circumstances – virtually a certainty in all practical applications – guarantees that these estimates of I.Op. 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 (5) is a reduced-form equation, where any effect of circumstances on incomes through their effects on efforts is already captured by the regression coefficients, and hence reflected in I(x̂BT). Equation (5) therefore captures the reducedform influence of circumstances on advantages, both directly and indirectly through efforts. By construction, the only omitted variables that matter for IEO are omitted circumstances, and those variables can only bias estimates of IEOL and IEOR downwards.<sup>23</sup> Recently, an upper-bound estimator of between-types inequality of opportunity has also been proposed, which requires panel data sets where the same individuals – or households – are observed at different points in time. If all circumstances are time-invariant (e.g., because they are determined at birth), then Peichl (2014) shows that the share of inequality associated with individual fixed effects in such data can provide an upper-bound estimate for between-types I.Op.. Intuitively, if all circumstances are time-invariant, their contribution to inequality is fully captured by the fixed effects. If, in addition, some component of individual responsibility (or effort) is also time-invariant, then that estimate is upwardly-biased – hence an upper bound. One limitation of this approach, of course, is that it rules out the existence of time-varying circumstances, such as later brute luck.

Table 7 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 and overall income inequality. 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-I index, 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, GNI per capita comes from the World Bank’s World Development Indicators database.

# [PLEASE PLACE TABLE 7 HERE]

Table 7 should be read in close conjunction with Table 8, which provides some basic information on each of the eight studies used to construct the inequality of opportunity estimates in Table 7. Table 8 describes which countries are studied in each paper; the specific data sets (including survey year); the precise income and circumstance variables used; whether the

> 23 Of course, this does not hold for the estimates of the individual coefficients β̂. First, these coefficients are reducedform, rather than structural, estimates. In addition, they are likely to be biased (upwards or downwards) even as reduced-form estimates, due to the omission of unobserved circumstances. The lower-bound result applies only to the overall measures of inequality of opportunity, IEO-L and IEO-R, whether parametrically estimated or not. See Ferreira and Gignoux (2011).

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