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 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
where . 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, , is negatively skewed and represents technology inefficiency. If , the error term is positively skewed and represents cost inefficiency. PROC QLIM models the error component as a half normal, exponential, or truncated normal distribution.
In case of the normal-half normal model, is iid , is iid with and independent of each other. Given the independence of error terms, the joint density of v and u can be written as
Substituting into the preceding equation gives
Integrating u out to obtain the marginal density function of results in the following form:
where and .
In the case of a stochastic frontier cost model, and
The log-likelihood function for the production model with producers is written as
Under the normal-exponential model, is iid and is iid exponential with scale parameter . Given the independence of error term components and , the joint density of v and u can be written as
The marginal density function of for the production function is
and the marginal density function for the cost function is equal to
The log-likelihood function for the normal-exponential production model with producers is
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 and is iid . The joint density of and can be written as
The marginal density function of for the production function is
and the marginal density function for the cost function is
The log-likelihood function for the normal-truncated normal production model with producers is
For more detail on normal-half normal, normal-exponential, and normal-truncated models, see Kumbhakar and Lovell (2000); Coelli, Prasada Rao, and Battese (1998).