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
- Estimate the model by OLS and obtain the residuals $\hat{\mu}_1, \hat{\mu}_2+\cdots$
- Estimate the variance of the residuals i.e. $\hat{\sigma}^2=\frac{\sum e_i^2}{(n-2)}$
- 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
- Test the statistical significance of ESS/2 by $\chi^2$-test with 1 df at appropriate level of significance (α).
- 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
