Stochastic frontiers with a Rayleigh distribution

Abstract

We introduce a stochastic frontier model with a one-parameter distribution known as the Rayleigh distribution which has a non-zero mode and yet it is easy to estimate and use. We show how this model can be estimated using various estimation methods. The Rayleigh model with environmental variables and time-varying features is also considered. It is also tested against exponential and half-normal models using two real data sets.

Appendix: Bayesian estimation and model selection

The Bayesian approach to the estimation of stochastic frontier models is popular and has been described in Koop and Steel (2001) among others. Griffin and Steel (2007) have also shown how a wide range of stochastic frontier models can be estimated using the WinBUGS software. Here we consider the model

$$\begin{gathered} y_{it} = {\mathbf{x}}_{{{\mathbf{it}}}} {\varvec{\upbeta}} \mp u_{i} + v_{it} \hfill \\ u_{i} = Rayleigh(\sigma_{u} ) \hfill \\ \end{gathered}$$

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For Bayesian analysis, we need priors for all parameters. We assume the following standard priors:

$$p({\varvec{\upbeta}}) \propto 1\quad \sigma_{v}^{ - 2} \sim G(A,B)\quad \sigma_{u}^{ - 2} \sim G(a_{u} ,b_{u} )$$

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where G(.,.)denotes a gamma distribution. Using Bayes theorem we can obtain the log of the posterior as follows

$$\begin{aligned} \ln p = & C_{1} + (\frac{NT}{2} + A - 1)\ln \sigma^{ - 2} - \frac{{\sigma^{ - 2} }}{2}\left[ {\sum\limits_{i = 1}^{N} {\sum\limits_{t = 1}^{T} {(y_{it} - {\mathbf{x}}_{it} {\varvec{\upbeta}} \pm u_{i} )^{2} + 2B} } } \right] \\ + (N + a_{u} - 1)\ln \sigma_{u}^{ - 2} + \sum\limits_{i = 1}^{N} {\ln u_{i} } - \left( {\frac{1}{2}\sum\limits_{i = 1}^{N} {u_{i}^{2} } + b_{u} } \right)\sigma_{u}^{ - 2} \\ \end{aligned}$$

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where C₁ is an unknown constant. It is straightforward to obtain the conditional densities and set-up the following Gibbs sampler

$$\left\{ \begin{gathered} {\varvec{\upbeta}}|{\mathbf{u}},\sigma_{v}^{ - 2} ,\sigma_{u}^{ - 2} ,{\mathbf{y}}\sim N\{ ({\mathbf{x'x}})^{ - 1} {\mathbf{x}}'({\mathbf{y}} \pm {\mathbf{u}} \otimes {\mathbf{i}}),\sigma_{v}^{2} ({\mathbf{x'x}})^{ - 1} \} \; \hfill \\ \sigma_{v}^{ - 2} |{\mathbf{u}},{\varvec{\upbeta}},\sigma_{u}^{ - 2} ,{\mathbf{y}}\sim G\left\{ {A + NT/2,B + \frac{1}{2}\sum\limits_{i = 1}^{N} {\sum\limits_{t = 1}^{T} {\left[ {(y_{it} - {\mathbf{x}}_{{{\mathbf{i}}t}} {\varvec{\upbeta}} \pm u_{i} )^{2} } \right]} } } \right. \hfill \\ \sigma_{u}^{ - 2} |{\mathbf{u}},\sigma_{v}^{ - 2} ,{\varvec{\upbeta}},{\mathbf{y}}\sim G(N + a_{u} ,b_{u} + \frac{1}{2}\sum\limits_{i = 1}^{N} {u_{i}^{2} } ) \hfill \\ p(u_{i} |\sigma_{v}^{ - 2} ,{\varvec{\upbeta}},\sigma_{u}^{ - 2} ,{\mathbf{y}}) = \frac{1}{{C_{i} }}u_{i} \exp \left( { - \frac{{\sigma_{{}}^{ - 2} }}{2}\left( {u_{i}^{{}} - \mu_{i} } \right)^{2} } \right) \hfill \\ \end{gathered} \right.$$

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where i is a vector of ones (with size T), ⊗ denotes Kronecker product, σ ⁻² = σ ⁻² _u  + Tσ ⁻² _v , $\mu_{i} = \pm \frac{{\sigma_{v}^{ - 2} \sum_{t = 1}^{T} {({\mathbf{x}}_{it} {\varvec{\upbeta}} - y_{it} )} }}{{\sigma_{u}^{ - 2} + T\sigma_{v}^{ - 2} }}$ and C _i is the constant that turns the last distribution into a proper density. By direct integration one can show that $C_{i} = \sqrt {2\pi } \sigma \left\{ {\sigma_{i} \phi ({{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } {\sigma )}}} \right. \kern-0pt} {\sigma )}} + \mu_{i} \varPhi ({{\mu_{i} } \mathord{\left/ {\vphantom {{\mu_{i} } {\sigma )}}} \right. \kern-0pt} {\sigma )}}} \right\}$.

