that serves as a lower bound to the “real” level of inequality of opportunity that could be measured if _all_ relevant circumstances were observable from the data and included in the analysis.<sup>6</sup> DD The HOI can be considered as a welfare function à la Sen, where mean outcomes are adjusted by one minus a measure of inequality (Brunori et al., 2013). However, given that is a direct measure of inequality, we focus on to assess inequality of opportunity in educational achievement. DD _2.1.1. Decomposition of the D-index_ DD Given that is sensitive to the set of circumstances chosen for the analysis, we identify the impact of each circumstance on it using Shapley Decompositions. We do so by considering all the changes DD that occur when we add a circumstance to all possible subsets of pre-existing variables, and then taking the average of all these possible changes. Thus, if we have two sets of circumstances , the impact (on _D_ ) of adding the set of circumstances is given by: AA, BB∈ <u>| |!( | |−1)!</u> XX = ∑ AA , ss nn− ss where is the set of all circumstances, which includes DDAA SS⊆NN\{AA} nn! [DD(SS∪{AA}) −DD(SS)] circumstances in total; is a subset of (containing circumstances) that does not contain ; is the dissimilarity index estimated with the set of circumstances NN ; and is the dissimilarity index calculated with set of nn SS NN circumstances ss and . Hence, we can define the contribution of the set of circumstances AA DD(SS) to the SS DD(SS∪{AA}) inequality index as = ~~,~~ where ∑ = 1.<sup>7</sup> SS AA DDAA AA AA ii∈NN ii MM DD(NN) MM **3. Results** The -index or _inequality of opportunity index_ and the results of the Shapley Decompositions by country can be found in the Appendix (Table A2). In this section we analyze these results. DD First, we ask ourselves whether the level of inequality of opportunity is related to development. In a recent study, Brunori et al. (2013) find that inequality of opportunity is highly correlated to Gross National Income per-capita. In the same spirit, we compare the -index to a popular measure of the level of development: the Human Development Index (HDI) published by United Nations DD Development Programme (UNDP). The HDI is a national average of human development achievements in the three basic dimensions of health, education and standard of living, measured by the geometric mean of indicators of life expectancy, Gross National Income, and the mean and expected years of schooling.<sup>8</sup>

We expect the HDI to be negatively correlated to our inequality of opportunity (or D) index for educational achievement since countries with high HDI have highly educated populations and

> 6 For further methodological details see Barros et al. (2009, 2010).

> 7 For further methodological details see Hoyos and Narayan (2011).

> 8 The HDI is the geometric mean of normalized (using minimum and maximum value for each) indices for each of the three dimensions: (i) the health dimension measured by life expectancy at birth; (ii) the education dimension measured by (the arithmetic mean of) mean of years of schooling for adults aged 25 years and expected years of schooling for children of school entering age; and (iii) the standard of living dimension measured by (the logarithm of) gross national income per capita. (UNDP, 2014)

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