19 schooling) and the more novel quality of education indicators<sup>15</sup> substantially improve the explanatory power of the model. Additionally, the quantity indicators turn out statistically insignificant (Hanushek and WoBmann, 2007). Recognizing the limitations in the use of standardized testing as proxies for the quality of education, it is essential to explain how to achieve improvements in the quality of education. This leads us to public expenditure and institutional issues.

The literature, in general, shows that additional public spending is not necessarily associated with improved education outcomes (Pritchett 2001, 2006; Hanushek and Kimko, 2000). Hanushek and WoBmann (20007) corroborate these results using the quality of education indicators for a sample of OECD and non-OECD economies. The authors verify graphically and with linear regression the absence of a relationship between educational spending per student and student performance measured as the math test scores in the PISA 2003. Though these results were robust for both the average OECD country and the non-OECD, it is legitimate to ask whether they are valid for an enlarged sample including more developing countries and whether policy design should be based on average behavior.

When the sample is enlarged to include more developing countries, the association between public spending and quality of education for the average country remains insignificant and may even turn negative. The comparison across countries of such a diverse nature deserves caution. For instance, there is a clear positive association between the GDP per capita level and the test scores, as Figure 7 shows. Similarly, there is a positive association between public spending in education and GDP. When both scores and educational spending are controlled for GDP, the negative association emerges for the average country as Figure 8 shows.

However, for policymaking purposes, working with the best-practice cases may be more informative and useful than working with average behavior. This is especially relevant when adopting a production-function approach to education that relates inputs (spending) and outcomes (education attainment). In this case, what is of interest is the maximum expected output for a given input level, or alternatively, the minimum amount of input required to achieve a determined output level. The countries that obtain the highest scores per unit of public spending (best-practice cases) are considered the most efficient and constitute what is known as the efficiency frontier, depicted in Figure 9. The other countries’ efficiency is estimated as the distance with respect to the frontier. When the relationship between public expenditure and quality of education is cast in these terms, an efficiency frontier emerges. Instead of a negative association, a step-wise linear function is obtained with the best-practice (most efficient) cases lying on the frontier.<sup>16</sup>

> 15 The indicator is based on results form cognitive achievement tests, such as the TIMSS, the PISA and the PIRLS among others.

> 16This method is called the Free Disposable Hull (FDH) method.  A similar technique, Data Envelopment Analysis (DEA) can also be used to examine efficiency. Appendix 3 presents a discussion of efficiency frontiers.