The 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 $E(u_i^2)=\sigma_i^2$, 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 $\hat{u}_1, \hat{u}_2+\cdots$
- Estimate the variance of the residuals i.e. $\hat{\sigma}^2=\frac{\sum e_i^2}{(n-2)}$
- Run the regression $\frac{e_i^2}{\hat{\sigma^2}}=\beta_1+\beta_2 Z_i + u_i$ and compute the explained sum of squares (ESS) from this regression
- Test the statistical significance of $\frac{ESS}{2}$ by $\chi^2$-test with 1 df at the appropriate level of significance ($\alpha$).
- Reject the hypothesis of homoscedasticity in favour of heteroscedasticity if $\frac{ESS}{2} > \chi^2_{(1)}$ at the appropriate level of $\alpha$.
Note that the
- The Breusch Pagan test is valid only if $u_i$’s are normally distributed.
- For k independent variables, $\frac{ESS}{2}$ has ($\chi^2$) Chi-square distribution with k degree of freedom.
- If the $u_i$’s (error term) are not normally distributed, the White test is used.
If heteroscedasticity is detected, remedies may include using robust standard errors, transforming the data, or employing weighted least squares estimation to adjust for heteroscedasticity.
The Breusch Pagan test is considered a useful tool for detecting the presence of heteroscedasticity in the regression models. The Breusch Pagan Test helps to ensure the validity of statistical inference and estimation.
A sample of Stata output related to the Breusch-Pagan Test for the detection of heteroscedasticity.
By analyzing the p-value of the chi-squared test statistic from the second regression, one can decide whether to reject the null hypothesis of homoscedasticity. If the p-value is lower than the chosen level of significance (say, 0.05), one has the evidence of heteroscedasticity.
The following are important points that need to be considered while using Breusch Pagan test of Heteroscedasticity.
- The Breusch-Pagan test can be sensitive to the normality of the error terms. Therefore, It is advisable to check if the residuals are normally distributed before running the Breusch-Pagan test.
- There are other tests for heteroscedasticity, but the Breusch-Pagan test is a widely used and relatively straightforward option.
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