Conditionals for ${\varvec{\upbeta}},\sigma_{v}^{ - 2} \;{\text{and}}\;\sigma_{u}^{ - 2}$ are normal and gamma distributed and drawing from them is straightforward. However, the distribution for u _i is not of a well-known form. Methods such as Metropolis–Hastings can be used to draw from this distribution but note that by direct integration we can obtain the cdf of this distribution as

$$G(u_{i} ) = \frac{{\sqrt {2\pi } \sigma }}{{C_{i} }}\left\{ {\sigma \left[ {\phi \left( {\frac{{ - \mu_{i} }}{\sigma }} \right) - \phi \left( {\frac{{u_{i} - \mu_{i} }}{\sigma }} \right)} \right] + \mu \left[ {\varPhi \left( {\frac{{u_{i} - \mu_{i} }}{\sigma }} \right) - \varPhi \left( {\frac{{ - \mu_{i} }}{\sigma }} \right)} \right]} \right\}$$

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Therefore, an inverse-cdf method (see e.g. Geweke 2005) can be used to draw from it. Solving the nonlinear equation G(u ^r _i ) = F ^r _i in terms of u ^r _i where F ^r _i is the rth random draw from uniform(0, 1) provides a draw from this distribution.

1.1 Model selection

Bayesian model selection relies on the Bayes factor defined below

$$BF = \frac{{ML({\mathbf{y}}\left| {M_{1} } \right.)}}{{ML({\mathbf{y}}\left| {M_{2} } \right.)}}$$

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where prior probabilities for models M ₁ and M ₂ are considered the same and ML represents the marginal likelihood. Unfortunately, calculating marginal likelihoods can be a challenging task, even when using Makov chain Monet Carlo (MCMC). A range of methods for calculating marginal likelihoods have been proposed (see e.g., Frühwirth-Schnatter (2006) for a review of various methods). In most cases, implementation of these methods needs a candidate distribution. Here we follow Hajargasht and Woźniak (2013) to use a variational Bayes posterior (see below for further information) as a candidate for computation of the marginal likelihood. Specifically, we use the framework proposed by Gelfand and Dey (1994) to write

$$ML = M\left[ {\sum\limits_{m = 1}^{M} {\frac{{{\mathbf{q}}({\varvec{\uptheta}}^{m} |{\mathbf{y}})}}{{L({\mathbf{y}}|{\varvec{\uptheta}}^{m} )p({\varvec{\uptheta}}^{m} )}}} } \right]^{ - 1}$$

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where the θ ^ms are MCMC draws from the posterior p(θ|y) and q(θ ^m|y) is the posterior from variational Bayes calculated at MCMC draw θ ^m. This method provides accurate estimates if the candidate density q(θ ^m|y) is a good approximation and has narrower tails than the true posterior which is normally the case with a variational Bayes posterior. In Sect. 5 we use this methodology to compare models.

1.2 Variational bayes

Variational Bayes (VB) is an approximate method for Bayesian inference (see e.g. Bishop 2006, Ormerod and Wand 2010 and for its application to stochastic frontier models see Hajargasht and Griffiths 2013). The idea behind VB is to approximate an intractable posterior p(θ|y) with a more tractable density q(θ ^m). The optimal q(θ) is obtained by minimizing the Kullback–Leibler distance between the true posterior and the simpler density as follows

$$\min_{q} KL(q) = \min_{q} \int {q({\varvec{\uptheta}})} \log [q({\varvec{\uptheta}})/p({\varvec{\uptheta}}\left| {\mathbf{y}} \right.)]d{\varvec{\uptheta}}$$

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To obtain a tractable approximation we have to make simplifying assumptions about q(θ). One common assumption in variational Bayes is

$$q({\varvec{\uptheta}}) = \prod\limits_{k = 1}^{K} {q_{k} (} {\varvec{\uptheta}}_{k} )$$

