A Coupled Dynamic Optimization and Entropy-Regularized PCA Framework for Dynamic Panels

Abstract

Analyzing the dynamic interplay between multidimensional policy instruments and environmental outcomes in regional economies poses formidable mathematical and statistical challenges. Specifically, traditional composite index methods lack rigorous theoretical foundations in dynamic panel settings, while conventional econometric estimators frequently falter when both treatments and outcomes are latent constructs plagued by measurement errors and persistent unobserved heterogeneity. To resolve these limitations, this paper establishes an integrated mathematical framework that unifies three core components: (i) a dynamic optimization model of technology adoption under financial constraints that characterizes the cost and constraint channels of policy transmission; (ii) an entropy-regularized factor construction procedure that extracts a time-varying composite index from high-dimensional indicators by prioritizing information-rich components; and (iii) a dynamic panel estimation strategy leveraging the System Generalized Method of Moments (GMM) to mitigate endogeneity and temporal persistence. The resulting framework yields well-defined equilibrium conditions and offers a coherent inference strategy for complex panel data environments. To demonstrate its empirical utility, the framework is applied to provincial panel data from China's real estate sector (2010-2022) to investigate the nexus between green finance and carbon emission intensity. The empirical results yield robust, economically meaningful estimates and validate the framework's capacity to handle high-dimensional treatments, dynamic adjustments, and endogenous regressors. Emphasizing its methodological contribution, this coupled framework provides a generalizable analytical toolkit for complex socio-economic systems where multifaceted interventions govern latent outcomes through dynamic channels, offering immediate extensions to health economics, education policy, and innovation studies.