science, math and reading. For circumstances, we use gender, whether the child has attended preschool or not, presence of parents at home, education level of the most educated parent/guardian, immigration status, quintile in the distribution of economic, social and cultural status, and city size, as these are all characteristics over which 15-year olds have no control.

Gender is a dummy that takes the value of 1 if the student is a boy. Preschool attendance is a dummy that takes the value of 1 if the student attended at least one year of preschool. Presence of parents is coded into two categories: i) the student lives with any of his/her parents, and ii) the student does not live with any of his/her parents for any reason whatsoever. Maximum level of education of the parents or guardians (in case that he/she does not live with his/her parents) is coded into four categories: a) primary education or less, b) secondary education, c) technical or technician, d) tertiary education (undergrad or postgrad). We compute the quintiles of the distribution of socioeconomic status on the basis of the country level distribution of social, economic and cultural status.<sup>5</sup> City size has the following categories: village (less than 3,000 habitants), small town (3,000 to 15,000 habitants), town (15,000 to 100,000 habitants), city (100,000 to 1 million habitants) and large city (more than 1 million habitants).

_2.1. Human Opportunity Index_ In a discrete population of size , let denote the probability of student of scoring at or above = level 2 proficiency. The share of students having access to the opportunity would be given by ii 1 nn pp ii . In practice, probabilities can be estimated econometrically from binary data on access using CC̅ a discrete choice model. Let this population be partitioned into ii ii types, so that for any type nn ∑pp defined by a set of circumstances: 1 2 ) , 1 2 and = 1 MM kk ∑ . Denoting the share of type in the entire sample by MM , the Human Opportunity Index mm kk XX= (XX , XX , ... , XX kk∈XX × XX × ... × XX CC can be defined as: nnkk ii∈kk ppii kk wwkk H , 1 where HH= CC̅(1 −DD)∑ |. MM kk=1 kk kk is a version of the dissimilarity index commonly used in sociology to measure segregation. In this DD= 2CC̅ ww |CC̅ −CC case might be seen as a measure of inequality of opportunity, more precisely as the share of total DD opportunities that are misallocated in favor of (or against) types that have coverage rates higher (or DD lower) than . Note also that for any , and , , which implies CC̅ AA, BB∈XX AA∩BB= ∅ DD( AA, BB) ≥DD( AA)

> 5 The index of economic, social and cultural status was constructed by the OECD on a basket of 13 household items: a dishwasher; a DVD player; number of cellular phones, televisions, computers, cars, rooms with a bath or shower; a room of their own (student); a computer that can be used for schoolwork; educational software; Internet; a desk; a quiet place to study; books to help with school work, technical reference books, dictionary, classic literature, books of poetry, works of art, and three country specific items. The index was computed using weighted likelihood estimation, and it has a mean of zero and a standard deviation of one for OECD countries as a whole.

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