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

Abstract

The presence of outliers in the data has implications for stochastic frontier analysis, and indeed any performance analysis methodology, because they may lead to imprecise parameter estimates and, crucially, lead to an exaggerated spread of efficiency predictions. In this paper we replace the normal distribution for the noise term in the standard stochastic frontier model with a Student's t distribution, which generalises the normal distribution by adding a shape parameter governing the degree of kurtosis. This has the advantages of introducing flexibility in the heaviness of the tails, which can be determined by the data, as well as containing the normal distribution as a limiting case, and we outline how to test against the standard model. Monte Carlo simulation results for the maximum simulated likelihood estimator confirm that the model recovers appropriate frontier and distributional parameter estimates under various values of the true shape parameter. The simulation results also indicate the influence of a phenomenon we term 'wrong kurtosis' in the case of small samples, which is analogous to the issue of 'wrong skewness' previously identified in the literature. We apply a Student's t-half normal cost frontier to data for highways authorities in England, and this formulation is found to be preferred by statistical testing to the comparator normal-half normal cost frontier model. The model yields a significantly narrower range of efficiency predictions, which are non-monotonic at the tails of the residual distribution.

Utilization of Student’s T Distribution to Handle Outliers in Technical Efficiency Measurement

“…If noise component ( ) is assumed to have a non-standardized Student's t distributionwhich includes a scale parameter and inefficiency component ( ) has half normal distribution (Student's t -Half Normal model), then the density function of and are as follows (Wheat et al, 2019):…”

“…The primary characteristics of stochastic frontier model is that its error is partitioned into random shock (noise) and inefficiency components (Wheat, 2017). The noise component is generally assumed to have a normal distribution, while the inefficiency component could be exponential, half normal, truncated normal or gamma distributions (Wheat et al, 2019).…”

“…The normality assumption of noise is not a problem if the outlier does not exist. If the outliers are present, Wheat et al (2019) showed that the standard stochastic frontier model can produce exaggrerated and implausible spreads of technical efficiency scores. Production frontier in stochastic frontier model consists of deterministic and stochastic parts.…”

“…Student's t distribution is one of the heavy-tailed distribution, so that the chance to cover extreme values is greater than the normal distribution. Wheat et al (2019) had proposed the use of Student's t distribution to solve the outlier issues in England highways maintenance costs. However, this approach has not been widely explored in Indonesia, especially in agricultural studies.…”

The case for the use of multiple imputation missing data methods in stochastic frontier analysis with illustration using English local highway data

“…and Wheat et al (2019) The and subscripts denote authority in year . Note the change from (1), since we are using panel data.…”