Quantifying Parameter Heterogeneity Under Endogeneity in Panel Data: An IV Mixed-Effects Approach

Thomas M. Menino Convention & Exhibition Center 

This paper studies linear mixed models for longitudinal data when covariates may be correlated with the idiosyncratic error and/or the random effects, biasing both fixed-effect estimates and random-effect predictions. We propose a two-stage linear mixed model instrumental variables estimator (LMMIV) that uses valid instruments to correct endogeneity while preserving the mixed-model structure. Its key novelty is the ability to quantify heterogeneity in all regression parameters, that is, both intercepts and slopes, even when endogeneity is present, rather than focusing only on mean effects or assuming random effects are uncorrelated with regressors. Under standard instrument exogeneity conditions, LMMIV yields consistent fixed effects and unbiased empirical BLUP-style predictions of random effects. Simulations confirm these theoretical results. Applied to a panel of 37 metro systems (1994–2019), we find substantial heterogeneity in demand elasticities: fare −0.86 to 0.12 (mean −0.29), income −0.07 to 0.57 (mean 0.18), and quality of service −0.03 to 0.91 (mean 0.32), with mean effects in line with the literature.

Slope heterogeneity

linear mixed models

endogeneity

instrumental variables 

Business and Economic Statistics Section