Modeling unobserved spatial heterogeneity in technical and environmental inefficiency: A spline-based by-production stochastic frontier approach

Measuring technical and environmental inefficiency jointly while accounting for unobserved spatial heterogeneity remains a key challenge in production analysis. Existing studies typically model these dimensions separately or ignore spatial variation, potentially biasing inefficiency estimates. This study presents a semiparametric, spline-based by-production stochastic frontier model that jointly models marketable and polluting outputs, captures their dependence using copulas, and incorporates spatial heterogeneity through thin plate splines. The model is applied to a panel dataset of Dutch dairy farms with detailed production, emission, and geographic information. The results reveal nonlinear input–output relationships, low inefficiency levels, and positive upper tail dependence between production and emissions. Environmental inefficiency exhibits distinct spatial patterns, with clusters of high and low inefficiency. Ignoring spatial heterogeneity inflates environmental inefficiency, underestimates its variance, and deteriorates model fit. These findings highlight the importance of jointly modeling production and emissions while accounting for spatial factors for accurate efficiency measurement and policy design.