PANEL Procedure

Restricted Estimation

The PANEL procedure can fit models that have linear restrictions, producing a Lagrange multiplier (LM) test for each restriction. Consider a set of J linear restrictions bold upper R bold-italic beta equals bold q, where bold upper R is upper J times upper K and bold q is upper J times 1.

The restricted regression is performed by minimizing the error sum of squares subject to the restrictions. In matrix terms, the Lagrangian for this problem is

upper L equals left-parenthesis bold y minus bold upper X bold-italic beta right-parenthesis prime left-parenthesis bold y minus bold upper X bold-italic beta right-parenthesis plus 2 bold-italic lamda left-parenthesis bold upper R bold-italic beta minus bold q right-parenthesis

The Lagrangian is minimized by the restricted estimator bold-italic beta Superscript asterisk, and it can be shown that

bold-italic beta Superscript asterisk Baseline equals ModifyingAbove bold-italic beta With caret minus left-parenthesis bold upper X Superscript prime Baseline bold upper X right-parenthesis Superscript negative 1 Baseline bold upper R Superscript prime Baseline bold-italic lamda

where ModifyingAbove bold-italic beta With caret is the unrestricted estimator.

Because bold upper R bold-italic beta Superscript asterisk Baseline equals bold q, you can solve for bold-italic lamda to obtain the Lagrange multipliers

bold-italic lamda Superscript asterisk Baseline equals left-bracket bold upper R left-parenthesis bold upper X Superscript prime Baseline bold upper X right-parenthesis Superscript negative 1 Baseline bold upper R Superscript prime Baseline right-bracket Superscript negative 1 Baseline left-parenthesis bold upper R ModifyingAbove bold-italic beta With caret minus bold q right-parenthesis

The standard errors of the Lagrange multipliers are the square roots of the diagonal elements of the variance matrix

normal upper V normal a normal r left-parenthesis bold-italic lamda Superscript asterisk Baseline right-parenthesis equals ModifyingAbove sigma With caret Subscript e Superscript 2 Baseline left-bracket bold upper R left-parenthesis bold upper X Superscript prime Baseline bold upper X right-parenthesis Superscript negative 1 Baseline bold upper R Superscript prime Baseline right-bracket Superscript negative 1

where ModifyingAbove sigma With caret Subscript e Superscript 2 is the mean square error (MSE) under the null hypothesis. A significant Lagrange multiplier indicates a restriction that is binding.

Last updated: June 19, 2025