from circumstance groups (or “types”, as defined in the inequality of opportunity literature) with coverage rate higher than p̅ to groups with coverage rate lower than p̅ to restore equal opportunity across all groups.

HOI has a number of desirable properties, including sensitivity to scale and inequality and its reward to Pareto improvements (see Barros et al. 2010). Of particular relevance is its sensitivity to inequality. HOI improves when inequality between types decreases with a fixed number of opportunities in a society, or when the number of opportunities increases and inequality among types stays constant.<sup>5</sup> The D-Index, on the other hand, improves if inequality among types, for a given number of total opportunities, declines. Importantly, while D is an indicator of inequality of opportunity, it is _not independent_ of the level of coverage. Clearly, as the coverage of an opportunity increases, the share of total opportunities allocated unequally among types is likely to fall, which is to say D is likely to fall. This is an inevitable consequence of the opportunity being a binary (zero-one) indicator, which implies that the coverage of an opportunity has an upper limit (100 percent). It also implies that the estimated effect of D on level of development will also reflect, in part, the effect of the level (coverage) of the opportunity, albeit to a much smaller extent than for the estimated effect of HOI.

Computing D and HOI when the number of circumstances is relatively large, as is the case here, requires an econometric exercise to obtain a prediction of HOI from observed access to the opportunity and circumstances among children (see Barros et al., 2010). This involves running a logistic regression model to estimate the relationship between access to a particular opportunity and the circumstances of the child, on the full sample of children for whom the HOI measure will be constructed. The estimated coefficients of the regression are used to obtain for each child his/her predicted probability of access to the opportunity, which is then used to estimate the coverage rate p̅, D and HOI (see Appendix 2).

We compute the HOI and the D-Index using data from the Program for International Student Assessment (PISA). PISA surveys 15-year-olds and assesses the extent to which students near the end of compulsory education have acquired the knowledge and skills essential for full participation in society. This is done through the use of standardized test scores. The scores range from 0 to 700, and six cut-off values or levels have been established by PISA. An “opportunity” is defined as achieving proficiency of Level 2 for the topics covered in all three tests (Mathematics, Reading and Science), which indicates a basic level of understanding of the topics covered.<sup>6</sup> The circumstances used are a measure of household wealth, calculated using principal component methodology based on asset ownership; gender of the child; parental education; school location;<sup>7</sup> and occupation of the head of the child’s household. For the child, these are all attributes s/he has no control over and are likely to have been determined at birth.

One of the drawbacks of using PISA is that the standardized tests are not carried out in all countries. To expand the sample of countries to a larger set, we also make use of Montenegro<sup>8</sup> et al. (2011)

> 5 The D-index and HOI are sensitive to the set of circumstances chosen for analysis. But this is mitigated by an additional property: the D-Index (HOI) will not be lower (higher) if more circumstances are added to the existing set of circumstances in the analysis.  This implies that the computed D-index serves as a lower bound to the “actual” inequality if all circumstances of interest could be included in the analysis.

> 6 The results are robust to the use of proficiency levels 3 or 4. See PISA 2009 for a description of what proficiency level 2 implies for each test.

> 7 The location of the school is classified as: a) Village, b) Small Town, c) Town, d) City or e) Large City.

> 8 Not all surveys had the necessary variables to compute the indicator. As a result, we end up with 80 surveys in which we could compute it.

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