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where $\{ {\varvec{\uptheta}}_{1} , \ldots ,{\varvec{\uptheta}}_{K} \}$ is some partition of ${\varvec{\uptheta}}$ and the q _k s are probability density functions. Using results from the calculus of variations, and some simple algebraic manipulations, it is shown that the optimal q _k (which we denote as $q_{\text{k}}^{*}$) are given by

$$\left\{ \begin{gathered} q_{1}^{*} ({\varvec{\uptheta}}_{1} ) = \frac{{\exp (E_{{ - {\varvec{\uptheta}}_{1} }} \log [p({\mathbf{y,\theta }})])}}{{\int {\exp (E_{{ - {\varvec{\uptheta}}_{1} }} \log [p({\mathbf{y,\theta }})])d{\varvec{\uptheta}}_{1} } }} \hfill \\ \quad \vdots \quad \quad \quad \quad \quad \quad \vdots \hfill \\ \quad \vdots \quad \quad \quad \quad \quad \quad \vdots \hfill \\ q_{K}^{*} ({\varvec{\uptheta}}_{K} ) = \frac{{\exp (E_{{ - {\varvec{\uptheta}}_{K} }} \log [p({\mathbf{y,\theta }})])}}{{\int {\exp (E_{{ - {\varvec{\uptheta}}_{K} }} \log [p({\mathbf{y,\theta }})])d{\varvec{\uptheta}}_{K} } }} \hfill \\ \end{gathered} \right.$$

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where $E_{{ - {\varvec{\uptheta}}_{i} }}$ denotes expectation with respect to the density $\prod\nolimits_{j \ne i} {q({\varvec{\uptheta}}_{j} )}$. An iterative procedure is required to find the densities that satisfy these equations.

To apply VB to our stochastic frontier model we consider the following factorization

$$q({\varvec{\upbeta}},\sigma_{v}^{ - 2} ,\sigma_{u}^{ - 2} ,{\mathbf{u}}) = q({\varvec{\upbeta}})q(\sigma_{v}^{ - 2} )q(\sigma_{u}^{ - 2} )q({\mathbf{u}})$$

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Note that for brevity we have suppressed the subscripts on the q densities. Using (14) and (21) we can derive the optimal densities as

$$\left\{ {\begin{array}{*{20}l} {q_{{}}^{*} ({\varvec{\upbeta}}) = N\left\{ {{\bar{\mathbf{\beta }}},\bar{V}_{{\varvec{\upbeta}}} } \right\}} \hfill & {{\bar{\mathbf{\beta }}} = \left( {{\mathbf{x'x}}} \right)^{ - 1} {\mathbf{x'}}({\mathbf{y}} \pm {\bar{\mathbf{u}}} \otimes {\mathbf{i}}_{T} )} \hfill & {\bar{V}_{{\varvec{\upbeta}}} = \frac{{\left( {{\mathbf{x'x}}} \right)^{ - 1} }}{{\overline{{\sigma_{v}^{ - 2} }} }}} \hfill \\ {q_{{}}^{*} (\sigma_{v}^{ - 2} ) = G\left( {\bar{A},\bar{B}} \right)} \hfill & {\bar{A} = A + {{NT} \mathord{\left/ {\vphantom {{NT} 2}} \right. \kern-0pt} 2}} \hfill & {\bar{B} = B + \frac{1}{2}\left( {\sum\limits_{i = 1}^{N} {\sum\limits_{t = 1}^{T} {\left[ {(y_{it} - {\mathbf{x}}_{{{\mathbf{i}}t}} {\bar{\mathbf{\beta }}} \pm \bar{u}_{i} )^{2} + V_{{u_{i} }} } \right]} + tr({\mathbf{x'x}}\bar{V}_{{\varvec{\upbeta}}} )} } \right)} \hfill \\ {q_{{}}^{*} (\sigma_{u}^{ - 2} ) = G\left( {\bar{a}_{u} ,\bar{b}_{u} } \right)} \hfill & {\bar{a}_{u} = N + a_{u} } \hfill & {\bar{b}_{u} = \frac{1}{2}\sum\limits_{i = 1}^{N} {\overline{{u_{i}^{2} }} + b_{u} } } \hfill \\ {q_{{}}^{*} (u_{i} ) = \frac{1}{{C_{i} }}u_{i} \exp \left( { - \frac{{\overline{{\sigma_{{}}^{ - 2} }} }}{2}\left( {u_{i}^{{}} - \tilde{\mu }_{i} } \right)^{2} } \right)} \hfill & {} \hfill & {} \hfill \\ \end{array} } \right.$$

