PANEL Procedure

Lagrange Multiplier (LM) Test for Cross-Sectional and Time Effects

For random one-way and two-way error component models, the Lagrange multiplier (LM) test for the existence of cross-sectional or time effects or both is based on the residuals from the restricted model (that is, the pooled model). For more information about the Breusch-Pagan LM test, see the section Tests for Random Effects.

Honda UMP Test and Moulton and Randolph SLM Test

The Breusch-Pagan LM test is two-sided when the variance components are nonnegative. For a one-sided alternative hypothesis, Honda (1985) suggests a uniformly most powerful (UMP) LM test for upper H 0 Superscript 1 Baseline colon sigma Subscript gamma Superscript 2 Baseline equals 0 (no cross-sectional effects) that is based on the pooled estimator. The alternative is the one-sided upper H 1 Superscript 1 Baseline colon sigma Subscript gamma Superscript 2 Baseline greater-than 0. Let ModifyingAbove u With caret Subscript i t be the residual from the simple pooled OLS regression, and let d equals left-parenthesis sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-bracket sigma-summation Underscript t equals 1 Overscript upper T Endscripts ModifyingAbove u With caret Subscript i t Baseline right-bracket squared right-parenthesis slash left-parenthesis sigma-summation Underscript i equals 1 Overscript upper N Endscripts sigma-summation Underscript t equals 1 Overscript upper T Endscripts ModifyingAbove u With caret Subscript i t Superscript 2 Baseline right-parenthesis. Then the test statistic is defined as

upper J identical-to StartRoot StartFraction upper N upper T Over 2 left-parenthesis upper T minus 1 right-parenthesis EndFraction EndRoot left-bracket d minus 1 right-bracket right-arrow Overscript upper H 0 Superscript 1 Baseline Endscripts script upper N left-parenthesis 0 comma 1 right-parenthesis

The square of J is equivalent to the Breusch and Pagan (1980) LM test statistic. Moulton and Randolph (1989) suggest an alternative standardized Lagrange multiplier (SLM) test to improve the asymptotic approximation of Honda’s one-sided LM statistic. The SLM test’s asymptotic critical values are usually closer to the exact critical values than are those of the LM test. The SLM test statistic standardizes Honda’s statistic by its mean and standard deviation. The SLM test statistic is

upper S identical-to StartFraction upper J minus upper E left-parenthesis upper J right-parenthesis Over StartRoot normal upper V normal a normal r left-parenthesis upper J right-parenthesis EndRoot EndFraction equals StartFraction d minus upper E left-parenthesis d right-parenthesis Over StartRoot normal upper V normal a normal r left-parenthesis d right-parenthesis EndRoot EndFraction right-arrow script upper N left-parenthesis 0 comma 1 right-parenthesis

Let upper D equals upper I Subscript upper N Baseline circled-times upper J Subscript upper T, where upper J Subscript upper T is the upper T times upper T square matrix of 1s. The mean and variance can be calculated by the formulas

upper E left-parenthesis d right-parenthesis equals StartFraction normal upper T normal r left-parenthesis upper D upper M Subscript upper Z Baseline right-parenthesis Over n minus k EndFraction
normal upper V normal a normal r left-parenthesis d right-parenthesis equals 2 StartFraction left-parenthesis n minus k right-parenthesis normal upper T normal r left-parenthesis upper D upper M Subscript upper Z Baseline right-parenthesis squared minus left-bracket normal upper T normal r left-parenthesis upper D upper M Subscript upper Z Baseline right-parenthesis right-bracket squared Over left-parenthesis n minus k right-parenthesis squared left-parenthesis n minus k plus 2 right-parenthesis EndFraction

