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 study the problem of efficiently estimating the coefficients of generalized linear models (GLMs) in the large-scale setting where the number of observations n is much larger than the number of predictors p, i.e. n p 1. We show that in GLMs with random (not necessarily Gaussian) design, the GLM coefficients are approximately proportional to the corresponding ordinary least squares (OLS) coefficients. Using this relation, we design an algorithm that achieves the same accuracy as the maximum likelihood estimator (MLE) through iterations that attain up to a cubic convergence rate, and that are cheaper than any batch optimization algorithm by at least a factor of O(p). We provide theoretical guarantees for our algorithm, and analyze the convergence behavior in terms of data dimensions. Finally, we demonstrate the performance of our algorithm through extensive numerical studies on large-scale real and synthetic datasets, and show that it achieves the highest performance compared to several other widely used optimization algorithms.

Estimation and inference in statistics pose significant challenges when data are collected adaptively.

Predictive coding is a framework for understanding the formation of low-dimensional internal representations mirroring the environment's latent structure. The conditions under which such representations emerge remain unclear. In this work, we investigate how the prediction horizon and network depth shape the solutions of predictive coding tasks. Using a minimal abstract setting inspired by prior work, we show empirically and theoretically that sufficiently deep networks trained with multi-step prediction horizons consistently recover the underlying latent structure, a phenomenon explained through the Ordinary Least Squares estimator structure and biases in learning dynamics. We then extend these insights to nonlinear networks and complex datasets, including piecewise linear functions, MNIST, multiple latent states and higher dimensional state geometries. Our results provide a principled understanding of when and why predictive coding induces structured representations, bridging the gap between empirical observations and theoretical foundations.