Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints

Abstract

The field of productive efficiency analysis is currently divided between two main paradigms: the deterministic, nonparametric Data Envelopment Analysis (DEA) and the parametric Stochastic Frontier Analysis (SFA). This paper examines an encompassing semiparametric frontier model that combines the DEA-type nonparametric frontier, which satisfies monotonicity and concavity, with the SFA-style stochastic homoskedastic composite error term. To estimate this model, a new twostage method is proposed, referred to as Stochastic Nonsmooth Envelopment of Data (StoNED). The first stage of the StoNED method applies convex nonparametric least squares (CNLS) to estimate the shape of the frontier without any assumptions about its functional form or smoothness. In the second stage, the conditional expectations of inefficiency are estimated based on the CNLS residuals, using the method of moments or pseudolikelihood techniques. Although in a cross-sectional setting distinguishing inefficiency from noise in general requires distributional assumptions, we also show how these can be relaxed in our approach if panel data are available. Performance of the StoNED method is examined using Monte Carlo simulations.

Semi-Parametric Measures of Scale Characteristics of German Natural Gas-Fired Electricity Generation

“…Thus, estimates of scale characteristics derived with DEA and SFA may depend on the assumptions by the researcher, which ultimately may influence policy implications (Bogetoft and Wang, 2005;Triebs et al, 2016). To overcome the limitations of DEA and SFA, a third approach known as Stochastic Non-Smooth Envelopment of Data (StoNED, Kuosmanen and Kortelainen, 2012) allows flexible estimation of production functions without an underlying functional form (similar to DEA) and stochastic treatment of inefficiency and noise (similar to SFA).…”

“…Since the accuracy of scale measures depends on the accuracy of the frontier estimator, several methodologies have been proposed to estimate transformation frontiers. Stochastic non-smooth envelopment of data (StoNED) as proposed by Kuosmanen (2008) and Kuosmanen and Kortelainen (2012) is a semi-parametric and stochastic approach combining characteristics of parametric SFA (Aigner et al, 1977;Meeusen and van den Broeck, 1977), and non-parametric DEA (Charnes et al, 1978;Banker et al, 1984). Similar to SFA, the approach differentiates noise and inefficiency to explain deviations from the estimated frontier based on distributional assumptions, and similar to DEA, the estimated transformation function has a piece-wise linear shape without any assumptions on an underlying functional form.…”

“…Similar to SFA, the approach differentiates noise and inefficiency to explain deviations from the estimated frontier based on distributional assumptions, and similar to DEA, the estimated transformation function has a piece-wise linear shape without any assumptions on an underlying functional form. Kuosmanen and Kortelainen (2012) suggest a two stage approach to estimate a transformation frontier y = f (x)exp(v − u), where f (x) is the function to estimate, v is a random two-sided disturbance, and u is positive inefficiency. In the first stage, an average function g(x) is estimated based on a quadratic programming problem (QP) to solve a convex non-parametric least squares problem (CNLS, Hildreth, 1954).…”

“…To estimate the average production function g(x), Kuosmanen and Kortelainen (2012) derive the following non-linear QP to obtain the intercept and slope estimates based on the log-transformed multiplicative model…”

“…Following Kuosmanen and Kortelainen (2012), a normal distribution is imposed for the noise term, v ∼ N (0, σ 2 v ). The inefficiency term is assumed to take only positive values and to follow a half-normal distribution, u ∼ |N (0, σ 2 u )|.…”