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

Abstract

Analyzing the dynamic interplay between multidimensional control parameters and latent state outcomes in complex multi-agent systems poses formidable mathematical and statistical challenges. This paper establishes a coupled mathematical framework that unifies three core components: (i) a dynamic optimization model whose steady-state equilibrium is characterized via a Banach fixed-point argument and proven unique; (ii) an entropy-regularized factor construction procedure with O(1/k) convergence under projected gradient descent; and (iii) a modified System Generalized Method of Moments (GMM) estimator designed to mitigate weak-instrument bias under high treatment persistence. The Karush–Kuhn–Tucker optimality conditions are explicitly derived, and the contraction mapping underlying the Bellman operator is proved in the Banach space of bounded continuous functions. Theoretical properties are validated through Monte Carlo simulations, and empirical feasibility is illustrated using a real-world dynamic panel dataset (N = 30, T = 13). The results demonstrate that the integrated methodology produces stable, internally consistent trajectories and outperforms standard benchmarks under structural persistence.