where normal upper T normal r denotes the trace of a particular matrix, Z represents the regressors in the pooled model, n equals upper N upper T is the number of observations, k is the number of regressors, and upper M Subscript upper Z Baseline equals upper I Subscript n Baseline minus upper Z left-parenthesis upper Z prime upper Z right-parenthesis Superscript negative 1 Baseline upper Z prime. To calculate normal upper T normal r left-parenthesis upper D upper M Subscript upper Z Baseline right-parenthesis, let upper Z equals left-parenthesis upper Z prime 1 comma upper Z prime 2 comma ellipsis comma upper Z Subscript upper N Baseline right-parenthesis prime. Then

normal upper T normal r left-parenthesis upper D upper M Subscript upper Z Baseline right-parenthesis equals upper N upper T minus normal upper T normal r left-parenthesis upper J Subscript upper T Baseline sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-bracket upper Z Subscript i Baseline left-parenthesis sigma-summation Underscript j equals 1 Overscript upper N Endscripts upper Z prime Subscript j Baseline upper Z Subscript j Baseline right-parenthesis Superscript negative 1 Baseline upper Z prime Subscript i right-bracket right-parenthesis

Honda (1991) further develops a new SLM test for two-way layout. To test for upper H 0 squared colon sigma Subscript alpha Superscript 2 Baseline equals 0 (no time effects), define d Baseline 2 equals left-parenthesis sigma-summation Underscript t equals 1 Overscript upper T Endscripts left-bracket sigma-summation Underscript i equals 1 Overscript upper N Endscripts ModifyingAbove u With caret Subscript i t Baseline right-bracket squared right-parenthesis slash left-parenthesis sigma-summation Underscript t equals 1 Overscript upper T Endscripts sigma-summation Underscript i equals 1 Overscript upper N Endscripts ModifyingAbove u With caret Subscript i t Superscript 2 Baseline right-parenthesis. Then the test statistic is modified as

upper J Baseline 2 identical-to StartRoot StartFraction upper N upper T Over 2 left-parenthesis upper N minus 1 right-parenthesis EndFraction EndRoot left-bracket d Baseline 2 minus 1 right-bracket right-arrow Overscript upper H 0 squared Endscripts script upper N left-parenthesis 0 comma 1 right-parenthesis

upper J Baseline 2 can be standardized by upper D equals upper J Subscript upper N Baseline circled-times upper I Subscript upper T, and other parameters are unchanged. Therefore,

upper S Baseline 2 identical-to StartFraction upper J Baseline 2 minus upper E left-parenthesis upper J Baseline 2 right-parenthesis Over StartRoot normal upper V normal a normal r left-parenthesis upper J Baseline 2 right-parenthesis EndRoot EndFraction equals StartFraction d Baseline 2 minus upper E left-parenthesis d Baseline 2 right-parenthesis Over StartRoot normal upper V normal a normal r left-parenthesis d Baseline 2 right-parenthesis EndRoot EndFraction right-arrow script upper N left-parenthesis 0 comma 1 right-parenthesis

To test for upper H 0 cubed colon sigma Subscript gamma Superscript 2 Baseline equals 0 comma sigma Subscript alpha Superscript 2 Baseline equals 0 (no cross-sectional and time effects), the test statistic is

upper J Baseline 3 equals StartFraction upper J plus upper J Baseline 2 Over StartRoot 2 EndRoot EndFraction
upper D equals StartRoot StartFraction n Over upper T minus 1 EndFraction EndRoot StartFraction upper I Subscript upper N Baseline circled-times upper J Subscript upper T Baseline Over 2 EndFraction plus StartRoot StartFraction n Over upper N minus 1 EndFraction EndRoot StartFraction upper J Subscript upper N Baseline circled-times upper I Subscript upper T Baseline Over 2 EndFraction

