Breusch Pagan Test for Heteroscedasticity

Breusch Pagan test (named after Trevor Breusch and Adrian Pagan) is used to check for the presence of 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. The 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 the 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, the White test is used.

References:

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

See the Numerical Example of the Breusch-Pagan Test for the Detection of Heteroscedasticity

Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, Python, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

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