Panel Stochastic Frontier Analysis with Positive Skewness

Abstract

This paper focuses on solving the problem of technical efficiency estimation for panel data when residuals are right-skewed. Indeed, there is an ambiguity in stochastic frontier analysis when the residuals of the ordinary least squares estimates are right-skewed, which might indicate that either there is no inefficiency, or that the model is misspecified. To overcome and avoid this problem, we propose a panel model in which the inefficiency term has an extended-half-normal distribution. Hence, our work is an extension of existing work for the cross-section case to panel data with time varying inefficiencies. We first propose estimators of the inefficiency under the extended-half-normal distribution assuming independence between the noise and the inefficiency term. A simulation study illustrates the good performance of our procedure. An application to drinking water for forty-two Moroccan municipalities in the period 2017 to 2019 favors our extended model. Results reveal that the performance of this public sector is generally medium and therefore the waste was significant.

Appendices

Appendix 1: Proof of Theorem 1: Error Term Density When the Inefficiency Component is an Extended-Half-Normal

Proof

Knowing that the inefficiency term $u_{i}$ is an extended-half-normal, that $0<u_i<B$ and $\epsilon _{it} = v_{it}- \eta \left( t\right) u_i$ and given that $f_{\epsilon ,u}$ is the $\left( \epsilon _i, u_i\right)$ joint density, $f_1$ is the $v_{it}$ density function and $f_{\gamma }^-$ is the $u_i$ density function, we have

$$ \begin{aligned} g^{ - } \left( {\epsilon_{i} } \right) &= \int\limits_{0}^{B} {f_{{\epsilon,u}} } \left( {\epsilon_{i} ,u_{i} } \right)du_{i} \\ &= \int\limits_{0}^{B} {f_{{\epsilon,u}} } \left( {\epsilon_{{i1}} , \ldots ,\epsilon_{{it}} , \ldots ,\epsilon_{{iT}} ,u_{i} } \right)du_{i} \\& = \int\limits_{0}^{B} {\prod\limits_{t} {f_{1} } } \left( {\epsilon_{{it}} + \eta \left( t \right) \cdot u_{i} )} \right) \cdot f_{\gamma }^{ - } \left( {u_{i} } \right)du_{i} \\& = \int\limits_{0}^{B} {\frac{1}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} }}} \exp \left\{ { - \frac{1}{2}\left[ {\frac{{\sum\limits_{t} {\left( {\epsilon_{{it}} + \eta \left( t \right) \cdot u_{i} } \right)^{2} } }}{{\sigma _{v}^{2} }}} \right]} \right\} \\ \cdot \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]^{{\left( { - 1} \right)}} \frac{1}{{\left| \gamma \right|}}\phi \left( {a_{0} - \frac{{u_{i} }}{{\left| \gamma \right|}}} \right)du_{i} \\ \end{aligned} $$

