A Proportional Scale-Invariant Measurement of Heterogeneous Directional Distances to the Technological Frontier

We consider a stochastic multiplicative-directional-distance-function-based formulation of the production technology in the presence of bad outputs. In contrast to the popular additive directional distance function, our approach preserves the highly desired scale invariance property of the proportional radial distance function and can easily assume popular translog specification. Our model can also meaningfully accommodate technological heterogeneity across individual firms by letting the direction in which the distance to frontier is measured be an unknown function of the firm's idiosyncratic characteristics reflective of the heterogeneous economic environment in which it operates. To mitigate the uncertainty associated with an arbitrary choice of the direction, we rely on a data-driven selection method to determine these varying heterogeneous directional vectors for individual firms. We showcase practical advantages of the proposed model by estimating the production technology of U.S. commercial banks in the presence of undesirable non-performing loans during the 2001–2010 period