18 Once a counterfactual density function ‐ like (19) or (20) ‐ has been constructed, the overall change in the distribution of incomes can be decomposed into the part accounted for by these specific changes, and a residual term, as follows: The decomposition in (22), which has been termed a generalized Oaxaca‐Blinder decomposition, is obviously analogous to the decomposition of GICs in equation (13). The latter can be uniquely obtained from (22) by integrating each density function to obtain cumulative distribution functions, inverting those to obtain the quantile functions, and then dividing the quantile function at t=0.

The basic idea of the generalized Oaxaca‐Blinder decompositions is to “break up” the complex and multilayered processes behind distributional change into individual building blocks – e.g. changes in returns, changes in personal characteristics, changes in labor market institutions – so that a sense of their relative importance can be gauged. The counterfactuals can be constructed in different ways: relying on specific functional forms and on importing parameters estimated for one year into another, as in JMP; using non‐parametric reweighting methods, as in DFL: or indeed using a combination of the two. The same basic idea has been extended in various directions, including the analysis of changes in the distribution of household incomes (as opposed to the simpler wage distributions studied by JMP and DFL), and the comparison of distributions across countries, rather than time periods.<sup>19</sup> Although mechanically one could apply a decomposition such as (13) or (22) to any distribution, whether of wages or of household incomes, the choice of the conditional distribution linking _y_ to the covariates _X_ is likely to differ substantially depending on the nature of the distribution under study. For a distribution of wages or earnings, the Mincerian equation, or some non‐parametric equivalent, clearly embodies the relevant features of the conditional distribution. If one were after causal parameters, one might want to correct for sample selection bias into employment, but if the objective is to provide an estimate of the conditional distribution of wages for workers actually employed, even the simple OLS estimate (in the parametric case) might suffice. When _y_ denotes household incomes, however, there are other linkages between observed characteristics (X) and final incomes (y). Individual characteristics affect the very composition of households – both through choice of partner and through fertility decisions. They affect occupational decisions within the household – both who works and who doesn’t, and the choice of sector and formality status. And they clearly still affect earnings in the usual way, depending on returns in different sectors and occupations.

Bourguignon, Fournier and Gurgand (2001) and Bourguignon, Ferreira and Lustig (BFL, 2005) combine the parametric and semi‐parametric approaches described above to the more complex problem of decomposing changes in household income distributions. They note that the vector of household covariates ( _X_ ) includes endogenous variables such as family size and individual occupation, which are

> 19 In this paper, we focus on the extension to household income distributions. But see Donald, Green and Paarsch (2000) for a comparison of wage distributions between Canada and the US, using a hazard‐function‐based estimator of cumulative distributions; and Bourguignon, Ferreira and Leite (2008) for a comparison of household income differences between Brazil and the US.