Flexible mixture modelling of stochastic frontiers

Abstract

This paper introduces new and flexible classes of inefficiency distributions for stochastic frontier models. We consider both generalized gamma distributions and mixtures of generalized gamma distributions. These classes cover many interesting cases and accommodate both positively and negatively skewed composed error distributions. Bayesian methods allow for useful inference with carefully chosen prior distributions. We recommend a two-component mixture model where a sensible amount of structure is imposed through the prior to distinguish the components, which are given an economic interpretation. This setting allows for efficiencies to depend on firm characteristics, through the probability of belonging to either component. Issues of label-switching and separate identification of both the measurement and inefficiency errors are also examined. Inference methods through MCMC with partial centring are outlined and used to analyse both simulated and real data. An illustration using hospital cost data is discussed in some detail.

Appendix: Some Details on the MCMC Sampler

1.1 A.1 Drawing the inefficiencies

The full conditional for the inefficiencies is, of course, different from the one in the existing literature. The firm-specific inefficiencies u _i will be independent given all the other parameters and the observations, with density function:

$$ p(u_i\vert \hbox{rest}) \propto u_i^{c\phi-1} \exp\left\{-\left[\frac{1}{2\sigma^2}(T_i u_i^2 + 2 \mu_i u_i) + \lambda u_i^c\right]\right\}, $$

(10)

where $\mu_i=T_i\alpha+\beta^{\prime} X_i^{\prime} \iota-y_i^{\prime} \iota$ for a cost frontier and its negative for a production frontier, if we define $y_i=(y_{i1},{\ldots},y_{i{T_i}})^\prime,\;X_i=(x_{i1},{\ldots},x_{i{T_i}})^\prime$ and $\iota$ is a T _i -dimensional vector of ones. We can easily show that the conditional in (10) is log-concave if

$$ u^c > \frac{1-c\phi}{\lambda c(c-1)}, $$

which is always satisfied if cϕ > 1 and c > 1. For this parameter combination, we use adaptive rejection sampling (see Gilks and Wild 1992) to sample directly from (10). In the other cases, we use random walk Metropolis-Hastings with a lognormal candidate generator. Random walk Metropolis-Hastings is also used to generate drawings for the parameters ϕ and c while λ can simply be drawn from a Gamma conditional. We use the reparameterization from (ϕ,c) to (ψ,c), where ψ = ϕc, since it leads to better mixing properties of the sampler.

1.2 A.2 Centring

As indicated in Sect. 6, we use a sampler which randomly mixes updates from the centred and the uncentred parameterizations. This basic idea is called “partial centring” in Papaspiliopoulos et al. (2003), who show that this generally leads to more robust sampling algorithms. We choose a centred update with probability 1/4, as we only need to “recentre” α once in a while. In addition, we found that centring works best if we integrate out λ while updating α (i.e. we effectively draw α and λ jointly). Thus, in the centred parameterization we use a random walk Metropolis-Hastings step to sample from the following conditional for α < min _i {z _i }:

$$ p(\alpha\vert z, \beta, \sigma^2, c, \phi, y_1,\ldots,y_n)\propto \prod_{i=1}^n(z_i-\alpha)^{c\phi-1} \left[\sum_{i=1}^n(z_i-\alpha)^c +(-\ln r^\star)^c\right]^{-\phi(n+1)}, $$

(11)

which can be shown to be log-concave whenever cϕ > 1 and c > 1. We use adaptive rejection sampling in the latter case and a random walk Metropolis-Hastings step otherwise.

1.3 A.3 The mixture model

In the case of the mixture inefficiency distribution, as described in Sect. 3.1, we extend the MCMC sampler of the basic case by augmenting with an indicator variable s _i ,i = 1,…,n which can take the values 0 or 1 and assigns firm i to one of the two efficiency groups (i.e. one of the two inefficiency components). For the mixture model the sampler will, thus, generate a chain on (α,β,σ²,w,θ,u,s), where s = (s ₁,…,s _n ). Inference about the parameters of each component (c _j , ϕ _j , λ _j ), j = 1,2 now only depends on those firms for which s _i  = j − 1. The full conditional distribution for w will be

$$ w\vert \hbox{rest}\sim\hbox{Be}\left(w_0+n-\sum_{i=1}^n s_i,w_1+\sum_{i=1}^n s_i\right), $$

and the values of s _i are updated through

$$ \hbox{P}(s_i=0\vert \hbox{rest})=\frac{w p_{GG}(u_i\vert c_1,\phi_1,\lambda_1)}{w p_{GG}(u_i\vert c_1,\phi_1,\lambda_1) + (1-w) p_{GG}(u_i\vert c_2,\phi_2,\lambda_2)}. $$

For the decomposition case, where we have included a prior probability of w = 0, the full conditional distribution of w has the same form as above except when $\sum_{i=1}^n s_i=n$. In this case w is zero with probability $q^{\star}$ and w∼Be(w ₀, w ₁ + n) otherwise where

$$ q^{\star}=\frac{q}{q+(1-q) \frac{\Gamma(w_0+w_1)\Gamma(w_1+n)}{\Gamma(w_1)\Gamma(w_0+w_1+n)}}. $$

Here we only considered switching to the one-component model when $\sum_{i=1}^n s_i=n$, which worked well. However, in applications with very large n it might be more efficient to also consider jumps between the models when $\sum_{i=1}^n s_i$ is relatively close to n.

In case we allow for the probability of being in the efficient group to depend on firm characteristics, we can use simple Gibbs sampling after data augmentation as in Albert and Chib (1993) to update the probit regression coefficients.