A Spatial Stochastic Frontier Model with Spill-In and Spillover Effects on Technical Inefficiency

This paper develops a spatial stochastic frontier framework for panel data that jointly accounts for spatial dependence and heteroskedastic technical inefficiency. Inefficiency and noise components are parameterized using scaling functions, while spatial dependence is modeled through both a spatial lag (SF-SLM) and a spatial Durbin specification (SF-SDM). Maximum likelihood estimation is implemented by explicitly incorporating the spatial autoregressive process into the log-likelihood function. A key innovation of this study is the use of the spatial multiplier to decompose estimated technical inefficiency into three components: (i) own inefficiency, (ii) spill-in effects (feedback of a unit’s inefficiency on itself through spatial interactions), and (iii) spillover effects (inefficiency transmitted from neighboring regions). This approach extends the stochastic frontier literature by showing that inefficiency is not purely local but can propagate across space. The method is applied to the Brazilian food manufacturing industry (2007–2018). Likelihood ratio tests confirm that spatial models outperform the nonspatial specification, with SF-SDM providing the best fit and more stable inefficiency estimates. Results reveal that, for an average region, approximately 9% of inefficiency is due to spillovers from neighbors, while 0.2% is explained by spill-in effects. Ignoring spatial structure would therefore overestimate region-specific inefficiency and underestimate the role of interregional linkages. The proposed framework offers a flexible tool for analyzing productive efficiency in spatially interconnected settings and provides new insights for regional policy and future research.