and obtaining coefficient estimates β<sup>�</sup> , these authors use predicted incomes as a parametric approximation to the smoothed distribution: Parametric estimates are also presented either as levels (IEO-L) or ratios (IEO-R), analogously. This approach follows Ferreira and Gignoux (2011), which in turn draws on Bourguignon et al. (2007). Empirically, parametric estimates of inequality of opportunity tend to be a little lower than their nonparametric counterparts but, at least in the case of Latin America, the differences are not great: proportional differences between the two average 6.6% in Ferreira and Gignoux (2011).

The fact that the parametric estimates are conservative – i.e. generally lower than the nonparametric ones – is consistent with another important property of these estimates of IEO-R and IEO-L. They are, in each and every case, _lower-bound_ estimates of inequality of opportunity. A formal proof of the lower-bound result is contained in Ferreira and Gignoux (2011), but the intuition is straight forward. The set of circumstances which is observed empirically - and used for partitioning the population into types - is a strict subset of the theoretical vector of all circumstance variables. The existence of unobserved circumstances – virtually a certainty in all practical applications – guarantees that these estimates of I.Op. – whether parametric or non-parametric – could only be higher if more circumstance variables were observed.

As discussed in Ferreira and Gignoux (2011), the existence of effort variables, observed or unobserved, is entirely immaterial to this result, since (6) is written as a reduced-form equation, where any effect of circumstances on incomes _through_ their effects on effort (such as years of schooling or hours worked) is captured by the regression coefficients, and hence influence the smoothed distribution. In a setting where some variables are treated as observed efforts (as in Bourguignon et al. 2007), Equations (6) and (7) capture the _reduced-form_ influence of circumstances on advantages, both directly and indirectly through efforts. By construction, therefore, the only omitted variables that matter for IEO are omitted circumstances.<sup>12</sup> Table 1 presents the estimates of IEO-L and IEO-R for each of the forty-one countries studied by the eight aforementioned papers. The table also lists their gross national income (GNI) per capita; overall inequality and, when available, a measure of intergenerational earnings elasticity (IGE) reported in the literature; a measure of the intergenerational correlation of education from Hertz at al. (2007); and the Human Opportunity Index. Overall inequality is measured by whatever index was used in the construction of the IEO indices for each country. Except where indicated, this measure was the mean logarithmic deviation, also known as the Theil-L index, and a member of the generalized entropy class of inequality measures. Whereas overall inequality, IEO-L and IEO-R come from the eight studies mentioned above, the other variables come from other sources. GNI per capita comes from the World Bank’s World Development Indicators database. Our measure of intergenerational correlation of

> 12 Of course, this does not hold for the estimates of the individual coefficients β� . First, these coefficients are reduced-form, rather than structural, estimates. In addition, they are likely to be biased (upwards or downwards) even as reduced-form estimates, by the omission of unobserved circumstances. The lower-bound result applies only to the overall measures of inequality of opportunity, IEO-L and IEO-R.

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