Interpreting and testing the scaling property in models where inefficiency depends on firm characteristics

In this paper, we are interested in a stochastic frontier model in which observable characteristics of the firms affect their levels of technical inefficiency. Let u ≥ 0 be the one-sided error reflecting technical inefficiency, and let z be a set of variables that affect u. We write u as u(z,δ) to reflect its dependence on z and some parameters δ. Various models in the existing literature specify the distribution of u(z,δ). We are interested in models that satisfy the scaling property, which says that u(z,δ) can be written as a scaling function h(z, δ) times a random variable u* that does not depend on z. This property implies that changes in z affect the scale but not the shape of u(z,δ). This paper reviews the existing literature and identifies models that do and do not have the scaling property. It also discusses practical advantages of the scaling property. The scaling property is argued to be intuitively appealing; it allows estimation by nonlinear least squares; it allows a distribution-free interpretation of the parameters δ that show how z affects inefficiency; and it underlies the model of Battese and Coelli, Journal of Productivity Analysis, 1992, which is currently the only model to allow correlation over time when inefficiency depends on firm characteristics. The paper shows how to test the hypothesis of scaling, and other interesting hypotheses, in the context of the model of Wang, Journal of Productivity Analysis, 2002. Finally, two empirical examples are given.