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**

- 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 appropriate level of α.

**Note that the**

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

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