Semi-partitioned Generalized Method of Moments for Longitudinal Data with Lagged and Feedback Covariates

TL;DR

This paper introduces a semi-partitioned GMM framework for longitudinal data that balances temporal detail and model stability, improving estimation of effects with feedback and lag structures.

Contribution

It presents a novel semi-partitioned GMM approach that effectively captures temporal effects while maintaining statistical efficiency and interpretability.

Findings

Achieves comparable coverage and efficiency to fully partitioned models.

Demonstrates robustness and interpretability in clinical applications.

Reduces instability issues common in fine-grained partitioning.

Abstract

We propose a semi-partitioned Generalized Method of Moments (GMM) framework for analyzing longitudinal data with time-dependent covariates, within a marginal modeling paradigm. This approach addresses limitations of both aggregated and fully partitioned GMM models. Aggregated methods obscure temporal dynamics by assuming constant effects, while fully partitioned approaches offer temporal specificity at the cost of increased model complexity and instability--particularly with moderate sample sizes or deep lag structures. Our method distinguishes immediate from lagged effects by estimating contemporaneous coefficients separately and grouping lagged moment conditions into structured sets, while retaining flexibility in the lag-specific effects. This yields a model that is both statistically efficient and interpretable, capturing essential temporal variation while mitigating variance inflation and convergence challenges associated with full partitioning. The framework accommodates feedback, supports both continuous and binary outcomes, and utilizes the Broyden--Fletcher--Goldfarb--Shanno (BFGS) algorithm for reliable optimization. Through simulations, we demonstrate that the semi-partitioned GMM achieves coverage and competitive efficiency relative to fully partitioned models when the grouped-lag structure approximates the underlying lag pattern. Applications to clinical datasets on knee osteoarthritis and adolescent obesity confirm that the method recovers consistent, interpretable effects and avoids instability associated with finely grained partitioning.