development, as measured by GDP per capita.<sup>24</sup> Figure 4 plots the relationship between IOp and mean achievement in mathematics. The regression line and a 95% confidence interval are shown on the graphs. The regression coefficient is statistically insignificantly different from zero at the 10% level. Figure 5 plots IOp in mathematics against GDP per capita, again showing the regression line and a 95% confidence interval. No statistically significant relationship is found. In order to test whether outliers such as Azerbaijan or Macao-China drive the statistical relationship, the procedure proposed by Besley, Kuh and Welsch (1980) is implemented to identify outliers and the test of a linear relationship is performed again after the exclusion of the corresponding observations. In this case, the negative regression coefficient is significant at 10% for mathematics, but remains insignificant for reading and science (not shown in figure).

The exact decomposition of inequality of opportunity into partial shares by individual circumstance, described in equation (18), is presented in Table 5 for mathematics scores. The shares of the ten circumstances add up to the total IOp given in the first column. As may be seen from inspection of equation (18), these partial shares are functions of individual regression coefficients from (12). As noted earlier, these individual coefficient estimates are likely to be biased, have not been presented here, and are not the focus of the paper. These partial shares reflect them, and should _not_ be interpreted causally in any way. They are useful only as a description of the variables underpinning the overall (lower-bound) measure of inequality of opportunity.

With that caveat in mind, Table 5 suggests that family educational and cultural resources seem to be associated with the largest share of inequality of learning achievement. Mother’s and father’s education combined account for a mean of 3.7 and a maximum of 9.2 (in Hungary) percentage points of the overall shares of explained inequality in the set of 57 countries, which take the mean of 24.7. The number of books at home accounts for a mean of 7.2 and a maximum of 14.4 percentage points (in Austria). Add parental education, language at home, numbers of books, and cultural possessions, and this set of “educational and cultural variables” add up to a mean of 15.0 points. Family economic resources also appear as an important source of learning inequalities. Father’s occupation and the “durable assets” indicator account for means of 3.6 and 3.8, respectively. With immigration status, the set of “economic variables” explains a mean of 7.8 points. Finally, the type of area where schools are located accounts for a mean of 1.6 and a maximum of 10.7 (in Kyrgyzstan) points of the overall shares, whereas the student’s gender accounts for a rather limited mean of 0.6 and a maximum of 2.1 (in Chile) points of the overall shares. There are also interesting regional variations in these partial shares of learning inequality. For instance, the partial share associated with educational and cultural resources has a higher mean in Western and Eastern European countries than in other regions, whereas the share associated with economic resources has a higher mean in Latin America.

> 24 GDP per capita is measured at purchasing power parity exchange rates, in 2006 US prices; the data are from the World Development Indicators (WDI) database.

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