over loadings and factors. Because each predicted return is bilinear in a Gaussian state, the predictive covariance matrix and the cumulants of any portfolio have closed forms. In simulations, the estimator recovers the nonlinear maps and the variance parameters. For S&P 500 constituents over 1995-2024, nonlinear loadings rank stocks better than affine loadings and improve the Gaussian log score. Relative to IPCA, the resulting portfolios have higher Sharpe ratios, also net of trading costs, and the minimum-variance portfolio has lower volatility. The predicted variances rank stocks and months by risk but are too compressed.
Automated markets increasingly route order flow from learning trading agents through automated market makers (AMMs). We model trader archetypes as AI agents defined by their reward functions and analyze the feedback they create on a constant-product AMM. Herd-dominated and arbitrage-dominated markets of equal size have identical return distributions and turnover, so no rule based on volatility, value-at-risk, stress frequency, or volume can tell them apart; yet their stress episodes differ in length by a factor unbounded in the feedback-to-noise ratio $\beta/s$. Serial statistics identify the feedback sign from $\Theta((s/\beta)^2\log(1/\delta))$ observations, and this rate is tight. When agents trade together, profit-seeking agents ride herds rather than lean against them, leading to all-or-nothing participation. Spread-earning market makers restore stability but make liquidity fragile, with discontinuous collapse and hysteresis; maker--taker pricing, not a uniform fee, is the effective instrument.
In this paper, we develop a projected instrumental variable framework that relaxes conventional assumptions for causal identification with instrumental variables. Building on the interpretation of conditional expectation as an orthogonal projection in a Hilbert space, the framework replaces the conditional expectation operator with a general projection operator. We provide an alternative inverse problem formulation based on the Riesz representation. Drawing on semiparametric efficiency theory, we further develop debiased and adaptive machine learning methods for estimation and inference. The proposed framework connects invalid instrumental variables to instrumented difference-in-differences. We also examine invalid projected instrumental variables and discuss their relationship to existing methods. Simulation studies and real data analysis demonstrate that the proposed methods perform favorably compared with existing approaches.
A shortest-path decision is often made from edge costs that are only observed through noisy samples, such as the average of $m$ simulation replications. When the mean cost vectors of different instances lie near a $k$-dimensional subspace, projecting the averaged sample onto that subspace should remove noise. We make this precise. The mean squared error of any projection estimator splits exactly into a variance term and a bias term. The best projector is the positive eigenspace of $\Sigma_c - D/m$, where $\Sigma_c$ is the covariance of the true costs and $D$ is the noise covariance. For a rank-$k$ signal, the projection has the error of a sample mean with a factor $d/(\mu k)$ more replications, where $d$ is the number of edges and $\mu$ is an incoherence constant. For a signal that is only approximately low-rank, every rank-$k$ projection has an error floor, and the plain average wins beyond an explicit replication count. A margin bound shows that the chance of choosing a suboptimal path falls at the same improved rate. A projector learned from $n$ noisy instances loses at most an $O(d/n)$ term.
Simulated maximum likelihood is widely used in latent-variable models where likelihood evaluation requires intractable integrals or probabilities. With a fixed number of simulation draws, its estimating equation is generally biased. We develop Stochastic Approximation with Unbiased Simulated Scores (SAUSS), combining score simulation with stochastic gradient descent. The Hajivassiliou--McFadden identity, an instance of Fisher's identity, expresses the likelihood score as the conditional expectation of the complete-data score given the observed data. Exact conditional draws yield an unbiased simulated score for any fixed number of draws. The estimator averages stochastic gradient iterates computed using mini-batches of fixed size regardless of sample size, avoiding repeated full-sample likelihood evaluations. Under local regularity and initialization conditions, we establish asymptotic normality and a functional central limit theorem for the partial-sum process of the iterates under multi-pass sampling from an observed sample. These results support plug-in and random-scaling confidence intervals that remain asymptotically valid when algorithmic uncertainty is non-negligible relative to sampling uncertainty. We develop multinomial probit as a leading application. In simulations and two applications, SAUSS returns estimates in every reported run, including cases where the simulated maximum likelihood implementation returns none. When both return estimates, SAUSS gives comparable estimates in less time.