Basic Statistics and Data Analysis

Lecture notes, MCQS of Statistics

Breusch-Pagan Test for Heteroscedasticity

Breusch–Pagan test (named after Trevor Breusch and Adrian Pagan) is used to test for heteroscedasticity in a linear regression model.

Assume our regression model is $Y_i = \beta_1 + \beta_2 X_{2i} + \mu_i$ i.e we have simple linear regression model, and $E(\mu_i^2)=\sigma_i^2$, where $\sigma_i^2=f(\alpha_1 + \alpha_2 Z_{2i})$

That is $\sigma_i^2$ is some function of the non-stochastic variable Z‘s. f() allows for both the linear and non-linear forms of the model. The variable Z is the independent variable X or it could represent a group of independent variables other than X.

Step to Perform Breusch-Pagan test

  1. Estimate the model by OLS and obtain the residuals $\hat{\mu}_1, \hat{\mu}_2+\cdots$
  2. Estimate the variance of the residuals i.e. $\hat{\sigma}^2=\frac{\sum e_i^2}{(n-2)}$
  3. Run the regression $\frac{e_i^2}{\hat{\sigma^2}}=\beta_1+\beta_2 Z_i + \mu_i$ and compute explained sum of squares (ESS) from this regression
  4. Test the statistical significance of ESS/2 by $\chi^2$-test with 1 df at appropriate level of significance (α).
  5. Reject the hypothesis of homoscedasticity in favour of heteroscedasticity if $\frac{ESS}{2} > \chi^2_{(1)}$ at appropriate level of α.

Note that the

  • Breusch-Pagan test is valid only if μi‘s are normally distributed.
  • For k independent variables, ESS/2 have ($\chi^2$) Chi-square distribution with k degree of freedom.
  • If the μi‘s (error term) are not normally distributed, White test is used.


  • Breusch, T.S.; Pagan, A.R. (1979). “Simple test for heteroscedasticity and random coefficient variation”. Econometrica (The Econometric Society) 47 (5): 1287–1294.

The Author

Muhammad Imdadullah

Student and Instructor of Statistics and business mathematics. Currently Ph.D. Scholar (Statistics), Bahauddin Zakariya University Multan. Like Applied Statistics and Mathematics and Statistical Computing. Statistical and Mathematical software used are: SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis etc.

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