Estimation and efficiency measurement in stochastic production frontiers with ordinal outcomes

Abstract

We consider Bayesian estimation of a stochastic production frontier with ordered categorical output, where the inefficiency error is assumed to follow an exponential distribution, and where output, conditional on the inefficiency error, is modelled as an ordered probit model. Gibbs sampling algorithms are provided for estimation with both cross-sectional and panel data, with panel data being our main focus. A Monte Carlo study and a comparison of results from an example where data are used in both continuous and categorical form supports the usefulness of the approach. New efficiency measures are suggested to overcome a lack-of-invariance problem suffered by traditional efficiency measures. Potential applications include health and happiness production, university research output, financial credit ratings, and agricultural output recorded in broad bands. In our application to individual health production we use data from an Australian panel survey to compute posterior densities for marginal effects, outcome probabilities, and a number of within-sample and out-of-sample efficiency measures.

Appendices

Appendix 1: Density functions for relative efficiency

To derive the density function for RE _s,c  = u _c /u _s given in (34) we use the short hand notation r = RE _s,c and transform the variables as r = u _c /u _s and z = u _s , where

$$ p(u_{j} ) = \lambda_{j}^{ - 1} \exp \left( { - u_{j} \lambda_{j}^{ - 1} } \right)\quad j = s,c. $$

Thus, u _c  = rz and

$$ \begin{aligned} p(r,z) & = p(u_{c} ,u_{s} )\left| {\begin{array}{*{20}c} {{{\partial u_{c} } \mathord{\left/ {\vphantom {{\partial u_{c} } {\partial r}}} \right. \kern-0pt} {\partial r}}} \hfill & {{{\partial u_{c} } \mathord{\left/ {\vphantom {{\partial u_{c} } {\partial z}}} \right. \kern-0pt} {\partial z}}} \hfill \\ {{{\partial u_{s} } \mathord{\left/ {\vphantom {{\partial u_{s} } {\partial r}}} \right. \kern-0pt} {\partial r}}} \hfill & {{{\partial u_{s} } \mathord{\left/ {\vphantom {{\partial u_{s} } {\partial z}}} \right. \kern-0pt} {\partial z}}} \hfill \\ \end{array} } \right| \\ & = \lambda_{c}^{ - 1} \lambda_{s}^{ - 1} \exp \left( { - \lambda_{c}^{ - 1} rz - \lambda_{s}^{ - 1} z} \right)\left| {\begin{array}{*{20}c} z \hfill & r \hfill \\ 0 \hfill & 1 \hfill \\ \end{array} } \right| \\ & = z\lambda_{c}^{ - 1} \lambda_{s}^{ - 1} \exp \left\{ { - z\left( {\lambda_{c}^{ - 1} r + \lambda_{s}^{ - 1} } \right)} \right\}. \\ \end{aligned} $$

The density for r is given by

$$ p(r) = \lambda_{c}^{ - 1} \lambda_{s}^{ - 1} \int\limits_{0}^{ + \infty } {z\exp \left\{ { - z\left( {\lambda_{c}^{ - 1} r + \lambda_{s}^{ - 1} } \right)} \right\}dz} = \frac{{\lambda_{c}^{ - 1} \lambda_{s}^{ - 1} }}{{\left( {\lambda_{c}^{ - 1} r + \lambda_{s}^{ - 1} } \right)^{2} }}. $$

Now we consider the density function for

$$ r_{sJ} = \frac{{1 - \Upphi \left( {\frac{{1 - \varvec{x}_{s} \varvec{\beta} + u_{s} }}{{\sigma_{v} }}} \right)}}{{p_{0} }}, $$

(41)

where $p_{0} = 1 - \Upphi \left( {{{(1 - \varvec{x}_{s} \varvec{\beta} )} \mathord{\left/ {\vphantom {{(1 - \varvec{x}_{s} \beta )} {\sigma_{v} }}} \right. \kern-0pt} {\sigma_{v} }}} \right)$. We wish to derive the density for r _sJ from that for u _s which is given by $p(u_{s} ) = \lambda_{s}^{ - 1} \exp \left( { - u_{s} \lambda_{s}^{ - 1} } \right)$. Solving (41) for u _s yields $u_{s} = \varvec{x}_{s} \varvec{\beta} - 1 + \sigma_{v} \Upphi^{ - 1} \left( {1 - r_{sJ} p_{0} } \right).$ Then, noting that, if $x = \Upphi^{ - 1} (p)$, then ${{dx} \mathord{\left/ {\vphantom {{dx} {dp}}} \right. \kern-0pt} {dp}} = {1 \mathord{\left/ {\vphantom {1 {\phi_{SN} \left( {\Upphi^{ - 1} (p)} \right)}}} \right. \kern-0pt} {\phi_{SN} \left( {\Upphi^{ - 1} (p)} \right)}}$, we have ${{du_{s} } \mathord{\left/ {\vphantom {{du_{s} } {dr_{sJ} }}} \right. \kern-0pt} {dr_{sJ} }} = {{ - \sigma_{v} p_{0} } \mathord{\left/ {\vphantom {{ - \sigma_{v} p_{0} } {\phi_{SN} \left( {\Upphi^{ - 1} \left( {1 - r_{sJ} p_{0} } \right)} \right)}}} \right. \kern-0pt} {\phi_{SN} \left( {\Upphi^{ - 1} \left( {1 - r_{sJ} p_{0} } \right)} \right)}}.$ Thus,

