Economics of Education Review 89 (2022) 102251
7Assumption 1. (Random Assignment) Children and teachers are as-
good-as randomly assigned to classes such that any systematic differences
occur at the school level: φjk = φk.
Importantly, Assumption 1 does not preclude random assignment of
unobserved teacher attributes (e.g., didactic capabilities) or teachers’
behaviors (Araujo et al., 2016). These unobserved characteristics may
correlate with the teacher’s overall capacity to form positive relation-
ships, affecting a child’s learning. Fortunately, we have access to a rich
set of student and classroom-related covariates such as the teacher’s
experience, age, sex, education, and class size. Therefore, we assume:
Assumption 2. (Exogeneity) Conditional on the child, peer, teacher, and
classroom observables, the teacher relationship skills, Xjk, do not correlate
with the error term, εijk.
A simple class average (i.e., Xjk ≡ 1/Ijk
∑Ijk
i=1Xijk) is a noisy measure
when unobserved preferences for a particular teacher correlate with the
children’s level of effort and learning and shape their perceptions and
hence their evaluations. For instance, as Dee (2004) pointed out, some
children may prefer a specific teacher identity bringing about a
role-model effect that increases their learning engagement and causes
them to evaluate their teacher more positively. In other words, there
would be an “own-observation problem” (Chetty et al., 2011, p. 1635).
Formally, denote θjk as the effect that is due to the teacher and denote
ρijk as child i’s unobserved preferences such that:
Xjk ≡ 1
Ijk
∑Ijk
i=1
Xijk = θjk + 1
Ijk
∑Ijk
i=1
ρijk (3)
We assume no peer effects (i.e., σ(θjk, ρijk ) = 0 and σ(ρijk, ρi′ jk) = 0 for
i ∕= i′
, where σ(⋅) is the covariance operator). If child i, who favors the
teacher for some unobserved reason, subsequently evaluates the teacher
more positively, σ(Xjk, ρijk) > 0, and as a result increases effort and
learning, σ(YG1
ijk , ρijk) > 0, then we have an upward bias in β with finite
class size (see Web Appendix D).18 Therefore, the leave-out-mean ad-
dresses the bias due to unobserved preferences assuming:
Assumption 3. (No Peer Effects) A child’s unobserved preferences do
not correlate with the teacher’s overall capacity to form positive relationships,
σ(θjk, ρijk) = 0, and unobserved preferences of child i do not affect the un-
observed preferences of child i′
, σ(ρijk, ρi′ jk) = 0 for i ∕= i′
.
Even under Assumption 3, there is still a bias. Intuitively, child i’s
unobserved preference, ρijk, still correlates with X( i)jk because our es-
timate is relative to the school mean and, consequently, ρijk correlates
with ρk. Therefore, following Chetty et al. (2011), we also omit child i
from the school mean such that ΔX( i)jk ≡ X( i)jk