contributions ranging from 0.19 to 0.32 standard-deviations a year.<sup>2</sup> Notably, the lagged valueadded model estimated by OLS gives similar results for the private school effect as our more data intensive dynamic panel methods. This is due to the countervailing heterogeneity and measurement error biases on _β_ and because lagged achievement can act as a partial proxy for omitted heterogeneity in learning.<sup>3</sup> Finally, towards an economic interpretation of low persistence, we use question-level exam responses as well as household expenditure and time-use data to explore whether psychometric testing issues, behavioral responses, or forgetting contribute to low persistence—causes that have different welfare implications. This investigation suggests that measurement error, mechanical psychometric testing, and behavioral response based explanations are insufficient. Understanding the behavioral or technological reasons for low persistence remains a critical issue in the literature.

The value-added of our contribution is several fold. To begin with, we show that the restricted value-added estimates based on longitudinal data may be _worse_ than the naive crosssectional OLS estimates. Second, we demonstrate how unobserved heterogeneity in learning and measurement error in test scores can bias estimates of persistence. The low persistence we find implies that long-run extrapolations from short-run impacts are fraught with danger. In the model above, the long-run impact of continued treatment is _α/_ (1 _− β_ ); with estimates of _β_ around 0.2 to 0.5, these gains may be much smaller than those obtained by assuming that _β_ is close to 1.<sup>4</sup> Third, we find that students in Pakistan’s private schools learn significantly more each year than their public school counterparts but that the popular gain-score or restricted value-added model would have detected no effect of private schools on learning.<sup>5</sup> From a public 2Harris and Sass (2006) find the that the persistence parameter makes little difference to estimates of teacher effects, while we find it starkly affects the estimates of school type. This can be explained by the relative gaps in baseline achievement. It is likely that a child does not continue with the same teacher, or an equally good teacher, over time. Hence, even if we don’t observe children’s educational history, two children who currently have different teachers may have been exposed to a similar quality teachers in the past. As such, children with different teachers often do not differ substantially in their baseline learning levels. In contrast, given that there is little switching across school types, children currently in different schools differ substantially in baseline learning levels.

> 3This results suggests that correcting for measurement error alone may do more harm than good. For example, Ladd and Walsh (2002) correct for measurement error in the lagged value-added model of school effects by instrumenting using double-lagged test scores but don’t address potential omitted heterogeneity. They show this correction significantly changes school rankings and benefits poorly performing districts. Given that we find unobserved heterogeneity in learning rates, rankings that correct for measurement error may be poorer than those that do not.

> 4For example, Krueger and Whitmore (2001), Angrist et al. (2002), Krueger (2003), and Gordon, Kane and Staiger (2006) calculate the economic return of various educational interventions by citing research linking test scores to earnings of young adults (e.g. Murnane, Willett and Levy, 1995; Neal and Johnson, 1996). Although effects on learning as measured by test-scores may fade, non-cognitive skills that are rewarded in the labor market could persist. For instance, Currie and Thomas (1995), Deming (2008) and Schweinhart et al. (2005) provide evidence of long run effects of Head Start and the Perry Preschool Project, even though cognitive gains largely fade after children enroll in regular classes.

> 5An alternative identification strategy to value-added models that we do not pursue here is instrumental

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