|Autonomy ‐ establishing student assessment|0.84|0.44|0.77|0.76|
|---|---|---|---|---|
|Autonomy ‐ determining courses content|0.73|0.41|0.29|0.32|
|Autonomy‐ decidingwhich courses are offered|0.73|0.49|0.11|0.15|

Notes: authors’ own computation with data from PISA. All means take into account PISA survey design (see Annex A for details) The selection of variables to be included in the estimation of the production function was restricted to those which were included in all four rounds of PISA. Some variables that were included in the four rounds were excluded from the estimation when 20 percent or more of the observations had missing values. The parameters of the production function were estimated with a variance-covariance matrix accounting for the additional variation in PISA results due to the plausible values methodology followed in the test design (see Annex A for details).<sup>9</sup> All the estimations and subsequent simulations are based on reading learning outcomes since it is the only subject area for which a 2000-2012 comparison is valid.

The estimated results are presented in Table 4. The proportion of explained variance by the production function (as captured by the R<sup>2</sup> ) is relatively high at around 40 percent, a level close to the one reported in Santos (2007). The model’s goodness of fit varies little across years. This might be the outcome of restricting the estimation of the production function to the same variables across surveys and using only those variables that have relatively low levels of missing values (less than 20 percent).

As expected, student characteristics and socioeconomic background variables are important determinants of student learning outcomes. In all four years, girls get significantly better reading results than boys, averaging close to a quarter of a standard deviation, which confirms the international evidence on gender differences in learning outcomes (Guiso et al., 2008). Grade-forage is also significantly correlated with learning outcomes across all years and with very large effects: 15-year-old test takers who are not yet in grade 10 (the accurate grade-for-age), capturing either late entry into the education system or, more likely, repetition of grades, achieve learning outcomes that are close to three quarters of a standard deviation lower than 15-year-olds who attend 10<sup>th</sup> grade.

In terms of family background, mother’s education is correlated with better results but only in 2009 and 2012, and father’s education is not statistically significant in most years. In both 2006 and 2009, family wealth is a significant predictor, but insignificant in 2000 and 2012. With regards to household factors, having books at home is a consistently important predictor of students’ performance, in line with many other studies in the literature. There is a substantial degree of multicolinearity between the number of books at home, wealth index, and parents’ education. Therefore it is not surprising that, contrary to what was expected _ex-ante_ , the wealth index or mothers’ education is for some years not statistically significant. We do not attempt to address the issue of multicollinearity, since the variables included in students’ characteristics and socioeconomic background are simple controls of exogenous inputs, whereas the paper’s main focus lies on the effects on learning of school characteristics and institutional settings.

> 9 The parameters of the production function were estimated in Stata using the user-written command pv developed by MacDonald (2008, revised in 2014).

15