For each observation system, and then for all of them combined, we first estimate a model with all Level 1 variables and then a model with the principal components that emerge from a principal components analysis of them.<sup>40</sup> Last, in order to investigate heterogeneity, we separate the sample into teachers who score below and above 0 on the (normalized) teacher test (we refer to this below as the “threshold”).<sup>41</sup> The coefficient estimates on the teacher observation and teacher score variables from all of these regressions are reported in Tables 12 to 16.<sup>42</sup> The tables also report p-values for F-tests of whether the coefficients on the variables in each of the sets of control variables are jointly equal to zero.

The data we use for this analysis are slightly different from those used above to investigate the properties of the various instruments. That analysis was carried out at the classroom observation level, whereas this analysis is carried out at the teacher level. Specifically, we aggregate all information from each observation tool to the level of a teacher. For example, if a teacher was observed with the SDI tool in Rounds 1 and 2, we average across the two rounds; or if a teacher has two videos that were coded using _Teach_ , then we average across those. This teacher level file is then merged with the student test scores and it is on this data that we run the regression analysis (and report standard errors that cluster at the teacher level). In order to ensure comparability across the models, we estimate these regressions on the sample for which we have all four observation types.

It is important to recognize that we cannot provide a causal interpretation to these estimates. While controlling for student, family, teacher, and school characteristics might help to identify the link from teacher skills and behaviors to student test scores, there are a variety of potential selection and reverse causation issues that we cannot rule out. The findings are therefore only indicative of (conditional) associations in the data.

# **_6.1 Instrument-by-instrument results_**

> 40 Note that this PCA analysis is slightly different from that reported in Section 5.  First, that analysis was done with all individual observations; this analysis is done at the level of each teacher—with all observations for that teacher with a particular instrument averaged.  Second, in this analysis we exclude teaching and learning time from the principal components analysis and treat it as a qualitatively different type of variable. In addition, we remove redundant variables and so, for example, include only one variable for share of time on teaching and learning. The results from this analysis are reported in Appendix 1 Tables 6 and 7 and are discussed further below.

> 41 The mean score across teachers is -0.05 and the standard deviation is 0.63 (the median score is 0.03).

> 42 We also estimate models where each variable enters one at a time since there is a degree of correlation between them and associations might get masked in the multivariate regression context. The results of these variable-byvariable models (first including only dummy variables for the subject observed, and then including for all the control variables) are reported in Appendix 1 Table 9. The results are generally consistent, although in the case of CLASS more variables emerge as being individually significantly associated with student test scores than in the multivariate models. This is consistent with the fact that they were highly correlated with each other, as discussed in Section 5.

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