classes completed and ability through linear regressions, lack of effort measures might bias the estimates. We investigate this issue in Table 8. We begin with the regression of log classes completed, ln( _xt_ ) _,_ on log ability using the observed data (column 1) and the simulated baseline (column 2). The coefficients on log ability in columns (1) and (2) are very close, which is additional evidence of our good fit. In both cases, log ability explains about 20 percent of the variation in log classes completed.

**Table 8: Classes Completed Per Year.**

||Actual data|Si|mulated Dat|a|
|---|---|---|---|---|
||(1)|(2)|(3)|(4)|
|ln(ability)|0_._166<sup>_∗∗∗_</sup>|0_._156<sup>_∗∗∗_</sup>|0_._090<sup>_∗∗∗_</sup>|0_._085<sup>_∗∗∗_</sup>|
||(0_._015)|(0_._005)|(0_._005)|(0_._000)|
|ln(efort)|||0_._854<sup>_∗∗∗_</sup>|0_._915<sup>_∗∗∗_</sup>|
||||(0_._004)|(0_._000)|
|ln(shock to classes completed)||||1_._000<sup>_∗∗∗_</sup><br>(0_._000)|
|Constant|2_._060<sup>_∗∗∗_</sup>|2_._748<sup>_∗∗∗_</sup>|2_._197<sup>_∗∗∗_</sup>|2_._996<sup>_∗∗∗_</sup>|
||(0_._012)|(0_._004)|(0_._004)|(0_._000)|
|_R_<sup>2</sup>|0.213|0.204|0.518|1.000|
|Num. Obs.|123,101|127,044|127,044|127,044|

_Source_ : OLS estimation using SPADIES for actual data and model’s baseline predictions for simulated data. _Notes_ : Dependent variable is ln(classes completed per year), or ln( _xt_ ) in the model. An observation is a student-year; years 1-8 are included. Ability is _θ_ , effort is _e_<sup>_∗_</sup> _t_<sup>,andshocktoclassescompletedis</sup><sup>_zt_.All</sup> regressions include year fixed effects (not shown). Standard errors (in parentheses) are clustered by student. * p _<_ 0.10, ** p _<_ 0.05, *** p _<_ 0.01.

One advantage of our structural model is that we can recover variables—effort and shocks— not observed in the administrative data. This allows us to expand regression (2) in order to gauge the relative roles of effort, performance shocks, and ability. Adding effort (column 3), we explain an additional 30 percent of the variation in classes completed. As expected, we explain all the variation in classes completed when we add the shock as well (column 4), with a coefficient on log ability equal to our point estimate for _α_ . Further, in column 4 effort and the shock account for 47 and 52 percent of the variation in classes completed, respectively, leaving a mere 1 percent explained by log ability. In other words, in our model almost the whole variation in classes completed is due to effort and shocks.

Consistent with estimates from the exogenous effort model, when effort is not controlled for in these regressions the coefficient on ability is overestimated by about 75 percent. This large upward bias should be taken into account not only when estimating achievement regressions with administrative datasets, which typically do not include effort measures, but also when designing policy. By ignoring or underestimating effort’s role, policy might place too much weight on a student trait—ability—that cannot be changed rather than on a student choice —effort—that could, in principle be changed.

# **7.5.4 Policy implications**

The previous findings elicit two important questions. The first is whether policies could actually raise the number of classes completed. Combining equations (3) and (12), we arrive at the effort 29