# Teacher bias or measurement error?

Thomas van Huizen Department of Applied Economics. Utrecht University School
of Economics, Utrecht University, Utrecht, The Netherlands.
<EMAIL_ADDRESS>Madelon Jacobs Research Centre for Education and the
Labour Market (ROA), School of Business and Economics, Maastricht University,
Maastricht, The Netherlands<EMAIL_ADDRESS>Matthijs
Oosterveen †† We thank Sönke Matthewes, Joppe de Ree, Astrid Sandsør, and
Dinand Webbink for valuable comments. The paper has also benefited from the
comments of participants at the EQOP meeting at the University of Oslo, KVS
new paper sessions, the LEER conference, the LESE conference, LMG Lisbon, the
PPE conference, and the Utrecht University School of Economics seminar. The
data used in this study are housed within the secure virtual environment at
Statistics Netherlands and cannot be publicly shared due to privacy
restrictions. However, all the underlying data supporting the presented
results are accessible through Statistics Netherlands. National and
international researchers can request access via the Netherlands Cohort Study
on Education (NCO) and Statistics Netherlands. Requests for data access should
be directed to<EMAIL_ADDRESS>A comprehensive description
of the data and the access procedure can be found in Haelermans et al. (2020).
The authors have no relevant or material financial interests that relate to
the research described in this paper. All omissions and errors are our own.
Author names are ordered alphabetically.  Department of Economics, Lisbon
School of Economics and Management, and Advance/CSG, University of Lisbon,
Lisbon 1200-781, Portugal<EMAIL_ADDRESS>
###### Abstract

In many countries, teachers’ track recommendations are used to allocate
students to secondary school tracks. Previous studies have shown that students
from families with low socioeconomic status (SES) receive lower track
recommendations than their peers from high SES families, conditional on
standardized test scores. It is often argued that this indicates teacher bias.
However, this claim is invalid in the presence of measurement error in test
scores. We discuss how measurement error in test scores generates a biased
coefficient of the conditional SES gap, and consider three empirical
strategies to address this bias. Using administrative data from the
Netherlands, we find that measurement error explains 35 to 43% of the
conditional SES gap in track recommendations.

JEL: I24, C36
Keywords: Teacher evaluation, Track recommendation, Measurement error,
Instrumental variables, Errors-in-variables

##  1 Introduction

Teacher bias may constitute an important source of inequality in educational
attainment and later life outcomes. To study teacher bias, studies generally
rely on comparisons between subjective teacher evaluations, such as track
recommendations, and objective measures of student abilities, such as
performance on standardized tests (Botelho et al., 2015; Alesina et al., 2018;
Timmermans et al., 2018). However, the presence of systematic gaps in teacher
evaluations between groups that condition on objective measures is not
necessarily evidence of teacher bias. The identification of teacher bias with
this approach is complicated by two main concerns: omitted variable bias and
measurement error in the objective measure of student abilities. As omitted
variables could (at least partly) explain the potentially remaining
conditional gap, the literature tries to address this concern by controlling
for additional factors such as non-cognitive skills (Burgess and Greaves,
2013; Cornwell et al., 2013; Triventi, 2020).111For instance, Cornwell et al.
(2013) show that girls are graded more favorably by their teachers than boys,
even if they perform equally well on several standardized tests. However, this
differential treatment disappears after controlling for differences in non-
cognitive skills.

This paper focuses on the second methodological concern: measurement error in
the objective measure of ability. Ability is often measured by standardized
tests and classical measurement error in test scores leads to attenuation bias
in the test score coefficient. Moreover, to the extent that student background
characteristics are associated with ability, this bias contaminates the main
parameter of interest that measures the conditional gap. For instance, if
socioeconomic status (SES) of the family is positively related to student
ability, analyses that control for standardized test scores without addressing
measurement error will overestimate teacher bias.222This methodological
problem has been coined Kelley’s Paradox (Wainer and Brown, 2006), after
statistician Truman Kelley who described this phenomenon in the first half of
the twentieth century (Kelley, 1927). Hence, systematic differences in teacher
evaluations conditional on test scores could be merely a statistical artefact
and can therefore not automatically be interpreted as evidence of teacher
bias.

Following Pei et al. (2019), we first discuss how classical measurement error
in test scores generates a bias that contaminates the coefficient measuring
the conditional gap between groups. We then discuss three strategies to
address measurement error. First, we consider an instrumental variable (IV)
approach, a well-established method in econometrics to correct for measurement
error (Hausman, 2001; Gillen et al., 2019; Ward, 2023). Second, we discuss a
method in the spirit of errors-in-variables (EIV) models, an approach that is
common in psychometrics (Fuller, 2009; Culpepper, 2012). The additional
parameter required for the EIV strategy, compared to OLS, is the reliability
ratio of the test that is used to measure ability. We argue that, under weaker
assumptions than the IV strategy, the IV first stage can be used to identify
the reliability ratio. Third, we combine the EIV strategy with a novel
approach to estimate the reliability ratio under yet weaker assumptions, while
making use of students’ complete test score histories.

