finance point of view, these different estimates matter particularly since per pupil expenditures are lower in private relative to public schools.<sup>6</sup> Our results are consistent with growing evidence that relatively inexpensive, mainstream, private schools hold potential in the developing country context (Jimenez, Lockheed and Paqueo, 1991; Alderman, Orazem and Paterno, 2001; Angrist et al., 2002; Alderman, Kim and Orazem, 2003; Tooley and Dixon, 2003; Muralidharan and Kremer, forthcoming; Andrabi, Das and Khwaja, 2008). Overall, our general results support a movement towards long-run, experimental evaluations of educational interventions.

A final contribution of our work is that it applies a wider set of econometric tools from the dynamic panel data literature than have been typically used in the education literature. In the use of dynamic panel methods, our estimators bear greatest resemblance to those discussed by Schwartz and Zabel (2005) and Sass (2006). Both use simple dynamic panel estimators, in the first case using school-level data and in the second using the Arellano and Bond (1991) differences GMM approach. Santibanez (2006) also uses the Arellano and Bond (1991) estimator to analyze the impact of teacher quality. Our efforts extend to include system GMM approaches (Arellano and Bover, 1995) and to address measurement error in test scores and alternative assumptions regarding omitted heterogeneity. We find both are important.

The rest of the paper is organized as follows: Section 2 presents the basic education production function analogy and discusses the specification and estimation of the value-added approximations to it. Section 3 summarizes our data. Section 4 reports our main results, several robustness checks, and a preliminary exploration of the economic interpretation of persistence. Section 5 concludes by discussing implications for experimental and non-experimental program evaluation.

# 2 Empirical Learning Framework

The “education production function” approach to learning relates current achievement to all previous inputs. Boardman and Murnane (1979) and Todd and Wolpin (2003) provide two accounts of this approach and the assumptions it requires; the following is a brief summary.<sup>7</sup> Using notation consistent with the dynamic panel literature, we aggregate all inputs into a single vector x _it_ and exclude interactions between past and present inputs. Achievement for variables. We are exploring such strategies using plausible exogenous geographical variation in a companion paper focused on the returns to private school education in Pakistan and competition between public and private schools. Our emphasis here is on the challenges faced by popular value-added strategies.

> 6For details on the costs of private schooling in Pakistan see Andrabi, Das and Khwaja (2008).

> 7Researchers generally assume that the model is additively separable across time and that input interactions can be captured by separable linear interactions. Cunha, Heckman and Schennach (2006) and Cunha and Heckman (2007) are two exceptions to this pattern, where dynamic complementarity between early and late investments and between cognitive and non-cognitive skills are permitted.

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