The HPQLIM Procedure

Stochastic Frontier Production and Cost Models

The Normal-Half-Normal Model
The Normal-Exponential Model
The Normal–Truncated Normal Model

Stochastic frontier production models were first developed by Aigner, Lovell, and Schmidt (1977); Meeusen and van den Broeck (1977). Specification of these models allow for random shocks of the production or cost but also include a term for technical or cost inefficiency. Assuming that the production function takes a log-linear Cobb-Douglas form, the stochastic frontier production model can be written as

$ln({y_ i}) = \beta _0+\sum _{n} \bbeta _ n\ln (x_{ni})+\epsilon _ i$

where $\epsilon _ i=v_ i-u_ i$ . The term represents the stochastic error component, and the term represents the nonnegative, technical inefficiency error component. The error component is assumed to be distributed iid normal and independent from . If , the error term $\epsilon _ i$ is negatively skewed and represents technical inefficiency. If , the error term $\epsilon _ i$ is positively skewed and represents cost inefficiency. PROC HPQLIM models the error component as a half-normal, exponential, or truncated normal distribution.

The Normal-Half-Normal Model

When is iid $N(0,\sigma _ v^2)$ in a normal-half-normal model, is iid $N^+(0,\sigma _ u^2)$ , with and independent of each other. Given the independence of error terms, the joint density of and can be written as

$f(u,v) = \frac{2}{2\pi \sigma _ u\sigma _ v} \exp \left\{ -\frac{u^2}{2\sigma _ u^2} - \frac{v^2}{2\sigma _ v^2} \right\}$

Substituting $v=\epsilon +u$ into the preceding equation gives

$f(u,\epsilon ) = \frac{2}{2\pi \sigma _ u\sigma _ v} \exp \left\{ -\frac{u^2}{2\sigma _ u^2} - \frac{(\epsilon +u)^2}{2\sigma _ v^2} \right\}$

Integrating out to obtain the marginal density function of $\epsilon$ results in the following form:

$\displaystyle f(\epsilon )$	$\displaystyle =$	$\displaystyle \int ^\infty _0 f(u,\epsilon )du$
$\displaystyle$	$\displaystyle =$	$\displaystyle \frac{2}{\sqrt {2\pi }\sigma } \left[ 1-\phi \left( \frac{\epsilon \lambda }{\sigma } \right) \right] \exp \left\{ -\frac{\epsilon ^2}{2\sigma ^2} \right\}$
$\displaystyle$	$\displaystyle =$	$\displaystyle \frac{2}{\sigma }\phi \left( \frac{\epsilon }{\sigma } \right) \Phi \left( -\frac{\epsilon \lambda }{\sigma } \right)$

where $\lambda =\sigma _ u/\sigma _ v$ and $\sigma =\sqrt {\sigma _ u^2+\sigma _ v^2}$ .

In the case of a stochastic frontier cost model, $v=\epsilon -u$ and

$f(\epsilon ) = \frac{2}{\sigma }\phi \left( \frac{\epsilon }{\sigma } \right) \Phi \left( \frac{\epsilon \lambda }{\sigma } \right)$

The log-likelihood function for the production model with producers is written as

$\ln L = \textnormal{constant} - N \ln \sigma +\sum _ i{ \ln \Phi \left( -\frac{\epsilon _ i\lambda }{\sigma } \right) } -\frac{1}{2\sigma ^2}\sum _ i \epsilon ^2_ i$

The Normal-Exponential Model

Under the normal-exponential model, is iid $N(0,\sigma _ v^2)$ and is iid exponential. Given the independence of error term components and , the joint density of and can be written as

$f(u,v) = \frac{1}{\sqrt {2\pi }\sigma _ u\sigma _ v} \exp \left\{ -\frac{u}{\sigma _ u} - \frac{v^2}{2\sigma _ v^2} \right\}$

