tution effects. School expenditures do drop slightly as do the hours spent helping the child on his or her homework. Minutes spent playing increases, but tuition also increases. While some of these responses are in the direction of substitution, they are generally not statistically significant. Given the detailed household data we obtained, it suggests that household substitution is unlikely to be a main driver behind low persistence. This may not be particularly surprising given that very low achievement suggests that children may be below parents’ desired learning levels.

Table 9 explores the possibility that fade out captures teachers targeting poorly performing students. If teachers target poorly performing children in each classroom, persistence should be lower within schools than between schools. To test this hypothesis, we estimate a basic lagged value-added model with no controls and instrument for lagged achievement using lagged differences in alternate subjects. We estimate this model using average school scores (between school specification) and child scores measured in deviations for the school average (within school specification). If anything, the persistence coefficient is lower for the between school regressions, suggesting that within school targeting is not the primary reason for low persistence. Finally, Table 10 looks at whether heterogeneity in persistence can provide some hints about its origin(Semb and Ellis, 1994).<sup>17</sup> To obtain the most power possible, we estimate the valueadded model for specific sub-populations using the “predetermined inputs, uncorrelated effects, and conditionally stationary” based estimator (M10 of Tables 4, 5, and 6). Unfortunately, large standard errors make it difficult to find statistically different decay rates between groups. Learning in private schools seems to decay faster than learning in government schools, but the difference is not statistically significant. A similar pattern holds for richer families and children with educated parents. These results hint that learning decays faster for faster learners.

These candidate explanations do not yield a compelling story thus far; it could just be that children forget what they learned. Psychology and neuroscience provide some compelling evidence for this using laboratory experiments. Psychological research on the “curve of forgetting” dates back to Ebbinghaus’s (1885) seminal study on memorization. Rubin and Wenzel (1996) review the laboratory research spawned by this contribution. Semb and Ellis (1994) review classroom studies that test how much students remember after taking a course. Both literatures document the fragility of human memory. Cooper et al. (1996) studies the learning losses that children experience between spring and fall achievement tests. These losses are generally not as rapid as the effects we find, but the experiment is different: we estimate the depreciation with no inputs, whereas summer activities provide some stimulus, particularly for privileged

> 17To give one example, MacKenzie and White (1982) report that fade out for geographical knowledge was much higher for in-class exercises compared to field excursions (or, passive versus active learning.) Similarly, Rothstein (2008) finds heterogeneity in the long-run effects of teachers who produce equal short-run gains; Jacob, Lefgren and Sims (2008) estimate that teacher effects are only a third as persistent as achievement in general.

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