statsmodels.stats.diagnostic.het_breuschpagan(resid, exog_het, robust=True)[source]

Breusch-Pagan Lagrange Multiplier test for heteroscedasticity

The tests the hypothesis that the residual variance does not depend on the variables in x in the form

Homoscedasticity implies that \(\alpha=0\).


For the Breusch-Pagan test, this should be the residual of a regression. If an array is given in exog, then the residuals are calculated by the an OLS regression or resid on exog. In this case resid should contain the dependent variable. Exog can be the same as x.


This contains variables suspected of being related to heteroscedasticity in resid.

robustbool, default True

Flag indicating whether to use the Koenker version of the test (default) which assumes independent and identically distributed error terms, or the original Breusch-Pagan version which assumes residuals are normally distributed.


lagrange multiplier statistic


p-value of lagrange multiplier test


f-statistic of the hypothesis that the error variance does not depend on x


p-value for the f-statistic


Assumes x contains constant (for counting dof and calculation of R^2). In the general description of LM test, Greene mentions that this test exaggerates the significance of results in small or moderately large samples. In this case the F-statistic is preferable.


Chisquare test statistic is exactly (<1e-13) the same result as bptest in R-stats with defaults (studentize=True).


This is calculated using the generic formula for LM test using $R^2$ (Greene, section 17.6) and not with the explicit formula (Greene, section 11.4.3), unless robust is set to False. The degrees of freedom for the p-value assume x is full rank.



Greene, W. H. Econometric Analysis. New Jersey. Prentice Hall; 5th edition. (2002).


Breusch, T. S.; Pagan, A. R. (1979). “A Simple Test for Heteroskedasticity and Random Coefficient Variation”. Econometrica. 47 (5): 1287–1294.


Koenker, R. (1981). “A note on studentizing a test for heteroskedasticity”. Journal of Econometrics 17 (1): 107–112.