$$ \begin{aligned} &= \int\limits_{0}^{B} {\frac{1}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} }}} \exp \left\{ { - \frac{1}{2}\left[ {\frac{{\sum\limits_{t} {\epsilon_{{it}}^{2} } + \sum\limits_{t} {\eta ^{2} } \left( t \right)u_{i}^{2} + 2\sum\limits_{t} {\epsilon_{{it}} } \eta \left( t \right) \cdot u_{i} }}{{\sigma _{v}^{2} }}} \right]} \right\} \\ &\quad \cdot \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]^{{\left( { - 1} \right)}} \frac{1}{{|\gamma |}}\frac{1}{{\sqrt {2\pi } }}\exp \left\{ { - \frac{1}{2}\left( {a_{0} - \frac{{u_{i} }}{{\left| \gamma \right|}}} \right)^{2} } \right\}du_{i} \\ &= \frac{1}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]\left| \gamma \right|\sqrt {2\pi } }} \\ &\quad \cdot \int\limits_{0}^{B} {\exp } \left\{ { - \frac{1}{2}\left[ {\frac{{\sum\limits_{t} {\epsilon_{{it}}^{2} } + \sum\limits_{t} {\eta ^{2} } \left( t \right)u_{i}^{2} + 2\sum\limits_{t} {\epsilon_{{it}} } \eta \left( t \right) \cdot u_{i} }}{{\sigma _{v}^{2} }} + \frac{{\left( {a_{0} \left| \gamma \right| - u_{i} } \right)^{2} }}{{\gamma ^{2} }}} \right]} \right\}du_{i} \\ & = \frac{1}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]\left| \gamma \right|\sqrt {2\pi } }} \\ &\quad \cdot \int\limits_{0}^{B} {\exp } \left\{ { - \frac{1}{2}\left[ {\frac{{\sum\limits_{t} {\epsilon_{{it}}^{2} } + \sum\limits_{t} {\eta ^{2} } \left( t \right)u_{i}^{2} + 2\sum\limits_{t} {\epsilon_{{it}} } \eta \left( t \right).u_{i} }}{{\sigma _{v}^{2} }} + \frac{{B^{2} + u_{i}^{2} - 2Bu_{i} }}{{\gamma ^{2} }}} \right]} \right\}du_{i} \\ \end{aligned} $$

$$ \begin{aligned} &= \frac{1}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]|\gamma |\sqrt {2\pi } }}\exp \left\{ { - \frac{1}{2}\left[ {\frac{{\gamma ^{2} \sum\limits_{t} {\epsilon_{{it}}^{2} } + \sigma _{v}^{2} B^{2} }}{{\sigma _{v}^{2} \gamma ^{2} }}} \right]} \right\} \\ & \quad \cdot \int\limits_{0}^{B} {\exp } \left\{ { - \frac{1}{2}\left[ {\frac{{\gamma ^{2} \sum\limits_{t} {\eta ^{2} } \left( t \right)u_{i}^{2} + 2\gamma ^{2} \sum\limits_{t} {\epsilon_{{it}} } \eta \left( t \right) \cdot u_{i} + \sigma _{v}^{2} u_{i}^{2} - 2B\sigma _{v}^{2} u_{i} }}{{\sigma _{v}^{2} \gamma ^{2} }}} \right]} \right\}du_{i} \\ &= \frac{{\sigma _{*} }}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]|\gamma |}}\exp \left\{ { - \frac{1}{2}\left[ {\frac{{\gamma ^{2} \sum\limits_{t} {\epsilon_{{it}}^{2} } + \sigma _{v}^{2} B^{2} }}{{\sigma _{v}^{2} \gamma ^{2} }}} \right]} \right\} \\ & \quad \cdot \int\limits_{0}^{B} {\frac{1}{{\sigma _{*} \sqrt {2\pi } }}} \exp \left\{ { - \frac{1}{2}\left[ {\frac{{\gamma ^{2} \sum\limits_{t} {\eta ^{2} } \left( t \right) + \sigma _{v}^{2} }}{{\sigma _{v}^{2} \gamma ^{2} }}u_{i}^{2} - 2\frac{{ - \gamma ^{2} \sum\limits_{t} {\epsilon_{{it}} } \eta \left( t \right) + B\sigma _{v}^{2} }}{{\sigma _{v}^{2} \gamma ^{2} }}u_{i} } \right]} \right\}du_{i} \\ &= \frac{{\sigma _{*} }}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]\left| \gamma \right|}}\exp \left\{ { - \frac{1}{2}\left[ {\frac{{\gamma ^{2} \sum\limits_{t} {\epsilon_{{it}}^{2} } + \sigma _{v}^{2} B^{2} }}{{\sigma _{v}^{2} \gamma ^{2} }}} \right]} \right\} \\ & \quad \cdot \int\limits_{0}^{B} {\frac{1}{{\sigma _{*} \sqrt {2\pi } }}} \exp \left\{ { - \frac{1}{2}\left[ {\frac{1}{{\sigma _{*}^{2} }}u_{i}^{2} - 2\frac{{\mu _{{*i}} }}{{\sigma _{*}^{2} }}u_{i} + \frac{{\mu _{{*i}}^{2} }}{{\sigma _{*}^{2} }} - \frac{{\mu _{{*i}}^{2} }}{{\sigma _{*}^{2} }}} \right]} \right\}du_{i} \\ \end{aligned} $$

