Moment-Based Inference for Regression with Latent Dirichlet Covariates

Abstract: 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 $α_0$. We show that, for $k\ge3$ topics and under a generic finite-probe condition, $α_0$ 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 $\hatα_0$ 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.

• The paper develops a spectral moment method to correct bias from plug-in estimators in regression with latent Dirichlet covariates.
• It uses observed-space operators and operator commutativity to identify and estimate both regression coefficients and the Dirichlet concentration parameter.
• Empirical results confirm that the new estimator achieves near-nominal confidence coverage, outperforming standard plug-in approaches.

Moment-Based Inference for Regression with Latent Dirichlet Covariates

Problem Statement and Theoretical Framework

The paper "Moment-Based Inference for Regression with Latent Dirichlet Covariates" (2605.30718) addresses the inferential challenges that arise when using topic models as dimension-reduction tools prior to regression, particularly within the canonical Latent Dirichlet Allocation (LDA) framework. The standard plug-in approach replaces the latent document-level topic proportions with their empirical estimates and treats these as observed covariates in downstream regression. This widely used methodology suffers from two substantial inferential deficiencies:

1. Inferential Regularity: Plug-in procedures omit the first-stage estimation error from downstream confidence intervals, invalidating coverage unless this component is asymptotically negligible.
2. Generated Regressor Bias: When document length is fixed, individual latent topic mixtures cannot be consistently recovered from their own token sets—even with a known topic matrix—leading to a persistent non-vanishing document-level error in the constructed covariate, and thus finite-sample bias that cannot be ignored in any classical asymptotic regime.

The paper thereby centers the object of inference not on document-specific latent variables, but on the population-level regression coefficient $\beta$ in the latent model

$Y_i = \beta^\top h_i + \varepsilon_i,$

where $h_i$ is the unobserved Dirichlet-distributed topic mixture, conditioned on by the observed response $Y_i$, and $\varepsilon_i$ satisfies specific orthogonality restrictions with moment structure.

Methodological Innovations

Corrected Spectral Moment Methods Extended to Regression

The work leverages and extends corrected spectral moment methods for LDA—originally designed for structural recovery when the Dirichlet concentration parameter $\alpha_0$ is known. When $\alpha_0$ is given, low-order cross-token word moments, upon subtracting suitable lower-order terms, yield observable operators that are diagonal in the latent topic basis. The main theoretical contributions are:

• Observed-Space Spectral Operator: The authors derive an observed-space operator using response-weighted, corrected moments. After projecting onto the recovered topic basis, this operator becomes diagonal—with diagonal entries proportional to the elements of $\beta$. Critically, this procedure enables identification and estimation of $\beta$ directly from word–response moments, eliminating the need to estimate document-topic shares.
• Identification and Estimation of the Concentration Parameter $\alpha_0$: A crucial advance is the realization that $Y_i = \beta^\top h_i + \varepsilon_i,$0 is identifiable by the commutativity of a family of corrected word-moment operators. Specifically, for $Y_i = \beta^\top h_i + \varepsilon_i,$1, at the true $Y_i = \beta^\top h_i + \varepsilon_i,$2, all such operators commute; away from this value, they generically do not. This yields a feasible estimator for $Y_i = \beta^\top h_i + \varepsilon_i,$3, the uncertainty of which can be propagated into the inference for $Y_i = \beta^\top h_i + \varepsilon_i,$4.
• Asymptotic Inference under a Fixed-Dimensional Regime: The asymptotic regime considered is fixed document length and vocabulary, with the number of documents $Y_i = \beta^\top h_i + \varepsilon_i,$5. Under this regime, suitably corrected moment estimators for $Y_i = \beta^\top h_i + \varepsilon_i,$6 are shown to be asymptotically linear, with regular influence functions and sandwich variance estimators.

