Estimating Efficiency in the Presence of Extreme Outliers: A Logistic-Half Normal Stochastic Frontier Model with Application to Highway Maintenance Costs in England

Abstract

In Stochastic Frontier Analysis the presence of outliers in the data, which can often be safely ignored in other forms of linear modelling, has potentially serious consequences in that it may lead to implausibly large variation in efficiency predictions when based on the conditional mean. This motivates the development of alternative stochastic frontier specifications which are appropriate when the two-sided error has heavy tails. Several existing proposals to this effect have proceeded by specifying thick tailed distributions for both error components in order to arrive at a closed form loglikelihood. In contrast, we use simulation-based methods to pair the canonical inefficiency distributions (in this example half-normal) with a logistically distributed noise term. We apply this model to estimate cost frontiers for highways authorities in England, and compare results obtained from the conventional normal-half normal stochastic frontier model. We show that the conditional mean yields less extreme inefficiency predictions for large residuals relative to the use of the normal distribution for noise.

Robust stochastic frontier analysis: a Student’s t-half normal model with application to highway maintenance costs in England

“…Table 2 shows that, while the Student's t-half normal model tends to yield lower estimates of σ v when a is small, its estimates of the standard deviation of v in fact tend to be higher. This supports the expectation, following the discussion in Section 2 and in Stead et al (2018), that a model with a heavy-tailed distribution in v should attribute more of the overall error variance to noise than to inefficiency.…”

“…A review of the literature in this area provided by Stead et al (2018) covers several different approaches to outliers: the use of alternative efficiency predictors, thick frontier analysis, heteroskedastic SF models, and the use of alternative noise distributions. Other possible approaches include right truncation of the inefficiency distribution to restrict the range of possible efficiency predictions, and detecting and removing outliers.…”

“…Similarly, Horrace and Parmeter (2018) show that f u|ε (u|ε)and hence E(u|ε)is constant for positive values of sε in the Laplace-exponential case. Focusing more explicitly on outliers, Stead et al (2018) propose a model in which v follows a logistic distribution and u follows a half normal distribution, and show that the model results in a smaller estimate for VAR(u) and yields a considerably narrower spread of efficiency scores than the normal-half normal model, with little change in exp[E(−u|ε)] for large |sε|. Each of these cases suggest that the choice of a heavy tailed distribution for v significantly affects the prediction of efficiency, and the uncertainty in prediction, at one or both tails, producing less extreme efficiency predictions.…”

“…which lacks a convenient closed form, hindering ML estimation. As an alternative, we may use simulation to approximate the integral and arrive at a simulated log likelihood function-see Train (2009) for an introduction to maximum simulated likelihood-as proposed by Greene (2003) for the normal-gamma model and Stead et al (2018) for the logistic-half normal model. We begin by noting that (7) is the expectation of f v (ε + su), where u is drawn from a truncated-normal distribution,…”

“…In this section, we apply a Student's t-half normal model to the dataset on highway maintenance costs in England used by Stead et al (2018). In England, responsibility for road maintenance is divided between Highways England-until 2015 the Highways Agency-a publicly-owned company responsible for the strategic 'trunk road network', and the county councils and unitary authorities, who maintain the non-trunk roads within their boundaries.…”