Бройш-Паган тесті: гетероскедастиктік талдау әдісі

Statistical test

In statistics, the Breusch–Pagan test, developed in 1979 by Trevor Breusch and Adrian Pagan, is used to test for heteroskedasticity in a linear regression model. It was independently suggested with some extension by R. Dennis Cook and Sanford Weisberg in 1983 (Cook–Weisberg test). 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. In that case, heteroskedasticity is present. Suppose that we estimate the regression model

and obtain from this fitted model a set of values for , the residuals. 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. It is a chi squared test: the test statistic is distributed nχ2 with k degrees of freedom. 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. If the Breusch–Pagan test shows that there is conditional heteroskedasticity, one could either use weighted least squares (if the source of heteroskedasticity is known) or use heteroscedasticity consistent standard errors.

A variant of this test, robust in the case of a non Gaussian error term, was proposed by Roger Koenker. In this variant, the dependent variable in the auxiliary regression is just the squared residual from the Step 1 regression, , and the test statistic is from the auxiliary regression. 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."

In R, this test is performed by the function ncvTest available in the car package, the function bptest available in the lmtest package, the function plmtest available in the plm package, or the function breusch pagan available in the skedastic package. In Stata, one specifies the full regression, and then enters the command estat hettest followed by all independent variables. In SAS, Breusch–Pagan can be obtained using the Proc Model option. In Python, there is a method het breuschpagan in statsmodels. stats. diagnostic (the statsmodels package) for Breusch–Pagan test. In gretl, the command modtest breusch pagan can be applied following an OLS regression.