and test sets (Mullainathan and Spiess 2017).<sup>17</sup> However, in our case, LASSO conducts variable selection in a stable manner with the most important variables being selected in the vast majority of model runs.

# _2.2.3 OLS specification_

<mark>As mentioned above, we benchmark the performance of our two ML algorithms against OLS. The OLS model we use for comparison is as follows:</mark> where,

- _Yi_ is Average Student Learning Gain for Teacher i

- _X1_ is a vector of Student controls such as student age, baseline score, whether a student attends private tuitions in that subject<sup>18</sup> and percentage of students belonging to the lowest asset quartile

- _X2_ is a vector of school level covariates such as pupil teacher ratio, school location (urban/rural), annual entitlement grant per school etc.

- _X3_ is a vector of school level variables that affect the management and governance of teachers

- _X4_ is a vector of controls representing teacher characteristics (who teachers are) such as gender, experience and academic qualifications

- _X5_ is a vector representing our variables of interest divided into 3 categories: (i) what teachers do (measured through the TEACH Classroom Observation Tool and Student surveys) (ii) what teachers believe (measured through modules in Teacher surveys representing mental models) and (iii) what teachers know<sup>19</sup> (teacher assessment scores etc.)

- _εi_ represents an idiosyncratic error term

# **2.3 Limitations**

Our study suffers from three main limitations. First, we cannot fully address the issue of non-random matching of students, teachers, and schools. A rich set of teacher, student, and school level controls helps, as does the reliance on two cohorts (Grades 2 and 3). Nonetheless, despite our use of a very detailed set of teacher variables, the problem of unobserved variable bias remains. Non-random matching is sufficiently mitigated in models that control for a rich set of covariates (Koedel et al. 2015), and TVA estimates that control for lagged test scores exhibit little to no bias (Chetty et al. 2014a). Second, the total of number of teacher observations in our study (436) is not very high. This is partly because we were unable to match a significant share of teachers in our overall sample of 748 teachers because we could not match them to students due to challenges in data collection and entry. The key challenges in matching teachers and students were those of teacher and student turnover and also significant churn in the students assigned to teachers. Often this churn took place in a way that was not formally documented and was hard to establish credible data on. However, our analysis in Table 1 shows that this attrition of teachers is mostly random on observables. Finally, we rely on ML methods which are appropriate for predictions and associations but

> 17 This instability can arise due to multicollinearity or if the underlying true relationship is non-linear.

> 18 These are averaged for all the students taught by a given teacher since our unit of observation is the teacher.

> 19 A comprehensive list of variables along with their description is available in the Annex 1.

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