obtained by students similar to them (in terms of the variables in _X_ ). If they are, in fact, likely to perform somewhat less well because of unobserved differences, the procedure overstates their true performance.

By its very nature, of course, selection on unobservables is harder to account for. The ancillary household surveys used to construct the reweighting function do not contain information on test scores. To provide another sensitivity test for the possible magnitude of sample selection bias driven by unobservables, we consider the (rather extreme) assumption that all those students who are counterfactually “re-introduced” into the PISA sample by the above procedure – a proportion given by , for each _X_ – do no better than those who are actually in the sample. In practice, we ascribe to them the lowest observed score for their cell in the partition. As an illustration of the effects of these two re-weighting procedures on the distribution, Figure 3 shows the histograms and kernel density estimates of the distribution of mathematics test scores in Turkey, under each alternative sample selection correction scenario: no correction, correction for selection on observables, and correction for selection on observables and unobservables, under the assumption of no common support.

In order to provide a sense of how sensitive our estimates of educational inequality (reported in Table 1) might be to sample selection, Table 3 reports the results of both of the above scenarios for the four countries with the lowest PISA coverage ratios in Table 1.<sup>15</sup> To economize on space, Table 3 reports the effects of these ‘selection correction’ procedures both on the standard deviation of test scores and on our measure of inequality of educational opportunity, which is introduced in the next section. The first three columns report these measures (and standard errors) for the uncorrected, original PISA sample, for reading, math and science respectively. The next three report estimates for the correction that assumes selection on observables only (equation 8), and the final three for the correction that assumes selection on unobservables (with no common support).

The results in Table 3 provide a mixed message. Somewhat surprisingly, both inequality of achievement (measured by the standard deviation) and inequality of opportunity seem to be quite robust to selection on observables, despite very low coverage rates (of approximately 50% in these four countries). While this is encouraging, the same cannot be said for the estimates for selection on unobservables. Under these (admittedly extreme) assumptions, inequality in achievement increases by between 44% in Turkey and 92% in Mexico. Inequality of educational opportunity also rises in all countries, except Mexico.

It is possible to interpret these results as comforting, if one chooses to focus on the relative robustness of the measures to selection on observables, even in countries where PISA coverage is lowest. It seems most likely that, if these observed variables account for most of the sample selection process, the estimates of educational inequality in Table 1 are robust for all countries. The fact that those estimates are sensitive to selection on unobservables can be minimized by

> 15 Coverage in these four countries – Brazil, Indonesia, Mexico and Turkey – was described in some detail in Section 2 and Table 2 above.

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