<p>Envelope models are powerful tools for improving estimation efficiency in multivariate regression by identifying and excluding immaterial variation. However, existing envelope approaches typically rely on the assumption of exogenous predictors and thus fail to account for endogeneity arising from unobserved confounding. This paper introduces a novel framework that integrates instrumental variable (IV) techniques into envelope modeling, referred to as the Instrumental Variable Envelope (IVE) method. By unifying IV regression with dual-envelope dimension reduction, the IVE framework extends envelope methodology to accommodate endogenous regressors and high-dimensional covariates. The proposed two-stage procedure first constructs an IV-adjusted predictor envelope to eliminate confounding bias, and then estimates a response envelope on the transformed design. We establish the unbiasedness and asymptotic efficiency of the proposed estimator, and show that it achieves strictly lower variance than standard envelope estimators in the presence of endogeneity. Simulation studies confirm the superior predictive accuracy and robustness of the IVE method across a range of data-generating scenarios. An empirical application to automobile market share analysis further demonstrates the practical advantages of the proposed approach.</p>

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Instrumental variable envelope models for endogenous multivariate regression

  • Lexuan Zheng,
  • Yuehan Yang

摘要

Envelope models are powerful tools for improving estimation efficiency in multivariate regression by identifying and excluding immaterial variation. However, existing envelope approaches typically rely on the assumption of exogenous predictors and thus fail to account for endogeneity arising from unobserved confounding. This paper introduces a novel framework that integrates instrumental variable (IV) techniques into envelope modeling, referred to as the Instrumental Variable Envelope (IVE) method. By unifying IV regression with dual-envelope dimension reduction, the IVE framework extends envelope methodology to accommodate endogenous regressors and high-dimensional covariates. The proposed two-stage procedure first constructs an IV-adjusted predictor envelope to eliminate confounding bias, and then estimates a response envelope on the transformed design. We establish the unbiasedness and asymptotic efficiency of the proposed estimator, and show that it achieves strictly lower variance than standard envelope estimators in the presence of endogeneity. Simulation studies confirm the superior predictive accuracy and robustness of the IVE method across a range of data-generating scenarios. An empirical application to automobile market share analysis further demonstrates the practical advantages of the proposed approach.