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Probabilistic frontier regression model for multinomial ordinal type output data

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Abstract

This paper proposes a probabilistic frontier regression model for multinomial ordinal type output data. We consider some of the output categories as ‘categories of interest’ and the reduction in probability of an output falling into these categories is attributed to the lack in technical efficiency (TE) of the decision-making unit. A measure for TE is proposed to determine the deviations of individual units from the probabilistic frontier of ‘categories of interest’. Simulation results show that the average estimated TE is close to its true value. An application of the proposed model is provided to the data related to the Indian companies, where the categorical output variable is an indicator of return on equity (ROE). Individual TE is obtained for each of the decision-making units (companies under consideration).

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Acknowledgements

The authors are thankful to the two referees and the Associate Editor for their comments and suggestions which has helped to improve the paper.

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Authors and Affiliations

  1. Department of Statistics, Savitribai Phule Pune University, Pune, 411007, India

    Meena Badade & T. V. Ramanathan

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  1. Meena Badade
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  2. T. V. Ramanathan
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Appendix

Appendix

1.1 Appendix A: Conditional posterior densities for gamma distribution for J = 3 with 2, 3 being ‘categories of interest’ and 1 as non-interest category

When u follows gamma distribution, the simplified form of posterior distribution becomes

$$\begin{array}{ll}p({\boldsymbol{\beta }},{\boldsymbol{\alpha }},a,b,{\boldsymbol{u}}| {\boldsymbol{y}},{\boldsymbol{X}})\\ &=\mathop{\prod }\nolimits_{i = 1}^{I}\mathop{\prod }\nolimits_{t = 1}^{T}\left\{K{\left(1-{e}^{-{u}_{i}}\left(1-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)\right)}^{{n}_{i1t}}\right.{\left(\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)}^{{n}_{i2t}}\\ &\,\,\left.\times {\left(\left(1-\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right){e}^{-{u}_{i}}\right)}^{{n}_{i3t}}\right\}\frac{1}{{b}^{a(I+{c}_{1})}\,\Gamma {(a)}^{I+{c}_{2}}}{\left({c}_{3}\mathop{\prod }\nolimits_{i = 1}^{I}{u}_{i}\right)}^{a-1}{e}^{-\frac{1}{b}({\sum \nolimits_{i = 1}^{I}}{u}_{i}+{c}_{4})}\\ &\,\,\times \mathop{\prod }\nolimits_{k = 1}^{p}\frac{1}{\sqrt{2\pi }{\sigma }_{k}}{e}^{-\frac{1}{2}{\left(\frac{{\beta }_{k}-{\mu }_{k}}{{\sigma }_{k}}\right)}^{2}}\frac{1}{\sqrt{2\pi }{\sigma }_{\alpha }}{e}^{-\frac{1}{2}{\left(\frac{{\alpha }_{1}-{\mu }_{\alpha }}{{\sigma }_{\alpha }}\right)}^{2}}\frac{1}{\sqrt{2\pi }{\sigma }_{\alpha }[1-\Phi (\frac{{\alpha }_{1}-{\mu }_{\alpha }}{{\sigma }_{\alpha }})]}{e}^{-\frac{1}{2}{\left(\frac{{\alpha }_{2}-{\mu }_{\alpha }}{{\sigma }_{\alpha }}\right)}^{2}},\end{array}$$

