FRONTIER Procedure

The Normal–Half Normal Model

In case of the normal–half normal model, is iid and is iid , with and independent of each other. Given the independence of the 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 gives

f left-parenthesis u comma epsilon 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 left-parenthesis epsilon plus u right-parenthesis squared Over 2 sigma Subscript v Superscript 2 Baseline EndFraction right-brace

Integrating u out to obtain the marginal density function of results in the form

StartLayout 1st Row 1st Column f left-parenthesis epsilon right-parenthesis 2nd Column equals 3rd Column integral Subscript 0 Superscript normal infinity Baseline f left-parenthesis u comma epsilon right-parenthesis d u 2nd Row 1st Column Blank 2nd Column equals 3rd Column StartFraction 2 Over StartRoot 2 pi EndRoot sigma EndFraction left-bracket 1 minus normal upper Phi left-parenthesis StartFraction epsilon lamda Over sigma EndFraction right-parenthesis right-bracket exp left-brace minus StartFraction epsilon squared Over 2 sigma squared EndFraction right-brace 3rd Row 1st Column Blank 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

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

ln upper L equals c o n s t a n t minus upper N ln sigma plus sigma-summation Underscript i Endscripts ln normal upper Phi left-parenthesis minus StartFraction epsilon Subscript i Baseline lamda Over sigma EndFraction right-parenthesis minus StartFraction 1 Over 2 sigma squared EndFraction sigma-summation Underscript i Endscripts epsilon Subscript i Superscript 2

Last updated: July 09, 2026