Breusch–Pagan test

In statistics, the Breusch–Pagan test (named after Trevor Breusch and Adrian Pagan) 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

$y = \beta_0 + \beta_1 x + u.\,$

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:

$\hat{u}^2 = \beta_0 + \beta_1 x + v.\,$

If an F-test confirms that the independent variables are jointly significant then we can reject the null hypothesis of homoscedasticity.

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 Breusch–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.

Application in economics

Under the classical assumptions, including homoscedasticity, Ordinary Least Squares is the Best Linear Unbiased Estimator (BLUE), i.e., it is unbiased and efficient. The efficiency is lost, however, in the presence of heteroscedastic disturbances. Before deciding upon an estimation method, one may conduct the Breusch–Pagan test to determine the presence of heteroscedasticity. The Breusch–Pagan test is based on models of the type $\sigma_i^2 = h(z_i'\gamma)$ for the variances of the observations where $z_i = (1, z_{2i}, \dots, z_{pi})$ explain the difference in the variances. The null hypothesis is equivalent to the $(p - 1)\,$ parameter restrictions:

$\gamma_2 = \dots = \gamma_p = 0.$

The following Lagrange Multiplier (LM) yields the result to the Breusch–Pagan test:

$LM=\left (\frac{\partial l}{\partial\theta} \right )'\left (-E\left [\frac{\partial^2 l}{\partial\theta \partial\theta'} \right ] \right )^{-1}\left(\frac{\partial l}{\partial\theta} \right ).$

This test is analogous to the following simple three-step procedure:

• Step 1: Apply OLS in the model

$y = X\beta+\varepsilon.$

and compute the regression residuals.

• Step 2: Perform the auxiliary regression

$e_i^2=\gamma_1+\gamma_2z_{2i}+\dots+\gamma_pz_{pi}+\eta_i.$

• Step 3: The test statistic is the result of the coefficient of determination of the auxiliary regression in Step 2 and sample size $n\,$ with:

$LM=nR^{2}\, .$

The test statistic is asymptotically distributed as $\chi^2 \left (p - 1 \right )$ under the null hypothesis homoscedasticity.

Software

In R (programming language), this test is performed by function bptest, available in package lmtest.

In Stata, one specifies the full regression, and then enters the command "estat hettest" followed by all independent variables.

See also

• White test
• R documentation about bptest

References

• Breusch, T.S.; Pagan, A.R. (1979). "Simple test for heteroscedasticity and random coefficient variation". Econometrica (The Econometric Society) 47 (5): 1287–1294. doi:10.2307/1911963. JSTOR 1911963. MR545960. .

• Heij, C.; P. de Boer (2004). Econometric Methods with Applications in Business and Economics. Oxford University Press. pp. 334–353. .