Moment-Based Inference for Regression with Latent Dirichlet Covariates

Topic models are often used as dimension-reduction tools before regression, with estimated document-level topic shares treated as observed covariates. This plug-in workflow creates two inferential difficulties: valid inference requires a regular first-stage-to-second-stage expansion that propagates topic-estimation uncertainty, and, at fixed document length, a document's topic mixture cannot be consistently recovered from its own words even when the population topic matrix is known. Corrected spectral moment methods for latent Dirichlet allocation (LDA) offer a starting point: when the total Dirichlet concentration is known, low-order word moments can be corrected to yield operators diagonal in the latent topic basis. We extend this to downstream regression. Under a finite LDA model with response residuals orthogonal to the low-order token moments used for identification, response-weighted word moments admit the same correction, and the resulting supervised operator identifies the regression coefficient β directly, without estimating document-level topic shares. The main obstacle is that the correction depends on the unknown total concentration α0. We show that, for k≥3 topics and under a generic finite-probe condition, α0 is identified by commutativity: at the true value a family of corrected word-moment operators commute, whereas away from it they generically do not. This yields a feasible estimator and lets uncertainty in hatα0 propagate into inference for β. The estimator is asymptotically linear as the number of documents grows with fixed document length, with sandwich standard errors from document-level moment contributions. Simulations show near-nominal coverage where plug-in topic-share regressions can undercover, and an application to top economics journals illustrates contrast inference for latent topic effects.