Teacher-to-Classroom Assignment and Student Achievement

Figures & data

Table 1 Feasible teacher reassignments.

Classroom	Status quo	Counterfactual
type	$\mathrm{Pr}(W_i=1|X_i,{\overline{X}}_{p(i)})$	$\mathrm{Pr}({\tilde{W}}_i=1|X_i,{\overline{X}}_{p(i)})$	Xi	${\overline{X}}_{p(i)}$	$f(X_i,{\overline{X}}_{p(i)})$
000	$\frac{1}{2}$	$\frac{1}{3}$	0	0	$\frac{1}{4}$
			0	0	$\frac{1}{4}$
			0	0	$\frac{1}{4}$
001	$\frac{1}{2}$	$\frac{1}{2}$	0	$\frac{1}{2}$	$\frac{1}{6}$
			0	$\frac{1}{2}$	$\frac{1}{6}$
			1	0	$\frac{1}{12}$
011	$\frac{1}{2}$	$\frac{1}{2}$	0	1	$\frac{1}{12}$
			1	$\frac{1}{2}$	$\frac{1}{6}$
			1	$\frac{1}{2}$	$\frac{1}{6}$
111	$\frac{1}{2}$	$\frac{2}{3}$	1	1	$\frac{1}{4}$
			1	1	$\frac{1}{4}$
			1	1	$\frac{1}{4}$

Note: The population fraction of type X_i = 1 students is $\frac{1}{2}$ and that of type W_i = 1 teachers is also $\frac{1}{2}$. Classrooms of three students each are formed, such that the frequency of each of the four possible classroom configurations is $\frac{1}{4}$ in the population of classrooms of size 3. Under the status quo teachers are assigned to classrooms at random; in the counterfactual teachers are assigned more assortatively. See the main text for more information.

Table 2 IV regression results of the 3 × 3 model.

		(1)	(2)	(3)	(4)	(5)	(6)
		A. Only teacher effects		B. Full model		C. Without teacher × peer interactions
		Math	ELA	Math	ELA	Math	ELA
δ	FFT middle	0.069	−0.029	−0.145	−0.219	0.027	−0.082
		(0.053)	(0.048)	(0.138)	(0.173)	(0.060)	(0.059)
	FFT high	0.038	−0.058	−0.547	−0.137	−0.155	−0.154*
		(0.067)	(0.065)	(0.380)	(0.203)	(0.103)	(0.083)
η	FFT middle
	× baseline middle			0.040	0.057	0.052	0.067
				(0.060)	(0.067)	(0.060)	(0.063)
	× baseline high			0.015	0.076	0.050	0.097
				(0.084)	(0.081)	(0.076)	(0.082)
	FFT high
	× baseline middle			0.184**	0.150**	0.196**	0.127*
				(0.082)	(0.074)	(0.084)	(0.072)
	× baseline high			0.226**	0.187**	0.265**	0.149
				(0.099)	(0.092)	(0.100)	(0.095)
λ	FFT middle
	× fraction peers middle			0.318	0.288
				(0.299)	(0.401)
	× fraction peers high			0.239	0.161
				(0.205)	(0.288)
	FFT high
	× fraction peers middle			0.641	0.227
				(0.513)	(0.389)
	× fraction peers high			0.460	−0.355
				(0.409)	(0.343)
β	Baseline middle	0.888***	0.739***	0.843***	0.699***	0.840***	0.682***
		(0.077)	(0.084)	(0.092)	(0.096)	(0.092)	(0.094)
	Baseline high	1.622***	1.555***	1.578***	1.503***	1.550***	1.470***
		(0.103)	(0.105)	(0.124)	(0.118)	(0.119)	(0.115)
R^2		0.617	0.554	0.616	0.552	0.617	0.555
N		8534	9641	8534	9641	8534	9641

Note: The dependent variables are subject-specific test score outcomes in math and ELA. The instrumental variables are based on assigned teacher FFT (Panels A–C) and assigned peer baseline test scores (Panel B). All regressions control for the $h(x,\overline{x})$ function (see Section 2) and for randomization block fixed effects (239 randomization blocks in math and 252 randomization blocks in ELA). Analytic standard errors, clustered by randomization block, are in parentheses. ***significant at the 1%-level **significant at the 5%-level *significant at the 10%-level.

Table 3 Average reallocation gains in math.

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
	Panel A. Optimal versus status quo
	A.I Full sample				A.II Conditional on being reallocated
	All students	High	Middle	Low	All students	High	Middle	Low
Gain	0.017	0.028	0.012	0.014	0.036	0.059	0.028	0.026
SE	(0.006)	(0.014)	(0.008)	(0.011)	(0.012)	(0.029)	(0.019)	(0.019)
N	8534	2332	3108	3094	4107	1121	1380	1606
	Panel B. Optimal versus worst allocation
	B.I Full sample				B.II Conditional on being reallocated
	All students	High	Middle	Low	All students	High	Middle	Low
Gain	0.040	0.072	0.019	0.038	0.060	0.103	0.032	0.053
SE	(0.012)	(0.034)	(0.015)	(0.026)	(0.019)	(0.048)	(0.022)	(0.037)
N	8534	2332	3108	3094	5746	1626	1912	2208

Note: The table shows average reallocation gains (optimal versus random assignment in Panel A and optimal versus worst assignment in Panel B). The gains are expressed in test score standard deviations. The computations are based on a 3 × 3 model without teacher-by-peer interactions (see Table 2, column 5). The results are presented separately for the full sample of students (columns 1–4) and for the sample of classrooms that would get a new teacher as a result of the reallocation (columns 5–8). The reassignments are carried out within school types and districts. Standard errors are in parentheses and computed using the Bayesian bootstrap with 1000 replications (see Section 3.3.3). High/middle/low: students in the top/middle/bottom tercile of the baseline test score distribution.