[論文レビュー] Two-stage Estimation of Latent Variable Regression Models: A General, Root-N Consistent Solution

Yang Liu, Xiaohui Luo+5 more|arXiv (Cornell University)|Jan 24, 2026

Psychometric Methodologies and Testing被引用数 0

ひとこと要約

The paper develops a general bias-correction framework for two-stage estimation in latent variable models, showing sqrt(n)-consistency and providing practical algorithms for bias-corrected FSR (factor score regression) alongside Monte Carlo variance estimation.

ABSTRACT

Latent variable (LV) models are widely used in psychological research to investigate relationships among unobservable constructs. When one-stage estimation of the overall LV model is challenging, two-stage factor score regression (FSR) serves as a convenient alternative: the measurement model is fitted to obtain factor scores in the first stage, which are then used to fit the structural model in the subsequent stage. However, naive application of FSR is known to yield biased estimates of structural parameters. In this paper, we develop a generic bias-correction framework for two-stage estimation of parametric statistical models and tailor it specifically to FSR. Unlike existing bias-corrected FSR solutions, the proposed method applies to a broader class of LV models and does not require computing specific types of factor scores. We establish the root-n consistency of the proposed bias-corrected two-stage estimator under mild regularity conditions. To ensure broad applicability and minimize reliance on complex analytical derivations, we introduce a stochastic approximation algorithm for point estimation and a Monte Carlo-based procedure for variance estimation. In a sequence of Monte Carlo experiments, we demonstrate that the bias-corrected FSR estimator performs comparably to the ``gold standard'' one-stage maximum likelihood estimator. These results suggest that our approach offers a straightforward yet effective alternative for estimating LV models.

研究の動機と目的

• Motivate the use of two-stage estimation as a practical alternative when one-stage maximum likelihood is challenging in LV models.
• Develop a general bias-correction strategy to adjust two-stage estimators in the presence of multiple nuisance and focal parameters.
• Establish sqrt(n)-consistency of the bias-corrected estimator under mild regularity conditions.
• Provide implementable algorithms for bias-corrected point estimation and valid large-sample standard errors.
• Demonstrate performance through Monte Carlo studies across common LV models (simple latent regression, latent moderation, multidimensional IRT).

提案手法

• Propose a general, blackbox bias-correction framework for two-stage estimation using stochastic approximation to adjust focal parameters.
• Partition parameters into nuisance and focal parts to enable sequential estimation in two stages.
• Define the bias-correction mapping h, its inverse h^{-1}, and the Delta matrix to obtain the corrected estimator.
• Prove sqrt(n)-consistency of the bias-corrected estimator and derive its asymptotic covariance via implicit differentiation and the Delta method.
• Provide Algorithms 1 and 2: a Robbins-Monro style procedure for bias correction and a practical method for computing the asymptotic covariance matrix.
• Show applicability to factor score regression in LV models beyond linear measurement/structural forms.

Figure 1: Graphical illustration of the bias-correction strategy in a single-parameter model. The parameter is denoted by $\varphi$ . Panel A: The expected value of the initial (possibly biased) estimator $\hat{\varphi}(\mathbf{Y}_{1:n})$ , $h(\varphi)=\mathbb{E}_{\varphi}\hat{\varphi}(\mathbf{Y}_{1

実験結果

リサーチクエスチョン

• RQ1 Can a general bias-correction strategy be developed for two-stage LV model estimation that handles multiple nuisance and focal parameters?
• RQ2 Under mild regularity conditions, is the bias-corrected two-stage estimator sqrt(n)-consistent and how can its variance be estimated in practice?
• RQ3 How can practitioners implement bias-corrected FSR with minimal analytical derivations and only simple input programs?
• RQ4 Do Monte Carlo experiments demonstrate that bias-corrected FSR performs comparably to one-stage maximum likelihood in common LV models?

主な発見

• The proposed bias-correction framework yields sqrt(n)-consistent estimates for focal LV parameters under mild regularity conditions.
• The method relies on a general h function that maps biased initial estimators to their unbiased counterparts via h^{-1}, enabling bias correction.
• A practical Robbins-Monro algorithm and a Monte Carlo variance estimation procedure are provided for implementation.
• The approach applies to a broad class of LV models and does not require specialized factor scores or full likelihood evaluation in the second stage.
• Monte Carlo studies show the bias-corrected FSR estimator performs comparably to the one-stage ML estimator across various LV settings.