The QLIM 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) and 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 technological 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 is the nonnegative, technology inefficiency error component. The error component is assumed to be distributed iid normal and independently from . If , the error term, $\epsilon _ i$ , is negatively skewed and represents technology inefficiency. If , the error term $\epsilon _ i$ is positively skewed and represents cost inefficiency. PROC QLIM models the error component as a half normal, exponential, or truncated normal distribution.

The Normal-Half Normal Model

In case of the normal-half normal model, is iid $N(0,\sigma _ v^2)$ , 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

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

The Normal-Exponential Model

Under the normal-exponential model, is iid $N(0,\sigma _ v^2)$ and is iid exponential with scale parameter $\sigma _ u$ . 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

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

and 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

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

The Normal-Truncated Normal Model

The normal-truncated normal model is a generalization of the normal-half normal model by allowing 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

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

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

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

and 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

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

For more detail on normal-half normal, normal-exponential, and normal-truncated models, see Kumbhakar and Knox Lovell (2000) and Coelli, Prasada Rao, and Battese (1998).