factors explaining differences in student achievement across treatment groups. This assumption is called “selection on observables” in the econometric evaluation literature. Having in mind that PISA collects a rich set of background characteristics, which are strong predictors of student performance, we believe that this assumption is fulfilled and our approach is valid.

A crucial assumption in this model is that a difference between transitory shocks in time _t_ =0 and _t_ =1 is mean independent of the treatment (see Abadie 2005; Heckman, Ichimura and Todd 1998). It means that without the treatment the average outcome for the treated would experience the same change as the average outcome for the controls (or untreated). That assumption could be demanding if groups differ in important characteristics. Thus, usually conditional differences-indifferences estimator is employed which controls for the set of covariates: The crucial assumption here is that quasi-experimental groups differ only by observable covariates and this condition eliminates any bias caused by this. Typically, the differences-indifferences model is estimated using simple regression analysis when any characteristic one wants to control for could be entered into the equation and interacted with time and treatment (Meyer 1995; Gruber 1994). Another approach is to balance covariates across groups to make them more comparable, which can be achieved through matching methods (Rosenbaum and Rubin 1983; Heckman, Ichimura and Todd 1998).

In our case, we need to find counterparts for the treatment and control groups in 2000 among students in lower secondary schools in 2003 or 2006. This can be achieved with matching methods where counterfactual _t_ =1 scores are constructed using scores of students with similar characteristics to those observed in _t_ =0. Usually, matching methods are used to make control and treatment groups more comparable assuming that we have the same observations in each group in _t_ =0 and _t_ =1. In our case, we do not want to adjust for dissimilarities among treatment and control groups. We know that students who were in vocational schools differed from those in general schools but we are interested whether moving students from different tracks, who differ by assumption, into the one-type comprehensive lower secondary schools affected them similarly. Matching in our case is used to adjust in time by drawing comparable groups from 2003 or 2006 samples, not for adjustments across quasi-experimental groups.

As already mentioned, when dimension of _X_ is high, then exact matching on covariates is not possible (the “curse of dimensionality”). In this case, individuals can be matched on onedimensional propensity score _P_ = _P_ ( _D_ =1| _X_ ), where _D_ again indicates treatment and _P_ reflects the conditional probability of being treated (see Rosenbaum and Rubin 1983). However, as we noted above we have to balance covariates not between treatment and control groups, which differ by assumption, but between waves of the survey. Only in 2000 were students _treated_ , which means they were separated into different types of secondary schools. After the reform, in PISA 2003 and PISA 2006, all students were in lower secondary comprehensive schools. Nevertheless, one can draw from 2003 and 2006 samples to find good matches and construct

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