35.7 Motivating Examples

Do Teachers’ Qualifications Affect Student Test Scores? (Aaronson, Barrow, and Sander 2007)

1. Initial Model: $$Y_{ijt} = \delta_0 + Y_{ij(t-1)}\,\delta_1 + X_{it}\,\delta_2 + Z_{jt}\,\delta_3 + \epsilon_{ijt},$$ where $Y_{ijt}$ is the outcome (test score) of student $i$ in class $j$ at time $t$, $X_{it}$ are student-level covariates, and $Z_{jt}$ are teacher or classroom-level covariates.

2. Refining the Model: $$Y_{ijst} = \alpha_0 + Y_{ij(t-1)}\,\alpha_1 + X_{it}\,\alpha_2 + Z_{jt}\,\alpha_3 + \gamma_s + u_{isjt},$$ introducing school fixed effects ($\gamma_s$). Sorting of students and teachers often happens within schools, so including fixed effects can reduce bias.

3. Further Adjustment via Matching:
   Identify schools or classes that appear to allocate teachers as-if randomly (or close to it). Match students on prior scores ($Y_{ij(t-1)}$) and other observed characteristics. Then re-run a model with interactions: $$\begin{aligned} Y_{isjt} &= Y_{isj(t-1)}\,\lambda + X_{it}\,\alpha_1 + Z_{jt}\,\alpha_{21} + (Z_{jt} \times D_i)\,\alpha_{22} + \gamma_s + u_{isjt}, \end{aligned}$$ where $D_i$ is an indicator for “high-poverty” students. Testing $\alpha_{22} = 0$ checks for heterogeneous effects of teacher experience by student poverty level.

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References

Aaronson, Daniel, Lisa Barrow, and William Sander. 2007. “Teachers and Student Achievement in the Chicago Public High Schools.” Journal of Labor Economics 25 (1): 95–135.