Lagrange Multiplier Test: testing for Random Effects

The Breusch-Pagan Lagrange Multiplier Test is used to determine whether random effects are significant in panel data models.

The Random Effects Model is also known as the Error Components Model or the two error structure approach. The Fixed Effects Model has some clear problems in the use of dummy variables. Hence, the Random Effects Model provides a different approach to account for cross-sectional and time-specific effects in panel data.

Instead of using dummy variables like in fixed-effects models, the cross-sectional and time-specific effects are included as error terms. Download the Econometrics Learning Material to learn more about the concept of fixed effects, random effects and their estimation.

Specification of Random Effects Model

A random-effects model including cross-sectional effects can be specified as follows:

Random effects model specification

The random-effects term is not directly observable. However, we can obtain a joint distribution for both the error terms which is used to estimate the model. For details, see “Panel data models” in the Econometrics learning material.

Lagrange Multiplier Test

The null hypothesis of the test can be stated as follows:

The test statistic can be estimated as:

Lagrange multiplier test formula

This LM statistic follows a chi-square distribution with 1 degree of freedom because we are testing for one measure only (variance of random effects term). If we reject the null hypothesis using this test, we conclude that the random effects are significant in the model and the use of the Random Effects Model is appropriate.

Breusch-Pagan Lagrange Multiplier test is applied after estimating the Random Effects Model. The model and the test can be applied using statistical software packages. The results of the model show variance of the error terms, chi-square value and p-value and the results show the following:

Lagrange Multiplier Test for Random Effects
VarianceStandard deviation
Dependent variable0.1420.377
Usual Error term0.0720.268
Random Effects term0.060.244
Chi(1) = 3.98
P-value = 0.023

In the above table, we reject the null hypothesis because the p-value is less than 0.05. We can conclude that the Random Effects are significant. Hence, the use of the Random Effects model is appropriate.

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