where K is as given in (8). Thus the conditional distributions become,

$$\begin{array}{ll}p({\beta }_{k}| {\boldsymbol{\alpha }},a,b,{\boldsymbol{u}},{\boldsymbol{y}},{\boldsymbol{X}})\propto \mathop{\prod }\nolimits_{i = 1}^{I}\mathop{\prod }\nolimits_{t = 1}^{T}\left\{K{\left(1-{e}^{-{u}_{i}}\left(1-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)\right)}^{{n}_{i1t}}\right.\\ &\,\,\times \left.{\left(\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)}^{{n}_{i2t}}{\left(\left(1-\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right){e}^{-{u}_{i}}\right)}^{{n}_{i3t}}\right\}\\ &\,\,\times \frac{1}{\sqrt{2\pi }{\sigma }_{k}}{e}^{-\frac{1}{2}{\left(\frac{{\beta }_{k}-{\mu }_{k}}{{\sigma }_{k}}\right)}^{2}},\end{array}$$
$$\begin{array}{ll}p({\alpha }_{1}| {\boldsymbol{\beta }},{\alpha }_{2},a,b,{\boldsymbol{u}},{\boldsymbol{y}},{\boldsymbol{X}})\propto \mathop{\prod }\nolimits_{i = 1}^{I}\mathop{\prod }\nolimits_{t = 1}^{T}\left\{K{\left(1-{e}^{-{u}_{i}}\left(1-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)\right)}^{{n}_{i1t}}\right.\\ &\times \left.{\left(\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)}^{{n}_{i2t}}\right\}\frac{1}{\sqrt{2\pi }{\sigma }_{\alpha }}{e}^{-\frac{1}{2}{\left(\frac{{\alpha }_{1}-{\mu }_{\alpha }}{{\sigma }_{\alpha }}\right)}^{2}},\end{array}$$
$$\begin{array}{ll}p({\alpha }_{2}| {\boldsymbol{\beta }},{\alpha }_{1},a,b,{\boldsymbol{u}},{\boldsymbol{y}},{\boldsymbol{X}})\propto \mathop{\prod }\nolimits_{i = 1}^{I}\mathop{\prod }\nolimits_{t = 1}^{T}\left\{K{\left(\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)}^{{n}_{i2t}}\right.\\ &\,\,\times \left.{\left(\left(1-\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right){e}^{-{u}_{i}}\right)}^{{n}_{i3t}}\right\}\frac{1}{\sqrt{2\pi }{\sigma }_{\alpha }[1-\Phi (\frac{{\alpha }_{1}-{\mu }_{\alpha }}{{\sigma }_{\alpha }})]}{e}^{-\frac{1}{2}{\left(\frac{{\alpha }_{2}-{\mu }_{\alpha }}{{\sigma }_{\alpha }}\right)}^{2}},\end{array}$$
$$p(a| {\boldsymbol{\beta }},{\boldsymbol{\alpha }},b,{\boldsymbol{u}},{\boldsymbol{y}},{\boldsymbol{X}})\propto \frac{1}{{b}^{a(I+{c}_{1})}\,\Gamma {(a)}^{I+{c}_{2}}}{\left({c}_{3}\mathop{\prod }\nolimits_{i = 1}^{I}{u}_{i}\right)}^{a-1},$$
$$p(b| {\boldsymbol{\beta }},{\boldsymbol{\alpha }},a,{\boldsymbol{u}},{\boldsymbol{y}},{\boldsymbol{X}})\propto \frac{1}{{b}^{a(I+{c}_{1})}\,\Gamma {(a)}^{I+{c}_{2}}}{e}^{-\frac{1}{b}({\sum \nolimits_{i = 1}^{I}}{u}_{i}+{c}_{4})},$$

and

$$\begin{array}{ll}p({u}_{i}| {\boldsymbol{\beta }},{\boldsymbol{\alpha }},a,b,{\boldsymbol{y}},{\boldsymbol{X}})\propto \mathop{\prod }\nolimits_{t = 1}^{T}{\left(1-{e}^{-{u}_{i}}\left(1-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)\right)}^{{n}_{i1t}}\\ &\,\,\times {({u}_{i})}^{a-1}\exp \left\{-{u}_{i}\left(\mathop{\sum }\limits_{t=1}^{T}({n}_{i2t}+{n}_{i3t})+\frac{1}{b}\right)\right\}.\end{array}$$

1.2 Appendix B: Conditional posterior densities for half-normal distribution for for J = 3 with C = {2, 3} and 1 as non-interest category

