ABSTRACT Reduced rank regression provides a classical approach to multivariate regression by imposing a low‐rank structure on the regression coefficient matrix, thereby achieving dimension reduction in the predictor space. In parallel, sufficient dimension reduction methods based on inverse regression, such as the unstructured principal fitted components model, offer a likelihood‐based framework for identifying low‐dimensional sufficient predictors. This paper develops a generalized reduced rank regression framework by embedding reduced rank regression within the semiparametric inverse regression formulation of unstructured principal fitted components. We establish both structural and inferential connections between the two approaches, showing that when the inverse regression basis is linear, the likelihood ratio test for dimension selection in unstructured principal fitted components coincides exactly with Bartlett's likelihood ratio test for reduced rank regression. This result clarifies reduced rank regression as a constrained special case within a broader likelihood‐based dimension reduction framework. Our numerical experiments demonstrate that although the methods perform similarly in linear settings, the generalized framework significantly improves robustness and dimension selection consistency in nonlinear environments. An application to the 2017–2018 NHANES dataset further confirms that both methods yield interpretable and practically concordant sufficient predictors. This work bridges classical and modern perspectives, offering a unified likelihood‐based approach to dimension reduction in multivariate regression.
Jeong et al. (Tue,) studied this question.