The hinging hyperplanes: An alternative nonparametric representation of a production function

In this paper we propose hinging hyperplanes (HHs) as a flexible nonparametric representation of a concave or an S-shaped production function. We derive the HHs using expressions with focus on the distinction between hinge location and the bending along each hinge. We argue that the HHs approximation can be estimated using a fixed endogenous determined partitioning of the input space. Assuming a homothetic production function allows us to separate the S-shape scaling law and the underlying core function. We propose an estimation procedure where two HHs function approximations of the core function and the scaling law are estimated simultaneously. A closed form expression of the inverse of the piecewise linear inverse scaling law is proposed and proved. We stress the known result that the HHs formulation is equivalent to the Canonical Piecewise-Linear representation of a piecewise linear function and exploit the result that the HHs formulations for the core function and for the scaling law provide global representations of all piecewise linear functional forms. A simulation study evaluates the performance of the proposed estimation procedure of an S-shaped production function.