Timothy Christensen Publications

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 introduce two data-driven procedures for optimal estimation and inference in nonparametric models using instrumental variables. The first is a data-driven choice of sieve dimension for a popular class of sieve two-stage least-squares estimators. When implemented with this choice, estimators of both the structural function h₀ and its derivatives (such as elasticities) converge at the fastest possible (i.e. minimax) rates in sup-norm. The second is for constructing uniform confidence bands (UCBs) for h₀ 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 such, our UCBs are asymptotically more efficient than UCBs based on the usual approach of undersmoothing. As an application, we estimate the elasticity of the intensive margin of firm exports in a monopolistic competition model of international trade. Simulations illustrate the good performance of our procedures in empirically calibrated designs. Our results provide evidence against common parameterizations of the distribution of unobserved firm heterogeneity.

It has become common practice for researchers to use AI-powered information retrieval algorithms or other machine learning methods to estimate variables of economic interest, then use these estimates as covariates in a regression model. We show both theoretically and empirically that naively treating AI- and ML-generated variables as “data” leads to biased estimates and invalid inference. We propose two methods to correct bias and perform valid inference: (i) an explicit bias correction with bias-corrected confidence intervals, and (ii) joint maximum likelihood estimation of the regression model and the variables of interest. Through several applications, we demonstrate that the common approach generates substantial bias, while both corrections perform well.