The QLIM Procedure

Stochastic Frontier Production and Cost Models

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

$\text{[math]}$

where $\text{[math]}$ . The $\text{[math]}$ term represents the stochastic error component and $\text{[math]}$ is the nonnegative, technology inefficiency error component. The $\text{[math]}$ error component is assumed to be distributed iid normal and independently from $\text{[math]}$ . If $\text{[math]}$ , the error term, $\text{[math]}$ , is negatively skewed and represents technology inefficiency. If $\text{[math]}$ , the error term $\text{[math]}$ is positively skewed and represents cost inefficiency. PROC QLIM models the $\text{[math]}$ error component as a half normal, exponential, or truncated normal distribution.

The Normal-Half Normal Model

In case of the normal-half normal model, $\text{[math]}$ is iid $\text{[math]}$ , $\text{[math]}$ is iid $\text{[math]}$ with $\text{[math]}$ and $\text{[math]}$ independent of each other. Given the independence of error terms, the joint density of $\text{[math]}$ and $\text{[math]}$ can be written as

$\text{[math]}$

Substituting $\text{[math]}$ into the preceding equation gives

$\text{[math]}$

Integrating $\text{[math]}$ out to obtain the marginal density function of $\text{[math]}$ results in the following form:

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ .

In the case of a stochastic frontier cost model, $\text{[math]}$ and

$\text{[math]}$

The log-likelihood function for the production model with $\text{[math]}$ producers is written as

$\text{[math]}$

The Normal-Exponential Model

Under the normal-exponential model, $\text{[math]}$ is iid $\text{[math]}$ and $\text{[math]}$ is iid exponential. Given the independence of error term components $\text{[math]}$ and $\text{[math]}$ , the joint density of $\text{[math]}$ and $\text{[math]}$ can be written as

$\text{[math]}$

The marginal density function of $\text{[math]}$ for the production function is

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

and the marginal density function for the cost function is equal to

$\text{[math]}$

The log-likelihood function for the normal-exponential production model with $\text{[math]}$ producers is

$\text{[math]}$

The Normal-Truncated Normal Model

The normal-truncated normal model is a generalization of the normal-half normal model by allowing the mean of $\text{[math]}$ to differ from zero. Under the normal-truncated normal model, the error term component $\text{[math]}$ is iid $\text{[math]}$ and $\text{[math]}$ is iid $\text{[math]}$ . The joint density of $\text{[math]}$ and $\text{[math]}$ can be written as

$\text{[math]}$

The marginal density function of $\text{[math]}$ for the production function is

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

and the marginal density function for the cost function is

$\text{[math]}$

The log-likelihood function for the normal-truncated normal production model with $\text{[math]}$ producers is

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

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).