Breusch-Pagan test

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In statistics, the Breusch-Pagan test is used to test for heteroscedasticity in a linear regression model. It tests whether the estimated variance of the residuals from a regression are dependent on the values of the independent variables.

Suppose that we estimate the equation

$y = x_0 + x_1\beta + u.\,$

We can then estimate $\hat{u}$, the residual. Ordinary least squares constrains these so that their mean is 0, so we can calculate the variance as the average squared values. Even simpler is to simply regress the squared residuals on the independent variables, which is the Breusch-Pagan test:

$\hat{u}^2 = x_0 + x_1\beta + v.\,$

If an F-test confirms that the independent variables are jointly significant then we can reject the hypothesis of no heteroscedasticity.

The Breusch-Pagan test tests for conditional heteroscedasticity. It is a chi-squared test: the test statistic is nχ2 with k degrees of freedom. If the Breush-Pagan test shows that there is conditional heteroscedasticity, it can be corrected by using the Hansen method, using robust standard errors, or re-thinking the regression equation.