Empirical evidence on education production functions exists for both developed countries (for example, Hanushek, 1986 and 2002) and developing countries (for example, Glewwe 2002). Previous empirical studies do not always agree on which school and family inputs improve children’s achievement. For example, there is some disagreement about the role played by schooling inputs such as class size, teacher experience, teacher education, and mother’s employment. For a survey of related literature, see Todd and Wolpin (2003).

Nevertheless, although a child’s achievement is inherently individual in nature, a large body of evidence points to the existence of persistence effects in educational achievement across generations (Fertig 2003; Fertig and Schmidt 2002; Currie and Thomas 1999). Consequently, it is necessary to control for the characteristics of individual students as well as for their family backgrounds. Similarly, it is necessary to control for the characteristics of the school environment as well as its institutional arrangements. Recent evidence from the literature on early test score differentials suggests that differences in children’s cognitive ability among families appear at an early age, tend to persist, and may even widen with age. In general, ―good‖ families promote cognitive, social, and behavioral skills, while ―bad‖ families do not. This is important in determining what policy interventions can be successful (Carneiro and Heckman 2003). Evidence also suggests that socioeconomic and family background variables, such as the education levels of a student’s parents and the number of books a child has, are very important determinants of test scores at early ages (Fryer and Levitt 2002).

# **Methodology and Estimation**

Our first step was to specify and estimate cognitive achievement production functions that relate student achievement to individual, family, school, and institutional inputs. We then proceeded to decompose the over-time test score change into an explained component (accounting for student, family, school and institutional characteristics) and an ―unexplained‖ component (the efficiency by which the country is able to convert characteristics into student learning outcomes as measured by test scores), using the traditional Oaxaca-Blinder decomposition method (Oaxaca 1973; Blinder 1973).

The model specification that we used to estimate the production function for cognitive achievement is as follows: where _Tiaj_ is the observed test score (from the PISA math test) of student _i_ in household _j_ at time _a_ (the time of the test), _Aija_ is a vector of individual student characteristics, _Fija_ is a vector of parent inputs, _Sija_ is a vector of school-related inputs, _Iija_ is a vector of the school’s institutional characteristics, and _єija_ is an additive error, which includes all the omitted variables including those that relate to the history of past inputs, endowed mental capacity, and measurement error. Todd and Wolpin (2003) discuss in detail the assumptions that would satisfy the application of this specification, in which the achievement test score depends solely on contemporaneous measures of family, school, and other inputs. These assumptions state that: (i) current input measures capture the entire history of inputs or, alternatively, only contemporaneous inputs matter and (ii) contemporaneous inputs are unrelated to endowed mental capacity. Its linear specification (after dropping subscript _a_ ) is given by: 4