Correcting Bartlett factor-score error for latent covariates in partially linear double machine learning

Abstract

Substituting Bartlett factor scores for latent covariates leaves factor-score error that does not vanish as the sample grows and biases the adjusted structural coefficient. We propose Croon-DML, which subtracts the factor-score error covariance implied by the measurement model from the cross-fitted estimating equation of partially linear double/debiased machine learning (DML), and establish identification, Neyman orthogonality with respect to the conditional means, consistency, and asymptotic normality. Residual bias after correction comes from learning the conditional means of the factor scores: signal learning error, and noise fitting, in which the learner also fits the factor-score error. Signal learning error largely cancels when the three conditional means share a common penalty, and noise fitting by a linear smoother can be computed from its weights alone, so we also subtract it (Croon-DML-R). Across 12 Monte Carlo conditions, Croon-DML reduced the absolute bias of uncorrected DML by 66-95%, and the bias of Croon-DML-R with a common penalty differed from that of an oracle estimator using the true latent variables by at most 0.004. In an illustration with public longitudinal data, the correction was 15% of the latent-covariate adjustment; its size reflected the latent-covariate coefficients rather than the residualized reliabilities.