Breusch–Pagan test

Under the classical assumptions, ordinary least squares is the best linear unbiased estimator (BLUE), i.e., it is unbiased and efficient. It remains unbiased under heteroskedasticity, but efficiency is lost. Before deciding upon an estimation method, one may conduct the Breusch–Pagan test to examine the presence of heteroskedasticity. The Breusch–Pagan test is based on models of the type ${\displaystyle \sigma _{i}^{2}=h(z_{i}'\gamma )}$ for the variances of the observations where ${\displaystyle z_{i}=(1,z_{2i},\ldots ,z_{pi})}$ explain the difference in the variances. The null hypothesis is equivalent to the ${\displaystyle (p-1)\,}$ parameter restrictions:

${\displaystyle \gamma _{2}=\cdots =\gamma _{p}=0.}$

The following Lagrange multiplier (LM) yields the test statistic for the Breusch–Pagan test:

${\displaystyle {\text{LM}}=\left({\frac {\partial \ell }{\partial \theta }}\right)^{\mathsf {T}}\left(-E\left[{\frac {\partial ^{2}\ell }{\partial \theta \,\partial \theta '}}\right]\right)^{-1}\left({\frac {\partial \ell }{\partial \theta }}\right).}$

This test can be implemented via the following three-step procedure:

• Step 1: Apply OLS in the model

${\displaystyle y_{i}=X_{i}\beta +\varepsilon _{i},\quad i=1,\dots {},n}$

• Step 2: Compute the regression residuals, ${\displaystyle {\hat {\varepsilon }}_{i}}$, square them, and divide by the Maximum Likelihood estimate of the error variance from the Step 1 regression, to obtain what Breusch and Pagan call ${\displaystyle g_{i}}$:

${\displaystyle g_{i}={\hat {\varepsilon }}_{i}^{2}/{\hat {\sigma }}^{2},\quad {\hat {\sigma }}^{2}=\sum {{\hat {\varepsilon }}_{i}^{2}}/n}$

• Step 2: Estimate the auxiliary regression

${\displaystyle g_{i}=\gamma _{1}+\gamma _{2}z_{2i}+\cdots +\gamma _{p}z_{pi}+\eta _{i}.}$

where the z terms will typically but not necessarily be the same as the original covariates x.

• Step 3: The LM test statistic is then half of the explained sum of squares from the auxiliary regression in Step 2:

${\displaystyle {\text{LM}}={\frac {1}{2}}\left({\text{TSS}}-{\text{SSR}}\right).}$

where TSS is the sum of squared deviations of the ${\displaystyle g_{i}}$ from their mean of 1, and SSR is the sum of squared residuals from the auxiliary regression. The test statistic is asymptotically distributed as ${\displaystyle \chi _{p-1}^{2}}$ under the null hypothesis of homoskedasticity and normally distributed ${\displaystyle \varepsilon _{i}}$, as proved by Breusch and Pagan in their 1979 paper.

A variant of this test, robust in the case of a non-Gaussian error term, was proposed by Roger Koenker.[3] In this variant, the dependent variable in the auxiliary regression is just the squared residual from the Step 1 regression, ${\displaystyle {\hat {\varepsilon }}_{i}^{2}}$, and the test statistic is ${\displaystyle nR^{2}}$ from the auxiliary regression. As Koenker notes (1981, page 111), while the revised statistic has correct asymptotic size its power "may be quite poor except under idealized Gaussian conditions."

In R, this test is performed by the function ncvTest available in the car package,[4] the function bptest available in the lmtest package,[5][6] the function plmtest available in the plm package,[7] or the function breusch_pagan available in the skedastic package.[8]

In Stata, one specifies the full regression, and then enters the command estat hettest followed by all independent variables.[9][10]

In SAS, Breusch–Pagan can be obtained using the Proc Model option.

In Python, there is a method het_breuschpagan in statsmodels.stats.diagnostic (the statsmodels package) for Breusch–Pagan test.[11]

In gretl, the command modtest --breusch-pagan can be applied following an OLS regression.

• Glejser test
• Goldfeld–Quandt test
• Park test
• White test

• Gujarati, Damodar N.; Porter, Dawn C. (2009). Basic Econometrics (Fifth ed.). New York: McGraw-Hill Irwin. pp. 385–86. ISBN 978-0-07-337577-9.
• Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 292–298. ISBN 0-02-365070-2.
• Krämer, W.; Sonnberger, H. (1986). The Linear Regression Model under Test. Heidelberg: Physica. pp. 32–39. ISBN 9783642958762.
• Maddala, G. S.; Lahiri, Kajal (2009). Introduction to Econometrics (Fourth ed.). Chichester: Wiley. pp. 216–218. ISBN 978-0-470-01512-4.