Tim Christensen

Professor
Department of Economics
Yale University


curriculum vitae



Working Papers

Externally Valid Policy Choice
w/ C. Adjaho
revise and resubmit, Quantitative Economics
We consider the problem of estimating personalized treatment policies that are "externally valid" or "generalizable": they perform well in target populations that differ from the experimental (or training) population from which the data are sampled. We first show that welfare-maximizing policies for the experimental population are robust to a certain class of shifts in the distribution of potential outcomes between the experimental and target populations (holding characteristics fixed). We then develop methods for estimating policies that are robust to shifts in the joint distribution of outcomes and characteristics. In doing so, we highlight how treatment effect heterogeneity within the experimental population shapes external validity.
Inference for Regression with Variables Generated by AI or Machine Learning
w/ L. Battaglia, S. Hansen, and S. Sacher
[python package] [R package]
Researchers routinely use AI or other machine learning methods to estimate latent variables, then plug-in the estimates as covariates in a regression. We show both theoretically and empirically that naively treating AI/ML-generated variables as "data" leads to biased estimates and invalid inference. To restore valid inference, we propose two methods: (1) an explicit bias correction with bias-corrected confidence intervals, and (2) joint estimation of the regression parameters and latent variables. We illustrate these ideas through applications involving label imputation, dimensionality reduction, and index construction via classification and aggregation.
From Unstructured Data to Demand Counterfactuals: Theory and Practice
w/ G. Compiani
Empirical models of multi-product demand rely on low-dimensional product representations to capture substitution patterns, increasingly using proxies built from unstructured data. When proxies are imperfect, standard workflows yield biased counterfactuals and invalid inference. We develop a practical toolkit to address these issues. Our methods apply to market-level and/or individual data, require minimal additional computation, provide simple standard-error formulas, and accommodate proxies from fine-tuned models. Further, we propose diagnostics to assess proxy quality. Our methods yield meaningful improvements in predicting substitution in empirically calibrated simulations and in an application where we assess counterfactual prediction performance against a ground truth.
Bootstrapping with AI/ML-Generated Labels
w/ S. Gonçalves and B. Perron
AI/ML methods are increasingly used in economics to generate binary variables (or labels) via classification algorithms. When these generated variables are included as covariates in regressions, even small misclassification errors can induce large biases in OLS estimators and invalidate standard inference. We study whether the bootstrap can correct this bias and deliver valid inference. We first show that a seemingly natural fixed-label bootstrap, which generates data using estimated labels but relies on a corrupted version in estimation, is generally invalid unless a strong independence condition between the latent true labels and other covariates holds. We then propose a coupled-label bootstrap that jointly resamples the true and imputed labels, and show it is valid without this condition. Two finite-sample adjustments further improve coverage: a variance correction for uncertainty in estimated misclassification rates and a Hessian rotation for near-singular designs. We illustrate the methods in simulations and apply them to investigate the relationship between wages and remote work status.
Culling the Factor Zoo
w/ G. Bakshi, J. Crosby, and X. Gao
We propose a method to select and estimate the best low-dimensional linear factor model from a high-dimensional set of factors. This method selects the best subset of factors and estimates their loadings by mixed-integer optimization. The approach is based on a sound theoretical criterion, is supported by statistical guarantees, precludes unreasonably high rewards-for-risk, and enforces positive SDFs. Portfolios based on factors selected with our method yield superior out-of-sample alphas relative to standard benchmarks.


Published Papers

Optimal Decision Rules when Payoffs are Partially Identified
w/ H.R. Moon and F. Schorfheide
accepted for publication, Review of Economic Studies
We derive optimal statistical decision rules for discrete choice problems when payoffs depend on a partially-identified parameter θ and the decision maker can use a point-identified parameter μ to deduce restrictions on θ. Examples include treatment choice under partial identification and pricing with rich unobserved heterogeneity. We discuss implementation of optimal decision rules in both parametric and semiparametric models via the bootstrap and Bayesian methods. We provide detailed applications to treatment choice and optimal pricing. Our approach is well suited for realistic empirical settings in which the derivation of finite-sample optimal rules is intractable.

Performing Valid Inference with AI/ML-Generated Covariates: A Guide for Empirical Practice
w/ S. Hansen
AEA Papers and Proceedings, 2026, 116, 92-97

Adaptive Estimation and Uniform Confidence Bands for Nonparametric Structural Functions and Elasticities
w/ X. Chen and S. Kankanala
Review of Economic Studies, 2025, 92(1), 162-196
[paper] [supplement] [replication files] [R package]

Counterfactual Sensitivity and Robustness
w/ B. Connault
Econometrica, 2023, Vol. 91(1), 263-298
[paper] [supplement] [replication files] [arxiv version]

Existence and Uniqueness of Recursive Utilities without Boundedness
Journal of Economic Theory, 2022, Vol. 200, 105413

​Monte Carlo Confidence Sets for Identified Sets
w/ X. Chen and E. Tamer
Econometrica, 2018, Vol. 86(6), 1965-2018
[paper] [supplement] [replication files]

Optimal Sup-norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression
w/ X. Chen
Quantitative Economics, 2018, Vol. 9(1), 39-85
[paper] [supplement] [replication files] [online appendix]

Nonparametric Stochastic Discount Factor Decomposition
Econometrica, 2017, Vol. 85(5), 1501-1536
[paper] [supplement] [replication files]

Nonparametric Identification of Positive Eigenfunctions
Econometric Theory, 2015, Vol. 31(6), 1310-1330

Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions
w/ X. Chen
Journal of Econometrics, 2015, Vol. 188(2), 447-465

Forecasting Spikes in Electricity Prices
w/ S. Hurn and K. Lindsay
International Journal of Forecasting, 2012, Vol. 28(2), 400-411

Detecting Common Dynamics in Transitory Components
w/ S. Hurn and A. Pagan
Journal of Time Series Econometrics, 2011, Vol. 3(1), Article 3

Software

Python Package ValidMLInference
R Package MLBC
Bias corrections for regression with AI/ML-generated covariates

R Package NPIV
Uniform confidence bands for nonparametric IV + regression