It was independently suggested with some extension by R. Dennis Cook and Sanford Weisberg in 1983 (Cook–Weisberg test).
[2] Derived from the Lagrange multiplier test principle, it tests whether the variance of the errors from a regression is dependent on the values of the independent variables.
Ordinary least squares constrains these so that their mean is 0 and so, given the assumption that their variance does not depend on the independent variables, an estimate of this variance can be obtained from the average of the squared values of the residuals.
If the assumption is not held to be true, a simple model might be that the variance is linearly related to independent variables.
Such a model can be examined by regressing the squared residuals on the independent variables, using an auxiliary regression equation of the form This is the basis of the Breusch–Pagan test.
If the test statistic has a p-value below an appropriate threshold (e.g. p < 0.05) then the null hypothesis of homoskedasticity is rejected and heteroskedasticity assumed.
Before deciding upon an estimation method, one may conduct the Breusch–Pagan test to examine the presence of heteroskedasticity.
parameter restrictions: The following Lagrange multiplier (LM) yields the test statistic for the Breusch–Pagan test:[citation needed] This test can be implemented via the following three-step procedure: where the z terms will typically but not necessarily be the same as the original covariates x. where TSS is the sum of squared deviations of the
from their mean of 1, and RSS is the sum of squared residuals from the auxiliary regression.
A variant of this test, robust in the case of a non-Gaussian error term, was proposed by Roger Koenker.
As Koenker notes (1981, page 111), while the revised statistic has correct asymptotic size its power "may be quite poor except under idealized Gaussian conditions."
[8] In Stata, one specifies the full regression, and then enters the command estat hettest followed by all independent variables.
[9][10] In SAS, Breusch–Pagan can be obtained using the Proc Model option.
[11] In gretl, the command modtest --breusch-pagan can be applied following an OLS regression.