change in the indicator of interest, the estimated coefficients from a quantile regression on the 25<sup>th</sup> or 67<sup>th</sup> percentile can be used to generate predicted percentiles at the 25<sup>th</sup> or 67<sup>th</sup> percentile.

Quantile regressions have been used to generate student growth percentiles (Betebenner, 2009) and conditional growth charts (Wei and He, 2006). The work on student growth percentiles has generally conditioned only on past achievement levels (and not observable characteristics of the students or their teachers), as the effort of introducing the concept into educational management in such places as the states of Colorado and Massachusetts has been to separate the description of student progress (the SGP) from the attribution of responsibility for that progress<sup>7</sup> .

Quantile regressions have been used quite extensively in economics (see Koenker and Hallock, 2001 for a review). Three applications that are most closely related to controlling for heterogeneity in looking at the performance in percentiles are: a) Essama-Nassh et al (2010), which used quantile regressions<sup>8</sup> in an extension of the growth incidence curves that have been used to look at pro-poor growth (Ravallion and Chen, 2003): b) the World Bank Poverty Assessment for Guatemala (World Bank, 2009), which used quantile regressions to look at conditional percentile distributions for a set of poverty and social indicators; and c) (Azevedo and Pizzolitto, 2009) which used quantile regressions to assess the targets of the National Development Plan of the Dominican Republic.

The approaches followed with student growth percentiles and conditional growth charts share some similarities with the approach followed with growth incidence curves. They both look at the growth of an indicator at a given percentile in the distribution. They both can control for background characteristics. However, growth incidence curves are typically calculated from repeated cross-sections. They typically cannot provide information on the distribution of performance of those households or economic actors who began at a given percentile. In contrast, student growth percentiles or conditional growth charts (often of children’s height or weight) typically look at repeated observations of the same individual and looks at where the performance of that individual falls in the full distribution of performance of a comparable group. For the issue of target setting, one is typically concerned with repeated observations of the same agents, so the approach of the student growth percentiles or conditional growth charts would appear to be the most relevant.

# 4.  An illustration of target setting using School Education Data from Minas Gerais, Brazil

This section uses data on student achievement scores across different schools in the state of Minas Gerais, Brazil to illustrate some of the implications of using different approaches to set performance targets. The primary source of data for this exercise is from the PROEB (Program of Evaluation of Basic Education), which carries out Portuguese and math tests at the pupil level for municipal and state school students in 5<sup>th</sup> , 8<sup>th</sup> and 11<sup>th</sup> grade in the state of Minas Gerais. This analysis focuses only the State schools (which

> 7 See Betenner et al (2011).  See also the primers on student growth percentiles put together by the states of Colorado and Massachusetts. ((http://www.cde.state.co.us/cdedocs/Research/PDF/Aprimeronstudentgrowthpercentiles.pdf) and ( <u>http://www.doe.mass.edu/mcas/growth/InterpretiveGuide.pdf</u> )

> 8 They use unconditional quantile regressions developed by Firpo et al (2009), rather than the conditional quantile regressions used by Betebenner, Wei and He .

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