We apply these three strategies to estimate conditional gaps in teacher track
recommendations in the Netherlands, focusing on differences by family SES. The
Dutch early tracking regime provides an interesting case for several reasons.
First, existing studies claim that there is evidence of teacher bias in track
recommendations against low SES children, both in the Netherlands (Timmermans
et al., 2018) and other countries (Falk et al., 2020; Batruch et al., 2023).
Second, teacher bias in track recommendations may be highly consequential for
the educational paths students take and therefore may reduce equal
opportunities. Third, we can make use of detailed administrative data,
covering roughly half of the Dutch primary schools, to apply the three
different approaches mentioned above.

While almost all OECD countries have some form of educational tracking, the
Netherlands is considered an early tracking country. Students are assigned to
different educational programs in secondary school at the age of twelve. In
contrast to some other early tracking regimes, track assignment is based upon
binding track recommendations from the primary school teacher (WVO, 2014).
These recommendations may be highly consequential: although students are able
to move between tracks at least in the first grades of secondary school,
roughly 70% of the students do not experience track mobility four years into
secondary education (de Ree et al., 2023). Track placement determines among
other things the level of the curriculum and peer group abilities, and as such
may affect student achievement (Matthewes, 2021). As only the highest tracks
give access to college and university education, track placement may also have
long-term consequences in terms of attainment and income (Dustmann et al.,
2017; Borghans et al., 2019, 2020). Moreover, negatively biased track
recommendations signal low teacher expectations, which may by itself
negatively affect student outcomes (Rosenthal and Jacobson, 1968; Carlana,
2019; Papageorge et al., 2020; Hill and Jones, 2021).

Given the consequential nature of these recommendations, potential teacher
bias has been a major policy concern in the Netherlands. This concern gained
momentum in 2016, when a report by the Dutch Inspectorate of Education (2016)
demonstrated that students with lower family SES receive significantly lower
track recommendations, conditional on scores from a standardized end-of-
primary school test. This finding fueled an ongoing public debate on the role
of teachers as gatekeepers of opportunities and contributed to the
implementation of a national policy reform, effective in school year 2023-24.
The reform reduces the role of teachers in track placement, and increases the
role of the standardized end-of-primary school test (Ministry of Education,
2023b).333Moreover, the reform includes a revision of national guidelines for
the formulation of track recommendations (Ministry of Education, 2023a). These
guidelines pay explicit attention to (implicit) teacher bias in track
recommendations, especially with respect to SES. Concerns about unequal
opportunities have also been expressed in other countries based on similar
findings (Falk et al., 2020; Carlana et al., 2022).

Using new administrative data, we first replicate previous findings from the
Netherlands (Inspectorate of Education, 2016; Timmermans et al., 2018;
Zumbuehl et al., 2022) and find that children with lower family SES receive
systematically lower track recommendations, conditional on standardized test
scores. We then address concerns about measurement error using our three
strategies. The IV strategy uses the score from a standardized test
administered in the middle of the school year as an instrumental variable for
the standardized test administered at the end of the year. The two EIV
strategies use different methods to identify the reliability ratio of the end-
of-year test: the first stage from the IV strategy and a more novel approach
that uses a student’s standardized test scores across all primary school
grades. Our results indicate that measurement error can explain between 35 and
43% of the conditional SES gap.

Our paper contributes to two strands of literature. First, our study speaks to
the broader literature on measurement error (Pei et al., 2019; Ward, 2023;
Borjas and Hamermesh, 2024). In particular, we focus on contamination bias due
to measurement error. As Modalsli and Vosters (2022) point out, “[a]lthough
the notion of bias in one coefficient arising from error in another regressor
is a well-known econometric result, it is seldom addressed in practice with
empirical studies.” Modalsli and Vosters (2022) examine the implications of
measurement error in multigenerational income regressions and use an IV
approach to mitigate the resulting bias. They demonstrate that measurement
error in parents’ income leads to an upwards bias in the grandparent
coefficient. Gillen et al. (2019) replicate three classic experiments and show
that the findings change substantially when measurement error is addressed
with an IV approach. Gillen et al. (2019) argue that “these results show that
failing to properly account for measurement error may cause a field-wide bias
leading scholars to identify “new” phenomena.” A contribution of our study is
to discuss how different strategies to address measurement error are related
and how they produce valid estimates under different assumptions. This paper
thereby also contributes to a larger literature studying bias and
discrimination, where similar methodological concerns apply.

