Abstract
In this paper we consider a stochastic frontier model in which the distribution of technical inefficiency is truncated normal. In standard notation, technical inefficiency u is distributed as N +(μ, σ 2). This distribution is affected by some environmental variables z that may or may not affect the level of the frontier but that do affect the shortfall of output from the frontier. We will distinguish the pre-truncation mean (μ) and variance (σ 2) from the post-truncation mean μ * = E(u) and variance σ 2* = var(u). Existing models parameterize the pre-truncation mean and/or variance in terms of the environmental variables and some parameters. Changes in the environmental variables cause changes in the pre-truncation mean and/or variance, and imply changes in both the post-truncation mean and variance. The expressions for the changes in the post-truncation mean and variance can be quite complicated. In this paper, we suggest parameterizing the post-truncation mean and variance instead. This leads to simple expressions for the effects of changes in the environmental variables on the mean and variance of u, and it allows the environmental variables to affect the mean of u only, or the variance of u only, or both.
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Notes
The sign change for ε is needed because he has a cost function with error ε = v + u whereas we have ε = v − u. Changing the sign changes u to −u. It also changes v to −v, but this does not matter when v is normal with mean zero, hence symmetric.
The standard errors for this model, and for the next two models we will discuss, were calculated using the outer product of the gradient (OPG) version of the information matrix. Our attempts to calculate standard errors from the Hessian were not numerically stable, in the sense that small changes in starting values or details of the maximization led to small changes in the parameter estimates and in the likelihood values, but to substantial changes in the Hessian and the resulting standard errors. This did not occur with the OPG estimates.
A technical detail is that in practice we would evaluate the APE at the sample average value of \(\widehat{\mu }_{*}\) not \(\mu_{*}\) (or \(\widehat{\sigma }_{*}\) not \(\sigma_{*}\)). However, due to the extra level of averaging, the order in probability of (average value of \(\widehat{\mu }_{*}\) minus average value of \(\mu_{*}\)) is smaller than the order in probability of (\(\widehat{\gamma } - \gamma\)). This implies that we can treat the average value of \(\widehat{\mu }_{*}\) (or \(\widehat{\sigma }_{*}\)) as a constant in calculating an asymptotic standard error for the APE.
The partial effects of the \(z_{j}\) on \(\mu_{*}\) and \(\sigma_{*}^{2}\) are given by Wang (2002), pp. 244–245. To calculate the partial effect on the standard deviation \(\sigma_{*}\) we note that \(\frac{{d\sigma_{*}^{2} }}{dz} = 2\sigma_{*} \frac{{d\sigma_{*} }}{dz}\) and therefore \(\frac{{d\sigma_{*} }}{dz} = \frac{1}{{2\sigma_{*} }}\frac{{d\sigma_{*}^{2} }}{dz}\).
For the same reason given in footnote 3, we can treat the average values of \(\widehat{\mu }_{*}\) and \(\widehat{\sigma }_{*}\) as constants in calculating the asymptotic standard errors.
The fact that we have v + u instead of v − u is inconsequential. There are just a few sign changes in the likelihood.
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Amsler, C., Schmidt, P. & Tsay, WJ. A post-truncation parameterization of truncated normal technical inefficiency. J Prod Anal 44, 209–220 (2015). https://doi.org/10.1007/s11123-014-0409-8
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DOI: https://doi.org/10.1007/s11123-014-0409-8