Tim Christensen

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.

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.

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.

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.

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 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.

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.

Researchers increasingly use AI and machine learning to generate variables that are used in regression analysis. Ignoring measurement error in these variables can yield biased estimators and invalid inference. The methods that exist for bias correction require extensive validation data, which are typically not available in economic applications. We describe bias correction methods that do not require such data and show how empiricists can implement them via the Python package ValidMLInference. We illustrate with two applications: estimating the association between salary and remote work, and estimating long-run interest rate reactions to the sentiment expressed in Federal Open Market Committee statements.

We introduce two procedures for estimation and inference in nonparametric models using instrumental variables: a data-driven choice of sieve dimension and a method for constructing uniform confidence bands (UCBs) for the unknown function and its derivatives. Our UCBs guarantee coverage over a generic class of data-generating processes and contract at the minimax rate, possibly up to a logarithmic factor. As an application, we estimate the elasticity of the intensive margin of firm exports in a monopolistic competition model of international trade.

We propose a framework for analyzing the sensitivity of counterfactuals to parametric assumptions about the distribution of latent variables in structural models. We derive bounds on counterfactuals as the distribution of latent variables spans nonparametric neighborhoods of a given parametric specification while other “structural” features of the model are maintained. Our approach recasts the problem as a finite-dimensional convex program. We propose estimators of the bounds and two methods for inference. We present empirical applications to matching models with transferable utility and dynamic discrete choice models.

This paper derives primitive, easily verifiable sufficient conditions for existence and uniqueness of (stochastic) recursive utilities for several important classes of preferences. To accommodate models commonly used in practice, we allow both the state space and per-period utilities to be unbounded. For many of the models we study, existence and uniqueness is established under a single “thin tail” condition on the distribution of growth in per-period utilities. We present several applications to robust preferences, models of ambiguity aversion and learning about hidden states, and Epstein–Zin preferences.

We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of the full parameter vector and of subvectors in models defined through a likelihood or a vector of moment equalities or inequalities. The CSs are based on level sets of criterion functions and use cutoffs that are computed via Monte Carlo (MC) simulations from the quasi-posterior distribution of the criterion. We establish new Bernstein–von Mises type theorems for the quasi-posterior distributions of the quasi-likelihood ratio (QLR) and profile QLR in partially-identified models.

We derive sup-norm convergence rates for sieve NPIV (series two-stage least squares) estimators of structural functions and their derivatives. We also show that these rates are minimax when a spline or wavelet sieve is used. We then establish uniform Gaussian process strong approximations and the validity of score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of the structural functions. As applications, we show how to construct pointwise and uniform confidence bands for exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low-level conditions.

This paper introduces an empirical framework to analyze the permanent-transitory decomposition of SDF processes in dynamic asset pricing models. We show how to estimate nonparametrically the solution to the Perron–Frobenius eigenfunction problem of Hansen and Scheinkman (2009). Our empirical framework allows researchers to (i) construct time series of the estimated permanent and transitory components and (ii) estimate the yield and the change of measure which characterize pricing over long horizons. We establish consistency and convergence rates of the eigenfunction estimators and asymptotic normality of the eigenvalue estimator. We also introduce nonparametric estimators of the continuation value function in a class of models with recursive preferences. As an application, we study an economy where the representative agent is endowed with recursive preferences, allowing for general (nonlinear) consumption and earnings growth dynamics.

Important features of certain economic models may be revealed by studying positive eigenfunctions of appropriately chosen linear operators. Examples include long-run risk–return relationships in dynamic asset pricing models and components of marginal utility in external habit formation models. This paper provides identification conditions for positive eigenfunctions in nonparametric models. Identification is achieved if the operator satisfies two mild positivity conditions and a power compactness condition. Both existence and identification are achieved under a further nondegeneracy condition. The general results are applied to obtain new identification conditions for external habit formation models and for positive eigenfunctions of pricing operators in dynamic asset pricing models.

We show that spline and wavelet series regression estimators for weakly dependent regressors attain the optimal uniform (i.e. sup-norm) convergence rate of Stone (1982). The optimal rate is achieved even for heavy-tailed martingale difference errors. We also establish the asymptotic normality of t-statistics for possibly nonlinear, irregular functionals of the conditional mean function under weak conditions. The results are proved by deriving a new exponential inequality for sums of weakly dependent random matrices, which is of independent interest.

In many electricity markets, retailers purchase electricity at an unregulated spot price and sell to consumers at a heavily regulated price. Consequently, the occurrence of spikes in the spot electricity price represents a major source of risk for retailers, and the forecasting of these price spikes is important for effective risk management. Traditional approaches to modelling electricity prices have aimed to predict the trajectory of spot prices. In contrast, this paper focuses on the prediction of price spikes. The time series of price spikes is treated as a discrete-time point process, and a nonlinear variant of the autoregressive conditional hazard model is used to model this process. The forecasting performance of the model is evaluated against a benchmark that is consistent with the assumptions of commonly-used electricity pricing models.

This paper considers VECMs for variables exhibiting cointegration and common features in the transitory components. While the presence of cointegration between the permanent components of series reduces the rank of the long-run multiplier matrix, a common feature among the transitory components leads to a rank reduction in the matrix summarizing short-run dynamics. The common feature also implies that there exist linear combinations of the first-differenced variables in a cointegrated VAR that are white noise and traditional tests focus on testing for this characteristic. An alternative, however, is to test the rank of the short-run dynamics matrix directly. Consequently, we use the literature on testing the rank of a matrix to produce some alternative test statistics. We also show that these are identical to one of the traditional tests. This approach is applied to provide a check for the presence of common dynamics in DSGE models.