When u follows half-normal distribution, the posterior distribution simplifies to

$$\begin{array}{ll}p({\boldsymbol{\beta }},{\boldsymbol{\alpha }},\theta ,{\boldsymbol{u}}| {\boldsymbol{y}},{\boldsymbol{X}})\\ &=\mathop{\prod }\nolimits_{i = 1}^{I}\mathop{\prod }\nolimits_{t = 1}^{T}\left\{K{\left(1-{e}^{-{u}_{i}}\left(1-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)\right)}^{{n}_{i1t}}\right.{\left(\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}-{u}_{i}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)}^{{n}_{i2t}}\\ &\,\,\left.\times {\left(\left(1-\frac{{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{2}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right){e}^{-{u}_{i}}\right)}^{{n}_{i3t}}\right\}{\left(\frac{2\theta }{\pi }\right)}^{I}{e}^{-\frac{{\theta }^{2}}{\pi }{\sum \nolimits_{i = 1}^{I}}{u}_{i}^{2}}\frac{1}{{b}_{1}^{{a}_{1}}\Gamma ({a}_{1})}{\theta }^{{a}_{1}-1}{e}^{-\theta /{b}_{1}}\\ &\,\,\times \mathop{\prod }\nolimits_{k = 1}^{p}\frac{1}{\sqrt{2\pi }{\sigma }_{k}}{e}^{-\frac{1}{2}{\left(\frac{{\beta }_{k}-{\mu }_{k}}{{\sigma }_{k}}\right)}^{2}}\frac{1}{\sqrt{2\pi }{\sigma }_{\alpha }}{e}^{-\frac{1}{2}{\left(\frac{{\alpha }_{1}-{\mu }_{\alpha }}{{\sigma }_{\alpha }}\right)}^{2}}\frac{1}{\sqrt{2\pi }{\sigma }_{\alpha }[1-\Phi (\frac{{\alpha }_{1}-{\mu }_{\alpha }}{{\sigma }_{\alpha }})]}{e}^{-\frac{1}{2}{\left(\frac{{\alpha }_{2}-{\mu }_{\alpha }}{{\sigma }_{\alpha }}\right)}^{2}},\end{array}$$

the expression for K is as given in (8). The conditional distributions of β and α are the same as given in Appendix A. The conditional distributions of θ and u are given as follows:

$$p(\theta | {\boldsymbol{\beta }},{\boldsymbol{\alpha }},{\boldsymbol{u}},{\boldsymbol{y}},{\boldsymbol{X}})\propto {\left(\frac{2\theta }{\pi }\right)}^{I}{e}^{-\frac{{\theta }^{2}}{\pi }{\sum \nolimits_{i = 1}^{I}}{u}_{i}^{2}}\frac{1}{{b}_{1}^{{a}_{1}}\Gamma ({a}_{1})}{\theta }^{{a}_{1}-1}{e}^{-\theta /{b}_{1}}$$
$$\begin{array}{ll}p({u}_{i}| {\boldsymbol{\beta }},{\boldsymbol{\alpha }},\theta ,{\boldsymbol{y}},{\boldsymbol{X}})\propto \mathop{\prod }\nolimits_{t = 1}^{T}{\left(1-{e}^{-{u}_{i}}\left(1-\frac{{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}{1+{e}^{{\alpha }_{1}-{{\boldsymbol{x}}}_{{\boldsymbol{it}}}^{\prime}{\boldsymbol{\beta }}}}\right)\right)}^{{n}_{i1t}}\\ &\,\,\times \exp \left\{-{u}_{i}\mathop{\sum }\limits_{t=1}^{T}({n}_{i2t}+{n}_{i3t})-\frac{{\theta }^{2}{u}_{i}^{2}}{\pi }\right\}\end{array}$$

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Badade, M., Ramanathan, T.V. Probabilistic frontier regression model for multinomial ordinal type output data. J Prod Anal 53, 339–354 (2020). https://doi.org/10.1007/s11123-020-00581-x

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