## Breusch Pagan Test for Heteroscedasticity

**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.

**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**