FRONTIER Procedure

The Normal–Truncated Normal Model

The normal–truncated normal model is a generalization of the normal–half normal model in which the mean of is allowed 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

f left-parenthesis u comma v right-parenthesis equals StartFraction 1 Over 2 pi sigma Subscript u Baseline sigma Subscript v Baseline normal upper Phi left-parenthesis mu slash sigma Subscript u Baseline right-parenthesis EndFraction exp left-brace minus StartFraction left-parenthesis u minus mu right-parenthesis 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

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 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 1 Over StartRoot 2 pi EndRoot sigma normal upper Phi left-parenthesis mu slash sigma Subscript u Baseline right-parenthesis EndFraction normal upper Phi left-parenthesis StartFraction mu Over sigma lamda EndFraction minus StartFraction epsilon lamda Over sigma EndFraction right-parenthesis exp left-brace minus StartFraction left-parenthesis epsilon plus mu right-parenthesis squared Over 2 sigma squared EndFraction right-brace 3rd Row 1st Column Blank 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

and the marginal density function for the cost 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 minus mu Over sigma EndFraction right-parenthesis normal upper Phi left-parenthesis StartFraction mu Over sigma lamda EndFraction plus 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 log-likelihood function for the normal–truncated normal production model with N producers is

StartLayout 1st Row 1st Column ln upper L 2nd Column equals 3rd Column c o n s t a n t minus upper N ln sigma minus upper N ln normal upper Phi left-parenthesis StartFraction mu Over sigma Subscript u Baseline EndFraction right-parenthesis plus sigma-summation Underscript i Endscripts ln normal upper Phi left-parenthesis StartFraction mu Over sigma lamda EndFraction minus StartFraction epsilon Subscript i Baseline lamda Over sigma EndFraction right-parenthesis 2nd Row 1st Column Blank 2nd Column Blank 3rd Column minus one half sigma-summation Underscript i Endscripts left-parenthesis StartFraction epsilon Subscript i Baseline plus mu Over sigma EndFraction right-parenthesis squared EndLayout

For more information about normal–half normal, normal-exponential, and normal-truncated models, see Kumbhakar and Lovell (2000) and Coelli, Prasada Rao, and Battese (1998).

Last updated: July 09, 2026