$$ p(r_{sJ} ) = p(u_{s} )\left| {\frac{{du_{s} }}{{dr_{sJ} }}} \right| = \frac{{\sigma_{v} p_{0} \lambda_{s}^{ - 1} \exp \left\{ { - \lambda_{s}^{ - 1} \left[ {\varvec{x}_{s} \beta - 1 + \sigma_{v} \Upphi^{ - 1} \left( {1 - r_{sJ} p_{0} } \right)} \right]} \right\}}}{{\phi_{SN} \left( {\Upphi^{ - 1} \left( {1 - r_{sJ} p_{0} } \right)} \right)}}. $$

Appendix 2: Definition of variables

Variables	Definition
y
SRH	Self-reported health, 0 for poor, 1 for fair, 2 for good, 3 for very good and 4 for excellent
x
LT3EX	1 if doing exercise for less than 3 times but at least 1 time per week and 0 otherwise
MT3	1 if doing exercise for more than 3 times per week, including doing exercise every day and 0 otherwise
NOEX	1 if doing no exercise at all and 0 otherwise. This variable is used as the base for exercise level and is dropped off in the estimation
NOSM	1 if never smoke and 0 otherwise
LRA	1 if having low alcohol riska or alcohol risky and 0 otherwise
HIGHRA	1 if having high alcohol risk and 0 otherwise
NORA	1 if having no alcohol risk and 0 otherwise. This variable is used as the base for alcohol risk and is dropped off in the estimation
LONELY1	1 if sometime feel lonelyb and 0 otherwise
LONELY2	1 if always feel lonely and 0 otherwise
LONELY0	1 if never feel lonely and 0 otherwise. This variable is used as the base for social net work and is dropped off in the estimation
INNER	1 if living in inner region of Australia and 0 otherwise
OUTER	1 if living in outer region of Australia and 0 otherwise
REMOTE	1 if living in remote region of Australia and 0 otherwise
MAJOR	1 if living in major cities of Australia and 0 otherwise. This variable is used as the base for living region and is dropped off in the estimation
STUDENT	1 if full time study and 0 otherwise
PARTTIME	1 if part-time employed and 0 otherwise
UNEMP	1 if unemployed and 0 otherwise
RETD	1 if completely retired from labour market and 0 otherwise
NOTINLAB	1 if not in labour force and 0 otherwise
FULLTIME	1 if full-time employed and 0 otherwise. This variable is used as the base for major activity and is dropped off in the estimation
DEGREE	1 if the highest qualification is a tertiary degree and 0 otherwise
DIPLOMA	1 if the highest qualification is diploma or trade certificate and 0 otherwise
YEAR12	1 if the highest qualification is Year 12 and 0 otherwise
LOWER12	1 if still in school or cannot finish Year 12 and 0 otherwise. This variable is used as the base for education level and is dropped off in the estimation
w
GENDER	1 for male and 0 for female
AUSABO	1 if born in Australia and aboriginal and 0 otherwise
MAINENG	1 if born in other main English speaking countries and 0 otherwise
OTHERC	1 if born in other countries rather than Australia and main English speaking countries and 0 otherwise
AUSNABO	1 if born in Australia and not aboriginal and 0 otherwise. This variable is used as the base for country born status and is dropped off in the estimation
MARRIAGE	1 if living with somebody in a relationship for most of the time periods and 0 otherwise
AGEG1	1 if aged from 18 to 24 and 0 otherwise
AGEG2	1 if aged from 25 to 34 and 0 otherwise
AGEG3	1 if aged from 35 to 44 and 0 otherwise
AGEG4	1 if aged from 45 to 54 and 0 otherwise
AGEG5	1 if aged from 55 to 64 and 0 otherwise
AGEG6	1 if aged 65 or over and 0 otherwise. This variable is used as the base for age band and is dropped off in the estimation

1. ^aGenerally, no alcohol risk means 0 standard drinks per week; low alcohol risk means, for males, 1–6 standard drinks per week, or, for females, 1–4 standard drinks per week; high alcohol risk means, for males, at least 7 standard drinks per week, for females, at least 5 standard drinks per week
2. ^bThe information on loneliness is collected through the statement, ‘I often feel very lonely’. Respondents are assigned a number from 1 to 7 representing from strongly disagree to strongly agree; that is, the higher the number the individual chooses, the more she or he agrees with the statement. We re-classify the respondents into three groups as never feel lonely (1 or 2), sometimes feel lonely (from 3 to 5) and always feel lonely (6 or 7)