When Can We Work in Embedding Space? What Text Embeddings Preserve

When do text embeddings work as inputs to empirical analysis? Their use rests on an assumption: that we can trade text for its low-dimensional embedding, and lose little in doing so. I make that assumption precise under a generative model in which documents are mixtures of latent topics. I study two uses---clustering units in embedding space and controlling for high-dimensional text. A cluster of embeddings is a set of documents with similar topic mixtures; controlling for the embedding is equivalent to controlling for the topic mixture, so validity reduces to whether that mixture captures the confounding. In an application to 363 U.S. metropolitan areas, embedding-based clusters of LLM-generated economic descriptions recover interpretable economic archetypes and separate local employment dynamics more sharply than clustering on model residuals, or on a curated set of industry and demographic covariates.

SAUSS: Stochastic Approximation with Unbiased Simulated Scores for Limited Dependent Variable Models

Multinomial choice models allow flexible substitution patterns but become computationally demanding with many alternatives or observations. With a fixed per-observation simulation budget, simulated maximum likelihood introduces simulation bias, while each optimization step requires a full-sample likelihood evaluation. We propose Stochastic Approximation with Unbiased Simulated Scores (SAUSS), an averaged stochastic approximation based on conditionally unbiased mini-batch score estimates. Each iteration uses a fixed mini-batch regardless of sample size. For multinomial probit, accept-reject sampling provides exact conditional draws and unbiased score estimates for any fixed number of accepted draws. Under local conditions, asymptotic theory for the averaged estimator and the partial-sum process of the SAUSS iterates incorporates mini-batch and simulation variability and supports random-scaling and plug-in inference. In simulations and an application, SAUSS gives comparable results in less than 1% of the computation time of simulated maximum likelihood. SAUSS extends to limited dependent variable models with conditional-expectation score representations and exact conditional sampling.

Algorithm-Driven SVARs: Navigating the Wilderness of Big Data

Every SVAR result is conditional on two choices: the restrictions that identify the shock and the variables on which they operate. The literature disciplines the first; the second is chosen by hand. We develop a Bayesian methodology that constructs information sets, uses an out-of-sample criterion, and retains the largest system it admits. Under recursive identification, output rises with housing production rather than household credit alone. For monetary policy, an anchor-free joint Bayesian proxy SVAR with multiple instruments strengthens the credit spread channel. A core system augmented with the selected corporate spread identifies expected default risk as a potent transmission margin.

A fully nonlinear structural vector autoregressive model identified via independent innovation analysis

We develop a fully nonlinear structural vector autoregressive framework in which the contemporaneous structural mapping may be nonlinear and non-additive. Identification is achieved by exploiting variation in the conditional distributions of the mutually independent structural shocks induced by an observed exogenous variable. Specifically, a general contrastive learning framework that makes use of this variation together with the assumed exponential-family structure is employed to recover the shocks. Existing independent innovation analysis results identify such shocks only up to arbitrary componentwise invertible transformations, which is generally insufficient for structural econometric analysis. We strengthen this result by imposing a structured exponential-family specification for the conditional shock distributions. With the imposed sufficient statistics, the remaining ambiguity is reduced to a one-parameter transformed-scale map for each shock. We then show that, under a logistic specification used for the natural parameters, the identification is further strengthened up to permutation and componentwise sign changes. Once the shocks have been recovered, the fully nonlinear structural vector autoregression can be estimated using feed-forward neural networks, motivated by their universal approximation capabilities. The empirical application studies asymmetries in the responses of U.S. industrial production to the real oil price shock. We find modest asymmetries with respect to the sign of the shock and state of the economy. The accompanying R package iiasvar implements the introduced methods.

TorchDCM: A Unified PyTorch-Native Package for Discrete Choice Modeling

Estimating large and simulation-intensive discrete choice models (DCMs) requires repeated evaluation of utilities, probabilities, derivatives, and simulated likelihoods over many observations, alternatives, and draws. Existing DCM software provides mature econometric workflows, while recent GPU-oriented tools accelerate selected models, leaving a gap between econometric coverage and scalable differentiable computation. We introduce TorchDCM, an open Python package for discrete choice modeling that compiles choice data and model specifications into a unified PyTorch-native likelihood engine for estimation, inference, prediction, and structured reporting on CPU or CUDA devices. The package covers the principal econometric functionality available across Biogeme and Apollo, including multinomial, nested, mixed, ordered, latent-variable, and panel likelihoods. It also supports ragged choice sets, constrained parameters, covariance estimation, willingness-to-pay analysis, elasticities, and extensible likelihood components. We evaluate TorchDCM against seven other estimation packages in aligned synthetic and real-data full-estimation experiments. TorchDCM completes all 45 synthetic cases, runs fastest in every comparable synthetic case, and satisfies the prespecified final-log-likelihood tolerance in every comparison with at least two comparable solutions. More precisely, it reduces median runtime by 89.1%-99.7% relative to Biogeme and Apollo across model-data settings. CUDA provides an additional 12.0-71.0x speedup over single-core TorchDCM. These results establish a scalable and reproducible foundation for econometric estimation and differentiable choice-model development. The open-source package and executed examples are available at https://github.com/mbc96325/torchdcm.