The marginal density function of $\epsilon$ for the production function is

$\displaystyle f(\epsilon )$	$\displaystyle =$	$\displaystyle \int ^\infty _0 f(u,\epsilon )du$
$\displaystyle$	$\displaystyle =$	$\displaystyle \left( \frac{1}{\sigma _ u} \right) \Phi \left( -\frac{\epsilon }{\sigma _ v}-\frac{\sigma _ v}{\sigma _ u} \right) \exp \left\{ \frac{\epsilon }{\sigma _ u}+\frac{\sigma _ v^2}{2\sigma _ u^2} \right\}$

The marginal density function for the cost function is equal to

$f(\epsilon ) = \left( \frac{1}{\sigma _ u} \right) \Phi \left( \frac{\epsilon }{\sigma _ v}-\frac{\sigma _ v}{\sigma _ u} \right) \exp \left\{ -\frac{\epsilon }{\sigma _ u}+\frac{\sigma _ v^2}{2\sigma _ u^2} \right\}$

The log-likelihood function for the normal-exponential production model with producers is

$\ln L = \textnormal{constant} - N \ln \sigma _ u+N\left( \frac{\sigma _ v^2}{2\sigma _ u^2} \right) +\sum _ i\frac{\epsilon _ i}{\sigma _ u} +\sum _ i\ln \Phi \left( \frac{\epsilon _ i}{\sigma _ v}-\frac{\sigma _ v}{\sigma _ u} \right)$

The Normal–Truncated Normal Model

The normal–truncated normal model is a generalization of the normal-half-normal model that allows the mean of to differ from zero. Under the normal–truncated normal model, the error term component is iid $N^+(0,\sigma _ v^2)$ and is iid $N(\mu ,\sigma _ u^2)$ . The joint density of and can be written as

$f(u,v) = \frac{1}{\sqrt { 2\pi }\sigma _ u\sigma _ v\Phi \left( \mu /\sigma _ u \right) } \exp \left\{ -\frac{(u-\mu )^2}{2\sigma _ u^2}-\frac{v^2}{2\sigma _ v^2} \right\}$

The marginal density function of $\epsilon$ for the production function is

$\displaystyle f(\epsilon )$	$\displaystyle =$	$\displaystyle \int ^\infty _0 f(u,\epsilon )du$
$\displaystyle$	$\displaystyle =$	$\displaystyle \frac{1}{ \sqrt {2\pi }\sigma \Phi \left( \mu /\sigma _ u \right) } \Phi \left( \frac{\mu }{\sigma \lambda }-\frac{\epsilon \lambda }{\sigma } \right) \exp \left\{ -\frac{(\epsilon +\mu )^2}{2\sigma ^2} \right\}$
$\displaystyle$	$\displaystyle =$	$\displaystyle \frac{1}{\sigma }\phi \left( \frac{\epsilon +\mu }{\sigma } \right) \Phi \left( \frac{\mu }{\sigma \lambda }-\frac{\epsilon \lambda }{\sigma } \right) \left[ \Phi \left( \frac{\mu }{\sigma _ u} \right) \right]^{-1}$

The marginal density function for the cost function is

$\displaystyle f(\epsilon )$

$\displaystyle =$

$\displaystyle \frac{1}{\sigma }\phi \left( \frac{\epsilon -\mu }{\sigma } \right) \Phi \left( \frac{\mu }{\sigma \lambda }+\frac{\epsilon \lambda }{\sigma } \right) \left[ \Phi \left( \frac{\mu }{\sigma _ u} \right) \right]^{-1}$

The log-likelihood function for the normal–truncated normal production model with producers is

$\displaystyle \ln L$	$\displaystyle =$	$\displaystyle \textnormal{constant} - N \ln \sigma -N\ln \Phi \left( \frac{\mu }{\sigma _ u} \right) +\sum _ i\ln \Phi \left( \frac{\mu }{\sigma \lambda }+\frac{\epsilon _ i\lambda }{\sigma } \right)$
$\displaystyle$	$\displaystyle$	$\displaystyle -\frac{1}{2}\sum _ i{ \left( \frac{\epsilon _ i+\mu }{\sigma } \right)^2 }$

For more information about normal-half-normal, normal-exponential, and normal–truncated normal models, see: Kumbhakar and Lovell (2000); Coelli, Prasada Rao, and Battese (1998).