10.4 Estimation under heteroscedasticity
When the form of the heteroscedasticity is known up to a constant, \(\sigma _i^2 = \sigma ^2\omega _i\) with known \(\omega _i\), GLS is weighted least squares: divide each observation, including the constant, by \(\sqrt {\omega _i}\) and apply least squares. Two cases are common.
Example 10.4.1. Grouped data. Suppose only group averages are available: for group \(j\), the means \(\bar Y_j\) and \(\bar x_j\) of \(n_j\) individual observations each satisfying \(Y = x'\beta + u\) with \(\Var (u) = \sigma ^2\). Then \(\bar Y_j = \bar x_j'\beta + \bar u_j\) with \(\Var (\bar u_j) = \sigma ^2/n_j\). The group averages are heteroscedastic, with \(\omega _j = 1/n_j\), and GLS weights each group by \(n_j\): large groups, whose averages are precise, count for more. Least squares on the unweighted averages treats a group of \(5\) like a group of \(5\,000\).
Example 10.4.2. Aggregates and per-capita data. If a state’s total income is \(\text {TI}_i = \alpha + \beta \,\text {Pop}_i + u_i\) with \(\Var (u_i) = \sigma ^2\text {Pop}_i^2\), dividing through by \(\text {Pop}_i\) gives \[ \frac {\text {TI}_i}{\text {Pop}_i} = \beta + \alpha \frac {1}{\text {Pop}_i} + \frac {u_i}{\text {Pop}_i}, \qquad \Var \Bigl (\frac {u_i}{\text {Pop}_i}\Bigr ) = \sigma ^2 , \] a homoscedastic regression of income per head on the reciprocal of population. The slope of the original model becomes the intercept of the transformed one. Expressing an aggregate relation in per-capita terms is often exactly the right weighting.
When the \(\omega _i\) are unknown, they are modelled. A flexible and always-positive choice is the multiplicative model \(\sigma _i^2 = \exp (\gamma _0 + z_i'\gamma )\). Regressing \(\log e_i^2\) on \(z_i\) estimates \(\gamma \), apart from the intercept, consistently; the fitted values give \(\hat \omega _i\), and weighted least squares with weights \(1/\hat \omega _i\) is feasible GLS. Under correct specification of the variance function, feasible GLS is asymptotically as efficient as GLS. Under misspecification it remains consistent, but its conventional standard errors are wrong. A sensible compromise, increasingly common, is to use feasible GLS for efficiency and report robust standard errors for it.
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