$$ \begin{aligned} &= \frac{{\sigma _{*} }}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]\left| \gamma \right|}}\exp \left\{ { - \frac{1}{2}\left[ {\frac{{\gamma ^{2} \sum\limits_{t} {\epsilon_{{it}}^{2} } + \sigma _{v}^{2} B^{2} }}{{\sigma _{v}^{2} \gamma ^{2} }} - \frac{{\mu _{{*i}}^{2} }}{{\sigma _{*}^{2} }}} \right]} \right\} \\ & \quad \cdot \int\limits_{0}^{B} {\frac{1}{{\sigma _{*} \sqrt {2\pi } }}} \exp \left\{ { - \frac{1}{2}\left[ {\frac{1}{{\sigma _{*}^{2} }}\left( {u_{i} - \mu _{{*i}} } \right)^{2} } \right]} \right\}du_{i} \\ &= \frac{{\sigma _{*} \exp \left\{ { - \frac{1}{2}a_{{*i}} } \right\}}}{{\left( {2\pi } \right)^{{T/2}} \sigma _{v}^{T} \left[ {\Phi \left( {a_{0} } \right) - \Phi \left( 0 \right)} \right]\left| \gamma \right|}}\left[ {\Phi \left( {\frac{{B - \mu _{{*i}} }}{{\sigma _{*} }}} \right) - \Phi \left( { - \frac{{\mu _{{*i}} }}{{\sigma _{*} }}} \right)} \right] \\ \end{aligned} $$

$\square$

Appendix 2: Proof of Theorem 2: Technical Efficiency When the Inefficiency Component is an Extended-Half-Normal

Proof

If we denote $f_2$ the density of $(u_i\mid \epsilon _i )$, we have

$$\begin{aligned} TE_{it} & = E\left( \exp \left\{ -u_{it}\right\} \mid \epsilon _i\right) \\ &= E\left( \exp \left\{ -\eta \left( t\right) u_i\right\} \mid \left( \epsilon _{i1}, \ldots , \epsilon _{it}, \ldots , \epsilon _{iT}\right) \right) \\& = \int ^B_0\exp \left\{ -\eta \left( t\right) u_i\right\} f_2\left( u_{i} \mid \left( \epsilon _{i1}, \ldots , \epsilon _{it}, \ldots , \epsilon _{iT}\right) \right) d u_i \\&= \int ^B_0\exp \left\{ -\eta \left( t\right) u_i\right\} \frac{f_{\epsilon ,u}\left( \epsilon _{i1}, \ldots , \epsilon _{it}, \ldots , \epsilon _{iT}, u_{i} \right) }{g^-\left( \epsilon _i\right) } d u_i \\ & =\frac{1}{g^-\left( \epsilon _i\right) }\int ^B_0\exp \left\{ -\eta \left( t\right) u_i\right\} f_{\epsilon ,u}\left( \epsilon _{i1}, \ldots , \epsilon _{it}, \ldots , \epsilon _{iT}, u_{i} \right) d u_i \\ &=\frac{\sigma _* \exp \left\{ -\frac{1}{2}a_{*it}\right\} }{\left( 2\pi \right) ^{T/2}\sigma _v^T\left[ \Phi \left( a_0\right) - \Phi \left( 0\right) \right] |\gamma |} \left[ \Phi \left( \frac{B-\mu _{*it}}{\sigma _*}\right) -\Phi \left( -\frac{\mu _{*it}}{\sigma _*}\right) \right] \Big / g^-\left( \epsilon _i\right) \end{aligned}$$

$\square$