CQLIM Procedure

Stochastic Frontier Production and Cost Models

Stochastic frontier production models were first developed by Aigner, Lovell, and Schmidt (1977); Meeusen and van den Broeck (1977). Specification of these models allows for random shocks of the production or cost but also includes 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 left-parenthesis y Subscript i Baseline right-parenthesis equals beta 0 plus sigma-summation Underscript n Endscripts bold-italic beta Subscript n Baseline ln left-parenthesis x Subscript n i Baseline right-parenthesis plus epsilon Subscript i

where . 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 is negatively skewed and represents technical inefficiency. If , the error term is positively skewed and represents cost inefficiency. PROC CQLIM models the error component as a half-normal, exponential, or truncated normal distribution.

The Normal-Half-Normal Model

When is iid in a normal-half-normal model, 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

f left-parenthesis u comma v right-parenthesis equals StartFraction 2 Over 2 pi sigma Subscript u Baseline sigma Subscript v Baseline EndFraction exp left-brace minus StartFraction u squared Over 2 sigma Subscript u Superscript 2 Baseline EndFraction minus StartFraction v squared Over 2 sigma Subscript v Superscript 2 Baseline EndFraction right-brace

Substituting into the preceding equation and integrating u out gives

StartLayout 1st Row 1st Column f left-parenthesis epsilon right-parenthesis 2nd Column equals 3rd Column StartFraction 2 Over sigma EndFraction phi left-parenthesis StartFraction epsilon Over sigma EndFraction right-parenthesis normal upper Phi left-parenthesis minus StartFraction epsilon lamda Over sigma EndFraction right-parenthesis EndLayout

where and .

In the case of a stochastic frontier cost model, and

f left-parenthesis epsilon right-parenthesis equals StartFraction 2 Over sigma EndFraction phi left-parenthesis StartFraction epsilon Over sigma EndFraction right-parenthesis normal upper Phi left-parenthesis StartFraction epsilon lamda Over sigma EndFraction right-parenthesis

For more information, see SAS/ETS User's Guide.

The Normal-Exponential Model

Under the normal-exponential model, is iid and is iid exponential. Given the independence of the error term components and , the joint density of v and u can be written as

f left-parenthesis u comma v right-parenthesis equals StartFraction 1 Over StartRoot 2 pi EndRoot sigma Subscript u Baseline sigma Subscript v Baseline EndFraction exp left-brace minus StartFraction u Over sigma Subscript u Baseline EndFraction minus StartFraction v squared Over 2 sigma Subscript v Superscript 2 Baseline EndFraction right-brace

The marginal density function of for the production function is

StartLayout 1st Row 1st Column f left-parenthesis epsilon right-parenthesis 2nd Column equals 3rd Column left-parenthesis StartFraction 1 Over sigma Subscript u Baseline EndFraction right-parenthesis normal upper Phi left-parenthesis minus StartFraction epsilon Over sigma Subscript v Baseline EndFraction minus StartFraction sigma Subscript v Baseline Over sigma Subscript u Baseline EndFraction right-parenthesis exp left-brace StartFraction epsilon Over sigma Subscript u Baseline EndFraction plus StartFraction sigma Subscript v Superscript 2 Baseline Over 2 sigma Subscript u Superscript 2 Baseline EndFraction right-brace EndLayout

The marginal density function for the cost function is equal to

f left-parenthesis epsilon right-parenthesis equals left-parenthesis StartFraction 1 Over sigma Subscript u Baseline EndFraction right-parenthesis normal upper Phi left-parenthesis StartFraction epsilon Over sigma Subscript v Baseline EndFraction minus StartFraction sigma Subscript v Baseline Over sigma Subscript u Baseline EndFraction right-parenthesis exp left-brace minus StartFraction epsilon Over sigma Subscript u Baseline EndFraction plus StartFraction sigma Subscript v Superscript 2 Baseline Over 2 sigma Subscript u Superscript 2 Baseline EndFraction right-brace

For more information, see SAS/ETS User's Guide.

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 0. Under the normal–truncated normal model, the error term component is iid , and the error term component is iid . The joint density of and can be written as

The marginal density function of for the production function is

StartLayout 1st Row 1st Column f left-parenthesis epsilon right-parenthesis 2nd Column equals 3rd Column StartFraction 1 Over sigma EndFraction phi left-parenthesis StartFraction epsilon plus mu Over sigma EndFraction right-parenthesis normal upper Phi left-parenthesis StartFraction mu Over sigma lamda EndFraction minus StartFraction epsilon lamda Over sigma EndFraction right-parenthesis left-bracket normal upper Phi left-parenthesis StartFraction mu Over sigma Subscript u Baseline EndFraction right-parenthesis right-bracket Superscript negative 1 EndLayout

The marginal density function for the cost function is

For more information, see SAS/ETS User's Guide.

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).

Last updated: July 09, 2026