Stable Contexts Are Not Static Causes: A Latent Variable Approach to Time-Varying Consequences of Stable Unobserved Heterogeneity

Urban-health studies increasingly examine environmental co-occurrences, but longitudinal analyses often rely on fixed-effects (FE) models whose assumptions about stable unobserved heterogeneity are left implicit. This article uses a structural equation modeling (SEM) representation of FE to clarify a narrow but important point: stable, time-invariant, unobserved heterogeneity is controlled by conventional FE only when its contribution to repeated outcomes is also constant across waves. When stable histories of segregation, infrastructure, or governance have changing consequences as environmental co-occurrence configurations are reorganized, conventional FE models may leave residual confounding. I review the classical fixed-effects structural equation modeling (FE-SEM) representation, then present two diagnostic extensions: time-varying-loading FE-SEM (TVL-FE-SEM), a single-latent-dimension model with wave-specific loadings, and constant-and-deviation-loading FE-SEM (CDL-FE-SEM), a two-latent-dimension model that adapts the intercept/slope logic of latent growth modeling to separate common and deviation-loading profiles. A compact Monte Carlo simulation shows that classical FE-SEM recovers the true effect only when stable unobserved heterogeneity enters repeated outcomes with constant loadings, TVL-FE-SEM performs well when the same single latent source has changing loadings, and CDL-FE-SEM helps diagnose whether stable heterogeneity operates through more than one loading profile under a fixed-effects specification, though with finite-sample instability. An empirical illustration using Korean community-health data from 2021–2024 demonstrates the sequence as a diagnostic workflow.