A simple closed-form approximation for the cumulative distribution function of the composite error of stochastic frontier models

Abstract

This paper derives an analytic closed-form formula for the cumulative distribution function (cdf) of the composite error of the stochastic frontier analysis (SFA) model. Since the presence of a cdf is frequently encountered in the likelihood-based analysis with limited-dependent and qualitative variables as elegantly shown in the classic book of Maddala (Limited-dependent and qualitative variables in econometrics. Cambridge University Press, Cambridge, 1983), the proposed methodology is useful in the framework of the stochastic frontier analysis. We apply the formula to the maximum likelihood estimation of the SFA models with a censored dependent variable. The simulations show that the finite sample performance of the maximum likelihood estimator of the censored SFA model is very promising. A simple empirical example on the modeling of reservation wage in Taiwan is illustrated as a potential application of the censored SFA.

Appendix

This appendix shows the derivation of the approximated cumulative distribution function F _app . Since $a = \frac{\lambda }{\sigma } \ge 0$, we divide the derivation into two parts: for $\left( {Q \le 0,a \ge 0} \right)$ and $\left( {Q \ge 0,a \ge 0} \right)$. Furthermore, for ease of exposition, two equations from Abramowitz and Stegun (1970, Eqs. 7.11 and 7.4.32) are given:

$$ \begin{aligned} & erf\left( z \right) = \frac{2}{\sqrt \pi }\int\limits_{0}^{z} {e^{{ - t^{2} }} dt} = 2\int\limits_{0}^{\sqrt 2 z} {\phi \left( t \right)dt} , \\ & \int {e^{{ - \left( {kx^{2} + 2mx + n} \right)}} } dx = \frac{1}{2}\sqrt {\frac{\pi }{k}} \, e^{{\tfrac{{m^{2} - kn}}{k}}} \, erf\left( {\sqrt k x + \frac{m}{\sqrt k }} \right) + C, \, k \ne 0, \\ \end{aligned} $$

where C denotes a finite constant.

Given that $\left( {Q,a,b} \right) \in$ finite R, $b > 0,\,erf\left( { - x} \right) = - erf\left( x \right)$, and define $\varepsilon = \sqrt 2 v/a$, we have:

$$ \begin{aligned} I_{a \ge 0} & = \tfrac{\sqrt 2 }{a}\int\limits_{ - \infty }^{{\tfrac{a}{\sqrt 2 }Q}} {\left( {\int_{ - \infty }^{\sqrt 2 v} {\phi \left( \varsigma \right)d\varsigma } } \right)} \, \phi \left( {\sqrt 2 v\tfrac{b}{a}} \right)dv \\ \, & = \tfrac{\sqrt 2 }{2a}\int\limits_{ - \infty }^{{\tfrac{a}{\sqrt 2 }Q}} {\left( {1 + erf\left( v \right)} \right)} \, \phi \left( {\sqrt 2 v\tfrac{b}{a}} \right)dv \\ \, & = \tfrac{\sqrt 2 }{2a}\int\limits_{ - \infty }^{0} {1 + erf\left( v \right)} \, \phi \left( {\sqrt 2 v\tfrac{b}{a}} \right)dv + \tfrac{\sqrt 2 }{2a}\int\limits_{0}^{{\tfrac{a}{\sqrt 2 }Q}} {1 + erf\left( v \right)} \, \phi \left( {\sqrt 2 v\tfrac{b}{a}} \right)dv, \\ \end{aligned} $$

Note that erf(z) can be well approximated by a function, $g(x) = 1 - e^{{c_{1} x + c_{2} x^{2} }}$ for x ≥ 0, where c ₁ and c ₂ are chosen to ensure that g(x) is as close to erf(x) as possible. The choice of c ₁ and c ₂ is discussed in Sect. 2.

