Breusch-Pagan test

The Breusch-Pagan test and its special case, the White test , are statistical tests for testing heteroscedasticity . They are used in particular to check the assumption of homoscedasticity in regression analysis .

Breusch-Pagan test

If one considers the (multiple) linear model with normally distributed errors . Then the error variance is called ${\displaystyle Y_{t}=\beta _{0}+\sum _{j=1}^{p}\beta _{j}x_{tj}+U_{t}}$ ${\displaystyle U_{t}\sim {\mathcal {N}}(0;\sigma _{t}^{2})}$

${\displaystyle \sigma _{t}^{2}=h\left(\alpha _{0}+\sum _{k=1}^{q}\alpha _{k}z_{kt}\right)}$

modeled. If homoscedasticity ( ) is present, then the coefficients must be zero except for the constant term. ${\displaystyle \sigma _{t}^{2}=\sigma _{U}^{2}}$ ${\displaystyle \alpha _{k}}$

The hypotheses thus result as

${\displaystyle H_{0}:\alpha _{k}=0}$for everyone vs. ${\displaystyle k=1,\ldots ,q}$

${\displaystyle H_{1}:\alpha _{k}\neq 0}$for at least one . ${\displaystyle k}$

The test statistics are obtained as a score or Lagrange multiplier test using the maximum likelihood method and are thus distributed. ${\displaystyle \chi _{q}^{2}}$

In practice, the variables must either be specified or an estimate of the shape is considered. ${\displaystyle Z_{k}}$ ${\displaystyle {\hat {u}}_{t}^{2}=\epsilon _{0}+\epsilon _{1}{\hat {y}}_{t}}$

The Breusch-Pagan test reacts sensitively to a violation of the normal distribution assumption of the residuals .

Simple linear regression graphing the residuals and squared residuals of the Boston Housing data . The red line in the right diagram shows that the squared residuals depend non-linearly on the explanatory variable, i.e. that there is heteroscedasticity.

White test

The White test is a special case of the Breusch-Pagan test, since here the error variances as

${\displaystyle \sigma _{t}^{2}=\alpha _{0}+\sum _{k=1}^{p}\alpha _{k}x_{tk}+\sum _{k<l=1}^{p}\beta _{kl}x_{tk}x_{tl}}$

be modeled. The hypotheses are

${\displaystyle H_{0}:}$all coefficients except are zero vs. ${\displaystyle \alpha _{0}}$

${\displaystyle H_{1}:}$at least one coefficient other than is not equal to zero. ${\displaystyle \alpha _{0}}$

To carry out the White test, the number of observations should be significantly larger than the number of coefficients and . Otherwise you have to leave out the interaction terms in the model. Also, dummy variables are due to multicollinearity not included in the interaction terms. ${\displaystyle \alpha _{k}}$ ${\displaystyle \beta _{kl}}$ ${\displaystyle X_{k}X_{l}}$

The White test is less sensitive to a violation of the assumption of normal distribution of the residuals than the Breusch-Pagan test.

Individual evidence

1. ^ TS Breusch, AR Pagan: A simple test for heteroscedasticity and random coefficient variation. In: Journal of the Econometric Society, Econometrica. 1979, pp. 1287-1294, JSTOR 1911963 .
2. ^ H. White: A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. In: Journal of the Econometric Society, Econometrica. 1980, pp. 817-838, JSTOR 1912934 .