round such that the final standardized score has an overall mean of zero and standard deviation of one.<sup>15</sup> Table 2 reports regional means and standard deviations of the test scores, calculated at each step. The first (column I) are the weighted means of the raw competency tests (before any standardization); column II reports the age-standardized versions; and column III reports the final measures. As can be seen, movement from the second to the third metric constitutes a simple monotone transformation.

The test scores reported in Table 2 refer to the final sample used in the subsequent analysis, which pools all survey rounds. This is a slightly reduced sample of the original Uwezo data. Specifically, observations have been dropped that can be perfectly predicted using either household or school fixed effects – i.e., all singletons were removed. The objective of restricting the data in this way is to mitigate upward bias of the variance contribution estimates. As per the methodological discussion of Section 2.2, the analytical focus is on the variance components of the test score; and there is no evidence to suggest these dropped observations are distributed in a systematic pattern over regions or districts.<sup>16</sup> The (sample) standard deviations of the test scores, which can be directly interpreted as normalized measures of educational inequality (Van de Werfhorst and Mijs, 2010; Hanushek and Wößmann, 2006), are reported in parentheses in the table. It merits noting that the rank position of each region according to its test score variance is largely preserved, regardless of the transformation applied.

To implement the variance decomposition, the household and school indexes must be defined. The former is trivial – unique indexes are ascribed to all sibs in the same household in each year.<sup>17</sup> The school effects are less straightforward. In the present data, detailed information about the particular school each child attends is not provided. Nonetheless, we can identify the grade of attendance and the location of the household. Consequently, for each enumeration area (containing approximately 20 surveyed households), we categorize children into one of three school-grade categories based on their highest grade of completed or current schooling – namely: those attending grades 1-2; grades 3-4; and grades 5 or higher. For children who have never attended school, we hypothesize that the quality of schools (teachers) available in their locale may have an effect on their ability in numeracy and literacy. This may work directly, through the choice not to attend school, as well as indirectly through both sibling and peer effects – e.g., what other children learn can spill-over to nonattenders. In order to allow for these effects in the data, we allocate never-attenders to the

> 15  Due to the absence of equating or anchor items in the Uwezo tests, we cannot distinguish between changes in test difficulty over time and changes in (average) learning outcomes. However, from the perspective of a variance decomposition, standardization by year ensures that the variance contributions retain a consistent meaning in relation to the overall distribution of outcomes in each age cohort and year.

> 16 Details available on request from the authors.

> 17 The Uwezo surveys are cross-sectional in nature and no explicit attempt is made to track the same children over time.

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