Practical Workflow and Extensions

The workflow consists of constructing within-document empirical moments (including cross-token second and contracted third moments), estimating $Y_i = \beta^\top h_i + \varepsilon_i,$7 by minimizing the total non-commutativity over a finite family of probes, and then performing spectral decomposition to obtain both topic directions and the regression coefficient. This procedure is shown, both theoretically and empirically, to provide valid frequentist uncertainty quantification for $Y_i = \beta^\top h_i + \varepsilon_i,$8—unlike plug-in approaches, which fail even under known $Y_i = \beta^\top h_i + \varepsilon_i,$9.

Extensions to the methodology include:

• Compatibility with linear dimensionality reduction (PCA), provided the compressed topic subspace is admissible.
• Handling observed controls in regression via additional linear moment equations and a moment-based scale recovery procedure when spectral directions are not topic simplex normalized.
• Application to growing-vocabulary regimes via split-sample preprocessing.

Numerical Results and Empirical Illustration

Monte Carlo Evidence

Strong numerical results demonstrate the superiority of the proposed estimator in finite samples:

• Coverage: The moment-based estimator’s confidence intervals exhibit coverage near nominal levels (e.g., average coverage $h_i$0 to $h_i$1 as $h_i$2 increases from $h_i$3 to $h_i$4 in symmetric Dirichlet design with $h_i$5, $h_i$6). In contrast, plug-in regressions (even with the true topic matrix) suffer severe undercoverage (declining from $h_i$7 to $h_i$8 across the same range).
• Plug-In Failure: The performance gap holds both for symmetric and asymmetric Dirichlet priors. Notably, this underperformance of plug-in procedures is not remedied by supplying the true $h_i$9, directly confirming that the root problem lies in treating noisy, finite-document topic estimates as observed covariates.
• Estimation of $Y_i$0: The commutator-based estimator for $Y_i$1 is shown to be accurate, with empirical standard deviations closely matching analytic standard errors and negligible bias/variance as $Y_i$2 increases.

Application to Economics Article Citations

The paper applies the method to regression of log-citation counts on latent abstract topics (controlling for publication year) in top economics journals. The estimated topic effects (as contrasts) are significant, robust to choice of $Y_i$3, and connect to known field-level citation dynamics, providing a rigorous inferential framework for associating latent text structure with downstream outcomes.

Theoretical and Practical Implications

Theoretical Implications

• Plug-In Regret Cannot Be Neglected: The documented finite-document generated-regressor bias exposes a non-negligible inferential distortion in all classical asymptotic analyses employing plug-in approaches with fixed-length units and latent variable regression. This insight generalizes to a broad class of inference problems involving model-based lower-dimensional representations of high-dimensional or unstructured data.
• Spectral Commutativity as an Identification Tool: The use of operator commutativity for parameter identification in latent variable models presents a broadly applicable technique for scenarios where conventional likelihood-based or Bayesian approaches are infeasible or theoretically inadequate.

Practical Implications

• Robust Frequentist Inference: The proposed moment-based estimator delivers calibrated confidence intervals and valid inference for downstream regression coefficients without requiring accurate per-document recovery of latent variables—permitting reliable use in domains where document length is fixed or limited.
• Computational Transparency: As all steps are based on low-order moments and observable document statistics, the approach is amenable to auditing and diagnostic checking (e.g., through the shape, stability, and curvature of the commutator objective), which is beneficial for research transparency.

Future Directions

Potential extensions include:

• Incorporation of high-dimensional asymptotic regimes (growing $Y_i$4 or $Y_i$5 with $Y_i$6), to address massive corpora in practice.
• Generalization to non-Dirichlet latent mixture settings, correlated topic models, and relaxation of moment orthogonality constraints.
• Application to other latent variable models where regression targets or contrasts—not the latent structure itself—are the primary objects of inference.

Conclusion

This work establishes a rigorous and implementable alternative to plug-in regression for downstream inference with latent topic covariates, demonstrating theoretically and empirically that classical plug-in methods are not reliable at fixed document length, and that uncertainty must be propagated using corrected spectral moment techniques. The operator-commutativity identification of $Y_i$7 further broadens the applicability of moment-based inference in latent variable models. This framework provides a robust basis for empirical researchers employing topic models (and related latent factor structures) in econometric, social science, and machine learning workflows, and illuminates avenues for future inference-focused methodological developments.