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where $\overline{{\sigma^{ - 2} }} = \overline{{\sigma_{u}^{ - 2} }} + T\overline{{\sigma_{v}^{ - 2} }}$, $\tilde{\mu }_{i} = \pm \frac{{\overline{{\sigma_{v}^{ - 2} }} }}{{\overline{{\sigma_{{}}^{ - 2} }} }}\left( {\sum\limits_{t = 1}^{T} {({\mathbf{x}}_{{{\mathbf{i}}t}} \overline{{\varvec{\upbeta}}} - y_{it} )} } \right)$, V _ui is variance of u _i and C _i is the constant that makes the last distribution into a proper density and is equal to $C_{i} = \sqrt {\frac{2\pi }{{\overline{{\sigma^{ - 2} }} }}} \left\{ {\sqrt {\frac{1}{{\overline{{\sigma^{ - 2} }} }}} \phi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right) + \tilde{\mu }_{i} \varPhi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right)} \right\}$.

In (23) tr denote the trace of a matrix. The quantities $\bar{u}_{i} ,\;\overline{{\sigma^{ - 2} }}$, ${\bar{\mathbf{\beta }}}$, and $\bar{V}_{\beta }$ are the relevant means and variances from the q* densities in (23). They appear in the above distributions when we take expectations of the form $E_{{ - {\varvec{\uptheta}}_{k} }} \log [p({\mathbf{y,\theta }})]$, as described in Eq. (21). Using well-known results on expectations for normal and gamma distributions and direct integration in case of u _i , we can set up an iterative algorithm to find values of $\bar{u}_{i} ,\;\overline{{\sigma^{ - 2} }}$, ${\bar{\mathbf{\beta }}}$, V _ui , and V _β that simultaneously satisfy the equations in (23). The following parameters should be iterated until convergence.

$$\left\{ {{\bar{\mathbf{\beta }}},\;\bar{V}_{{\varvec{\upbeta}}} ,\;\;\overline{{\sigma_{v}^{ - 2} }} = {{\bar{A}} \mathord{\left/ {\vphantom {{\bar{A}} {\bar{B}}}} \right. \kern-0pt} {\bar{B}}},\;\;\overline{{\sigma_{u}^{ - 2} }} = \,\,\,{{\bar{a}_{u} } \mathord{\left/ {\vphantom {{\bar{a}_{u} } {\bar{b}_{u} }}} \right. \kern-0pt} {\bar{b}_{u} }},\;\;\bar{u}_{i} ,\;\;\overline{{u_{i}^{2} }} ,\,\;V_{ui} = \;\overline{{u_{i}^{2} }} - \bar{u}_{i}^{2} } \right\}$$

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where $\bar{u}_{i} = \frac{{\frac{{\tilde{\mu }_{i} }}{{\sqrt {\overline{{\sigma^{ - 2} }} } }}\phi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right) + (\tilde{\mu }_{i}^{2} + {1 \mathord{\left/ {\vphantom {1 {\overline{{\sigma^{ - 2} }} }}} \right. \kern-0pt} {\overline{{\sigma^{ - 2} }} }})\varPhi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right)}}{{\sqrt {\frac{1}{{\overline{{\sigma^{ - 2} }} }}} \phi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right) + \tilde{\mu }_{i} \varPhi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right)}}\quad \overline{{u_{i}^{2} }} = \frac{{\frac{{\tilde{\mu }_{i} + \frac{2}{{\overline{{\sigma^{ - 2} }} }}}}{{\sqrt {\overline{{\sigma^{ - 2} }} } }}\phi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right) + \tilde{\mu }_{i} \left( {\tilde{\mu }_{i}^{2} + \frac{3}{{\overline{{\sigma^{ - 2} }} }}} \right)\varPhi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right)}}{{\sqrt {\frac{1}{{\overline{{\sigma^{ - 2} }} }}} \phi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right) + \tilde{\mu }_{i} \varPhi \left( {\tilde{\mu }_{i} \sqrt {\overline{{\sigma^{ - 2} }} } } \right)}}$