To standardize, define d Baseline 3 equals StartRoot n slash left-parenthesis upper T minus 1 right-parenthesis EndRoot d slash 2 plus StartRoot n slash left-parenthesis upper N minus 1 right-parenthesis EndRoot left-parenthesis d Baseline 2 right-parenthesis slash 2, then

upper S Baseline 3 identical-to StartFraction upper J Baseline 3 minus upper E left-parenthesis upper J Baseline 3 right-parenthesis Over StartRoot normal upper V normal a normal r left-parenthesis upper J Baseline 3 right-parenthesis EndRoot EndFraction equals StartFraction d Baseline 3 minus upper E left-parenthesis d Baseline 3 right-parenthesis Over StartRoot normal upper V normal a normal r left-parenthesis d Baseline 3 right-parenthesis EndRoot EndFraction right-arrow script upper N left-parenthesis 0 comma 1 right-parenthesis

King and Wu LMMP Test and the SLM Test

King and Wu (1997) derive the locally mean most powerful (LMMP) one-sided test for upper H 0 Superscript 1 and upper H 0 squared, which coincides with the Honda (1985) UMP test. Baltagi, Chang, and Li (1992) extend the King and Wu (1997) test for upper H 0 cubed as follows:

upper K upper W identical-to StartFraction StartRoot upper T minus 1 EndRoot Over StartRoot upper N plus upper T minus 2 EndRoot EndFraction upper J plus StartFraction StartRoot upper N minus 1 EndRoot Over StartRoot upper N plus upper T minus 2 EndRoot EndFraction upper J Baseline 2 right-arrow Overscript upper H 0 cubed Endscripts script upper N left-parenthesis 0 comma 1 right-parenthesis

For the standardization, use upper D equals upper I Subscript upper N Baseline circled-times upper J Subscript upper T Baseline plus upper J Subscript upper N Baseline circled-times upper I Subscript upper T. Define d Subscript upper K upper W Baseline equals d plus d Baseline 2; then

upper S Subscript upper K upper W Baseline identical-to StartFraction upper K upper W minus upper E left-parenthesis upper K upper W right-parenthesis Over StartRoot normal upper V normal a normal r left-parenthesis upper K upper W right-parenthesis EndRoot EndFraction equals StartFraction d Subscript upper K upper W Baseline minus upper E left-parenthesis d Subscript upper K upper W Baseline right-parenthesis Over StartRoot normal upper V normal a normal r left-parenthesis d Subscript upper K upper W Baseline right-parenthesis EndRoot EndFraction right-arrow script upper N left-parenthesis 0 comma 1 right-parenthesis

Gourieroux, Holly, and Monfort LM Test

If one or both variance components (sigma Subscript gamma Superscript 2 and sigma Subscript alpha Superscript 2) are small and close to 0, the test statistics J and upper J Baseline 2 can be negative. Baltagi, Chang, and Li (1992) follow Gourieroux, Holly, and Monfort (1982) and propose a one-sided LM test for upper H 0 cubed, which is immune to the possible negative values of J and upper J Baseline 2. The test statistic is

upper G upper H upper M identical-to StartLayout Enlarged left-brace 1st Row 1st Column upper J squared plus left-parenthesis upper J Baseline 2 right-parenthesis squared 2nd Column if upper J greater-than 0 comma upper J Baseline 2 greater-than 0 2nd Row 1st Column upper J squared 2nd Column if upper J greater-than 0 comma upper J Baseline 2 less-than-or-equal-to 0 3rd Row 1st Column left-parenthesis upper J Baseline 2 right-parenthesis squared 2nd Column if upper J less-than-or-equal-to 0 comma upper J Baseline 2 greater-than 0 4th Row 1st Column 0 2nd Column if upper J less-than-or-equal-to 0 comma upper J Baseline 2 less-than-or-equal-to 0 EndLayout right-arrow Overscript upper H 0 cubed Endscripts left-parenthesis one-fourth right-parenthesis chi squared left-parenthesis 0 right-parenthesis plus left-parenthesis one-half right-parenthesis chi squared left-parenthesis 1 right-parenthesis plus left-parenthesis one-fourth right-parenthesis chi squared left-parenthesis 2 right-parenthesis

where chi squared left-parenthesis 0 right-parenthesis is the unit mass at the origin.

Last updated: June 19, 2025