Modelling Heterogeneous Dependence in a Copula‐Linked By‐Production Frontier: Evidence From French Dairy Farms

In agriculture, production technologies are often characterised by non‐linearities and stochastic dependence between outputs, particularly between desirable and undesirable outputs. This paper combines the copula‐based semi‐parametric framework for modelling heterogeneous dependence with the by‐production stochastic frontier approach and applies it to the analysis of milk production and greenhouse gas (GHG) emissions. The proposed approach models the dependence parameter as a function of farm‐level characteristics, addressing a key limitation of existing by‐production models that assume either no dependence or a constant dependence structure. Applied to panel data on French dairy farms observed from 2018 to 2022, the model jointly estimates technical and environmental inefficiency and their trade‐off. Model selection based on information criteria supports the presence of heterogeneous dependence between the two production frontiers. The results indicate a high level of both technical and environmental performance among French dairy farms and show that farm structure and management significantly influence inefficiency and the strength of the linkage between milk production and emissions.