With the preceding results of $I_{a \ge 0} \left( Q \right)$, we then have

$$ \begin{aligned} I_{a \ge 0,Q \ge 0} \left( Q \right) & = \tfrac{\sqrt 2 }{2a}\int\limits_{ - \infty }^{0} {1 + erf\left( v \right)} \, \phi \left( {\sqrt 2 v\tfrac{b}{a}} \right)dv + \tfrac{\sqrt 2 }{2a}\int\limits_{0}^{{\tfrac{a}{\sqrt 2 }Q}} {1 + erf\left( v \right)} \, \phi \left( {\sqrt 2 v\tfrac{b}{a}} \right)dv \\ & = \tfrac{\sqrt 2 }{2a}\int\limits_{0}^{\infty } {1 - erf\left( v \right)} \, \phi \left( {\sqrt 2 v\tfrac{b}{a}} \right)dv + \tfrac{\sqrt 2 }{2a}\int\limits_{0}^{{\tfrac{a}{\sqrt 2 }Q}} {\left( {1 + erf\left( v \right)} \right)} \, \phi \left( {\sqrt 2 v\tfrac{b}{a}} \right)dv \\ & \; \approx \tfrac{\sqrt 2 }{2a}\int\limits_{0}^{\infty } {e^{{c_{1} v + c_{2} v^{2} }} \tfrac{1}{{\sqrt {2\pi } }}} \, e^{{ - \tfrac{{v^{2} b^{2} }}{{a^{2} }}}} dv + \tfrac{\sqrt 2 }{a}\int\limits_{0}^{{\tfrac{a}{\sqrt 2 }Q}} { \, \tfrac{1}{{\sqrt {2\pi } }}e^{{ - \tfrac{{v^{2} b^{2} }}{{a^{2} }}}} } dv \\ & \; - \tfrac{\sqrt 2 }{2a}\int\limits_{0}^{{\tfrac{a}{\sqrt 2 }Q}} { \, \left( {e^{{c_{1} v + c_{2} v^{2} }} } \right)\tfrac{1}{{\sqrt {2\pi } }}e^{{ - \tfrac{{v^{2} b^{2} }}{{a^{2} }}}} } dv \\ & = \tfrac{1}{2\sqrt \pi a}\int\limits_{0}^{\infty } {exp\left( { - \left( {\tfrac{{b^{2} - a^{2} c_{2} }}{{a^{2} }}} \right)v^{2} + c_{1} v} \right)dv} \\ & \; + \tfrac{1}{\sqrt \pi a}\int\limits_{0}^{{\tfrac{b}{\sqrt 2 }Q}} { \, exp\left( { - v^{2} } \right)d\left( {\tfrac{a}{b}v} \right)} - \tfrac{1}{2\sqrt \pi a}\tfrac{a}{\sqrt 2 }Q\int\limits_{0}^{{}} {exp\left( { - \left( {\tfrac{{b^{2} - a^{2} c_{2} }}{{a^{2} }}} \right)v^{2} + c_{1} v} \right)dv} . \\ \end{aligned} $$

when we use (7.4.32) of Abramowitz and Stegun (1970), $I_{a \ge 0,Q \ge 0}$ can be approximated by:

$$ \begin{aligned} I_{a \ge 0,Q \ge 0} \left( Q \right) & \approx \exp \left( {\tfrac{{a^{2} c_{1}^{2} }}{{4b^{2} - 4a^{2} c_{2} }}} \right) \, \tfrac{1}{{4\sqrt {b^{2} - a^{2} c_{2} } }} \, \left[ {1 - erf \, \left( {\tfrac{{ - a \, c_{1} + \sqrt 2 Q\left( {b^{2} - a^{2} c_{2} } \right) \, }}{{2\sqrt {b^{2} - a^{2} c_{2} } }}} \right)} \right] \, \\ & \; + \tfrac{1}{2b}erf\left( {\frac{b \, Q}{\sqrt 2 }} \right). \\ \end{aligned} $$

Likewise, we can derive the approximation for $I_{a \ge 0,Q \le 0}$:

$$ I_{a \ge 0,Q \le 0} \left( Q \right) \approx \exp \left( {\tfrac{{a^{2} c_{1}^{2} }}{{4b^{2} - 4a^{2} c_{2} }}} \right) \, \tfrac{1}{{4\sqrt {b^{2} - a^{2} c_{2} } }} \, \left[ {1 - erf \, \left( {\tfrac{{ - a \, c_{1} - \sqrt 2 Q\left( {b^{2} - a^{2} c_{2} } \right) \, }}{{2\sqrt {b^{2} - a^{2} c_{2} } }}} \right)} \right]. $$

Combining the preceding results, we prove the result in (12).