10.3 Testing for heteroscedasticity
All the common tests examine whether the squared residuals are related to something they should not be related to under homoscedasticity.
Goldfeld–Quandt
If the variance is suspected to increase with one variable \(Z\), order the observations by \(Z\), omit \(c\) central observations, and fit the regression separately to the first and last \((n - c)/2\) observations. Under homoscedasticity and normality the ratio \[ \text {GQ} = \frac {\text {RSS}_2/(n_2 - k)}{\text {RSS}_1/(n_1 - k)} \sim F_{n_2-k,\ n_1-k} , \] exactly, because the two subsets have independent residual sums of squares, each \(\sigma ^2\chi ^2\) (Theorem 5.5.1). Omitting the middle observations sharpens the contrast between the two groups. The test is exact but needs a single ordering variable and gives no information about other forms of heteroscedasticity.
Breusch–Pagan
Suppose \(\sigma _i^2 = h(\alpha _1 + z_i'\alpha )\) for some smooth function \(h\) and a vector \(z_i\) of \(p\) variables, often some or all of the regressors. Homoscedasticity is \(\alpha = 0\). Breusch and Pagan derived the LM test of this hypothesis under normality. It is computed from the regression of \(g_i = e_i^2/\hat \sigma ^2\) (with \(\hat \sigma ^2 = e'e/n\)) on a constant and \(z_i\): \[ \text {LM} = \tfrac 12\,\text {ESS} \xrightarrow {d} \chi ^2_p . \] Remarkably, the statistic does not depend on the function \(h\). Koenker showed that the factor \(\frac 12\) relies on the normal kurtosis, since \(\Var (u^2) = 2\sigma ^4\) for normal \(u\), and that with fat-tailed disturbances the test rejects too often. His studentised version, \(nR^2\) from the regression of \(e_i^2\) on a constant and \(z_i\), is valid without normality and is the version most software now reports.
White’s test
White’s general test takes \(z_i\) to be all the regressors, their squares and their cross-products, and uses \(nR^2\) from the regression of \(e_i^2\) on them, which is asymptotically \(\chi ^2\) with degrees of freedom equal to the number of variables in \(z_i\). It tests whether the variance is related to the regressors in any way that would make the conventional covariance matrix inconsistent. With many regressors the number of terms grows quickly, and the power falls. A significant result can also signal a misspecified functional form rather than heteroscedasticity, because neglected curvature shows up in the squared residuals.
Theorem 10.3.1. In the auxiliary regression of \(e_i^2\) on a constant and \(z_i\), \(nR^2\) is the LM statistic for \(H_0\colon \alpha = 0\) in the model \(E(u_i^2 \mid z_i) = \alpha _1 + z_i'\alpha \), computed with a variance estimate for \(u_i^2\) that does not assume normality.
Sketch of proof. Treat \(e_i^2\) as the dependent variable in a linear regression on \((1, z_i')\). Under \(H_0\) the restricted model has only an intercept, whose residuals are \(e_i^2 - \overline {e^2}\). By Theorem 9.3.1 the LM statistic for the \(p\) slope restrictions in a linear model is \(n\) times the \(R^2\) of regressing these restricted residuals on all the regressors, which is the \(R^2\) of the auxiliary regression itself. Replacing \(u_i^2\) by \(e_i^2\) changes nothing asymptotically, because \(b\) is consistent. The variance of \(u_i^2\) is estimated from the data rather than set to \(2\sigma ^4\), which is why no normality is needed. □
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