Posterior Analysis of Stochastic Frontier Models with Truncated Normal Errors

Summary

Previous work in stochastic frontier models with exponentially distributed one-sided errors using both Gibbs sampling and Monte Carlo integration with importance sampling reveals the enormous computational gains that can be achieved using the former. This paper takes up inference in another interesting class of stochastic frontier models, those with truncated normal one-sided error terms, and shows that posterior simulation involves drawing from standard or log-concave distributions, implying that Gibbs sampling is an efficient solution to the Bayesian integration problem. The sampling behavior of the Bayesian procedure is investigated using a Monte Carlo experiment. The method is illustrated using US airline data.

Appendix. Some properties of posterior conditional distributions of ω −1 and ψ}

The posterior conditional distribution of ω is given by (8). A change of variables to v = ω⁻¹ yields the following distribution for v:

$$p(v|.) \propto {v^{N - 1}}\exp \left[ { - \sum\limits_{i = 1}^n {{{\left( {v{u_i} - \psi } \right)}^2}} /2} \right]$$

Taking logarithms and denoting Q(v) = ln p(v ∣.) we have

$$dQ(v)/dv = (N - 1){v^{ - 1}} - \sum\limits_{i = 1}^N {\left( {v{u_i} - \psi } \right)} $$

Setting to zero and multiplying by v yields the quadratic equation

$$\left( {\sum\limits_{i = 1}^N {u_i^2} } \right){v^2} - \left( {\psi \sum\limits_{i = 1}^N {{u_i}} } \right)v - (N - 1) = 0$$

This quadratic has a unique positive root, which determines an optimum of the posterior conditional distribution of v. Moreover to prove that this distribution is log-concave, we consider

$${d^2}Q(v)/d{v^2} = - (N - 1){v^{ - 2}} - \sum\limits_{i = 1}^N {{u_i}} $$

which is negative for all admissible values of v.

Regarding now the posterior conditional distribution of ψ it is given by (7). Assuming that the prior is log-concave consider concavity of the logarithm of the first term in the product of (7), namely

$$G(\psi ) \equiv - N\ln {\rm{\Phi }}(\psi ) - (N/2){\left( {\psi - \bar u{\omega ^{ - 1}}} \right)^2}$$

whose second derivative is

$${d^2}G(\psi )/d{\psi ^2} = N\left( {{{\psi \phi (\psi ){\rm{\Phi }}(\psi ) + \phi {{(\psi )}^2}} \over {{\rm{\Phi }}{{(\psi )}^2}}} - 1} \right)$$

after taking into account that dϕ(ψ)/dψ = −ψϕ(ψ). It can be easily shown that the function in parentheses is always non-positive, which proves log-concavity of the posterior conditional distribution of ψ.