A robust association between LLM use and scientific productivity: Assessing stopping-time selection

Renault, Bergeaud, and Bosquet (hereafter RBB) argue that dating LLM adoption as the first month in which an author's abstract is flagged induces a stopping-time selection that can produce a positive event-study path even when there is no causal effect. Although this mechanism is mathematically possible, it does not constitute proof of a null effect. Recalibrating RBB's own random placebo to the detector's realized flag rate, we show that the measured association stays well above this benchmark, so the artifact is too small to explain the productivity changes. We further re-estimate the association between LLM adoption and productivity with a series of complementary designs in which the timing artifact cannot bias the estimate: a before-and-after comparison that dates adoption in one year and measures output in another, a conservative control group for difference-in-differences, an intensity-based specification that never defines an adoption date, and a rank-based measurement holding the flag rate fixed. A positive productivity association persists across all of these estimates, while the same tests run on pre-ChatGPT placebo data return null effects. The artifact RBB identify is real but bounded, and it does not account for the pattern we report.

A Fixed-Effects Causal Forest for Staggered Adoption, with an Application to Medicaid Expansion

Difference-in-differences with staggered adoption identifies group-time average treatment effects ATT(g,t) by comparing each cohort to units not yet treated, which avoids the "forbidden comparisons" that bias two-way fixed-effects estimators when effects are heterogeneous. This paper studies the covariate-conditional version of that object, tau_{g,t}(x), and estimates it with a fixed-effects causal forest. Within each (g,t) comparison block, the outcome and treatment are residualized on unit and period fixed effects inside each tree node, and honest causal trees split on treatment-effect heterogeneity in the covariates. The estimand is not new: Hatamyar, Kreif, Rocha and Huber (2023) introduced it using a doubly-robust R-learner, and Imai, Qin and Yanagi (2023) study it for a single continuous covariate. What we add is a different way to estimate it. Where those methods remove confounding by modeling nuisance functions, we remove it by differencing out unit and period effects within each tree node, following the fixed-effects residualization of Kattenberg, Scheer and Thiel (2023) and Gavrilova, Langorgen and Zoutman (2025) and carrying it into the Callaway-Sant'Anna group-time structure. In Monte Carlo experiments the estimator is the only forest-based method that stays unbiased and correctly covered for the overall effect under staggered timing with cohort-varying effects; two-way fixed effects and a pooled causal forest inherit large forbidden-comparison bias. We apply the method to the Callaway-Sant'Anna minimum-wage panel as a validation and to the staggered county-level rollout of the ACA Medicaid expansion, where it recovers an average 2.25 percentage-point fall in the uninsured rate and a conditional surface on which poorer and lower-income counties gained substantially more coverage -- heterogeneity measured along socioeconomic covariates that are not lags of the outcome.

Econometrics with Pre-Trained Embeddings for Unstructured Data

Unstructured data, such as images and text, are increasingly used in empirical economics. Since training machine-learning models on unstructured data is costly, economists often use off-the-shelf pre-trained deep learning models developed by computer scientists to extract embeddings, which are then used as covariates in target economic analyses. Despite the popularity of this practice, its theoretical foundations remain limited. There are two main difficulties. First, the pre-trained model is usually trained on a different dataset and for a different task. Consequently, it is unclear when such a model can be used reliably for the target task. Second, the embedding function is subject to an identification problem, which makes it difficult to analyze the estimation error of the embedding function and its effect on the target task. In this paper, we provide sufficient conditions to overcome these difficulties and derive the convergence rate of machine learning models with pre-trained embeddings. We illustrate the theory through double machine learning applications for estimating parameters of interest, such as partially linear regression with unstructured controls, price elasticity in demand estimation considering the product quality measured by images and text, missing data imputation with unstructured data, and the average treatment effect with unstructured confounders.

Liquidity-Based Audit of Algorithmic Trading Strategies

We show that net demand for liquidity by algo strategies is identifiable from its trade and price history alone, with no knowledge of its signal or optimization problem. An exact multi-period regret decomposition implies that the sign of this statistic classifies a linear strategy as a net liquidity consumer or provider, recovering the Kyle (1985) informed-trader/market-maker dichotomy from observables alone. Under an AR(1) cost process, the same statistic equals the product of strategy size and the squared Roll (1984) implied spread, making the correction a direct proxy for prevailing illiquidity. Extending to endogenous price impact and aggregating across N correlated strategies yields a liquidity-balance condition whose violation produces welfare loss scaling as N squared, a closed-form fire-sale externality. We calibrate to CRSP equity data (2016-2025), tracking implied spreads through the COVID-19 and 2022 rate-shock episodes, with an estimator computable in O(Tnd) time.

Heterogeneous Effects of Green Finance on Urban Decarbonization: Evidence from 285 Cities in China

While green finance has become a key instrument for low-carbon city transitions, its actual decarbonization effects and transmission mechanisms remain unclear. This study employs econometric models and machine learning-based analysis to examine whether and how green finance reduces city-level carbon intensity. Results show that green finance significantly lowers carbon intensity, with green bonds and green investment having the strongest impacts and evident spatial spillovers. The effects vary by development level, being most pronounced in Fourth- and Fifth-tier cities. Mediation analysis reveals that green finance operates mainly through energy structure optimization, followed by industrial upgrading, foreign direct investment, and technological innovation. SHAP analysis confirms substantial differences across financial instruments, with green bonds, funds, and credit contributing most to decarbonization. Moreover, the marginal impact is stronger in cities with low technological capacity, high industrial dependency, and coal-based energy mixes. These findings provide theoretical support and policy guidance for building a multi-level, regionally differentiated green finance system to promote inclusive low-carbon transitions. Keywords: Green Finance; Carbon Intensity; Decarbonization Effect; Machine Learning; City