Understanding prediction intervals for firm specific inefficiency scores from parametric stochastic frontier models

Abstract

This paper makes two important contributions to the literature on prediction intervals for firm specific inefficiency estimates in cross sectional SFA models. Firstly, the existing intervals in the literature do not correspond to the minimum width intervals and in this paper we discuss how to compute such intervals and how they either include or exclude zero as a lower bound depending on where the probability mass of the distribution of $u_{i} |\varepsilon_{i}$ resides. This has useful implications for practitioners and policy makers, with greatest reductions in interval width for the most efficient firms. Secondly, we propose an ‘asymptotic’ approach to incorporating parameter uncertainty into prediction intervals for firm specific inefficiency (given that in practice model parameters have to be estimated) as an alternative to the ‘bagging’ procedure suggested in Simar and Wilson (Econom Rev 29(1):62–98, 2010). The approach is computationally much simpler than the bagging approach.

Appendices

Appendix 1: Derivation of minimum width predictive intervals for the truncated normal distributions

Consider $\left( {u_{i} |\varepsilon_{i} } \right)\sim N^{ + } \left( {\mu_{i*} ,\sigma_{*}^{2} } \right)$

There are two solutions to the Langrangean problem. Either:

$$ f(L^{*} ) = f(U^{*} ) \;{\text{exists}}\;{\text{such}}\;{\text{that}}\;\mathop \int \limits_{{L^{*} }}^{{U^{*} }} f\left( {u_{i} |\varepsilon_{i} } \right)du_{i} = \left( {1 - \alpha } \right)\;{\text{and}}\;L^{*} ,U^{*} \ge 0 $$

(11)

Otherwise

$$ L^{*} = 0\;{\text{and}}\;U^{*} \;{\text{such}}\;{\text{that}}\;\mathop \int \limits_{0}^{{U^{*} }} f\left( {u_{i} |\varepsilon_{i} } \right)du_{i} = 1 - \alpha $$

(12)

$U^{*}$ for the case in (12) is given by Horrace and Schmidt (1996) and reproduced in Eq. (3) as

$$ U^{*} = \mu_{i*} + \sigma_{*} \Upphi^{ - 1} \left[ {1 - \left( {1 - \left( {1 - \alpha } \right)} \right)\Upphi \left( {\frac{{\mu_{i*} }}{{\sigma_{*} }}} \right)} \right] $$

$$ U^{*} = \mu_{i*} + \sigma_{*} \Upphi^{ - 1} \left[ {1 - \alpha \cdot \Upphi \left( {\frac{{\mu_{i*} }}{{\sigma_{*} }}} \right)} \right] $$

(13)

Now consider (11).

Define, X ~ N $(\mu_{i*} ,\sigma_{*}^{2} )$

Then

$$ \mathop \int \limits_{{L^{*} }}^{{U^{*} }} f\left( {u_{i} |\varepsilon_{i} } \right)du_{i} = 1 - \alpha \leftrightarrow \mathop \int \limits_{{L^{*} }}^{{U^{*} }} f\left( X \right)dX = \left( {1 - \alpha } \right)\mathop \int \limits_{0}^{\infty } f\left( X \right)dX $$

$$ \mathop \int \limits_{{L^{*} }}^{{U^{*} }} f\left( X \right)dX = \left( {1 - \alpha } \right)\left( {1 - \Upphi \left( {\frac{{\mu_{i*} }}{{\sigma_{*} }}} \right)} \right) $$

(14)

Given the symmetry of $f\left( X \right)$, for $f(L^{*} ) = f(U^{*} )$,

$$ \mathop \int \limits_{ - \infty }^{{U^{*} }} f\left( X \right)dX = \left( {1 - \frac{\alpha }{2}} \right)\left( {1 - \Upphi \left( {\frac{{\mu_{i*} }}{{\sigma_{*} }}} \right)} \right) $$

(15)

$$ \mathop \int \limits_{ - \infty }^{{L^{ *} }} f\left( X \right)dX = \left( {\frac{\alpha }{2}} \right)\left( {1 - {{\Upphi}}\left( {\frac{{\mu_{i *} }}{{\sigma_{ *} }}} \right)} \right) $$

(16)

Yielding

$$ U^{*} = \mu_{i*} + \sigma_{*} \Upphi^{ - 1} \left[ {\left( {1 - \frac{\alpha }{2}} \right)\left( {1 - \Upphi \left( {\frac{{\mu_{i*} }}{{\sigma_{*} }}} \right)} \right)} \right] $$

(17)

$$ L^{*} = \mu_{i*} + \sigma_{*} \Upphi^{ - 1} \left[ {\left( {\frac{\alpha }{2}} \right)\left( {1 - \Upphi \left( {\frac{{\mu_{i*} }}{{\sigma_{*} }}} \right)} \right)} \right] $$

(18)

Intuitively, L* and U* in (11) are the boundaries of the central interval of the untruncated normal distribution with mean $\mu_{i*}$ and variance $\sigma_{*}^{2}$, since the normal distribution is symmetric. However they do not correspond to the usual $\frac{\alpha }{2}$ and $\left( {1 - \frac{\alpha }{2}} \right)$ percentiles of the normal distribution since the actual distribution is truncated and thus a correction is necessary for the untruncated distribution to integrate to unity.

Appendix 2: Model output for the empirical example

Table 3 gives the output for the preferred model in Smith and Wheat (2012) reestimated for a normal-half normal pooled model. See Smith and Wheat (2012) for more details on the model formulation and interpretation. We consider that the model parameter estimates are broadly in line with those from the Smith and Wheat model, which was a panel data model, but here we analyse the data as a pooled model. Importantly our conclusions regarding constant economies of scale are the same as that found in Smith and Wheat, although we no longer find economies of train density at the sample mean (see Smith and Wheat (2012) for details of computation in this context). The average point efficiency scores $\left( {exp\left( { - E\left[ {u_{i} |\varepsilon_{i} } \right]} \right)} \right)$ are 0.90 for the panel model and 0.91 for the pooled model, although the correlation between the scores is only 0.6 which is not surprising given the added structure imposed to efficiency variation in the panel model. Overall, while a pooled model is not our preferred model for modelling TOC costs, we consider that it is a reasonably credible alternative for the illustrative purpose of this paper.

Table 3 Model coefficient estimates