Second, we contribute to the literature on teacher bias by providing new
evidence on the conditional SES gap in track recommendations. Teacher bias in
track recommendations have been recently studied in Germany (Falk et al.,
2020), Italy (Carlana et al., 2022), and the Netherlands (Timmermans et al.,
2018; Zumbuehl et al., 2022). Previous studies have also examined teacher bias
in grading by gender (Cornwell et al., 2013; Lavy and Sand, 2018; Terrier,
2020), race and ethnicity (Burgess and Greaves, 2013; Botelho et al., 2015),
and migration background (Alesina et al., 2018; Triventi, 2020). Consistent
with the observation of Modalsli and Vosters (2022), the issue of
contamination bias due to measurement error has been largely ignored in this
literature. A notable exception is the study by Botelho et al. (2015) on
racial discrimination in grading in Brazil, who use an IV approach to address
contamination bias. Consistent with our findings, they find that the
conditional gap is reduced by around 50%. However, Botelho et al. (2015) do
not discuss or test the assumptions under which the IV approach is valid, and
do not examine alternative strategies to address measurement error.

The remainder of the paper is organized as follows. Section 2 discusses
contamination bias in the presence of measurement error and explains the three
strategies to address this bias. Section 3 describes the Dutch education
system and administrative data. Section 4 shows our empirical results. Section
5 concludes.

##  2 Methodology

###  2.1 Contamination bias

Let $y_{t}$ be a variable that measures a teacher’s subjective evaluation of a
student in time period $t$ (e.g., a track recommendation), $SES$ a variable
that measures the socioeconomic status of the student, and $s_{t}$ a variable
that reflects student ability in time period $t$. The long regression equation
equals,

$\displaystyle y_{t}=\beta SES+\gamma s_{t}+e_{t}.$ (1)

Throughout the paper we suppress the subscript for student $i$ and the
intercept in regression equations. Scholars and policy makers are generally
interested in the parameter $\beta$. That is, they are interested in whether
subjective teacher evaluations differ systematically by SES, or another group
characteristic such as gender or ethnicity, conditional on student ability.

We cannot estimate (1) as we do not observe ability. Instead we observe a test
score $s_{t}^{m}$ as an objective measure for student ability. This test score
is equal to ability plus measurement error $m_{t}$,

$\displaystyle s_{t}^{m}=s_{t}+m_{t}.$ (2)

The measurement error reflects, for instance, that students may guess several
answers, are more or less familiar with the specific selection of test items,
or feel (un)well on the day of the test. These processes could be random,
reflected by the assumption of classical measurement error.

######  Assumption 1.

Classical measurement error

  1. a.

(Zero mean) $\mathbb{E}[m_{t}]=0$ $\forall t$.

  2. b.

(Zero covariance) $\mathrm{Cov}[m_{t},x]=0$, $\forall t,x$.

The medium regression equation replaces $s_{t}$ with $s^{m}_{t}$,

$\displaystyle y_{t}=\beta^{m}SES+\gamma^{m}s_{t}^{m}+e_{t}^{m}.$ (3)

To analyze what $\beta^{m}$ and $\gamma^{m}$ identify, we also introduce the
balancing regression equation,

$\displaystyle\begin{pmatrix}s_{t}\\\
s_{t}^{m}\end{pmatrix}=\begin{pmatrix}s_{t}\\\
s_{t}+m_{t}\end{pmatrix}=\begin{pmatrix}\delta\\\
\delta^{m}\end{pmatrix}SES+\begin{pmatrix}u_{t}\\\ u^{m}_{t}\end{pmatrix}.$
(4)

Classical measurement error on the left-hand side does not affect OLS
estimates, and thus 1 guarantees that $\delta^{m}=\delta$ and
$u^{m}_{t}=u_{t}+m_{t}$.

The following proposition formalizes what OLS identifies under 1.

######  Proposition 1.

Under 1, it holds that:

$\displaystyle\gamma^{m}=$ $\displaystyle\gamma\lambda,$ (5)
$\displaystyle\beta^{m}=$
$\displaystyle\beta+\gamma\delta\big{(}1-\lambda\big{)}.$ (6)

With
$\lambda=\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}$,
where $\sigma_{\bullet}^{2}$ denotes the variance of the variable in the
subscript.

All proofs are provided in Appendix A to C. Equation 5 reflects the
attenuation bias in a multivariate setting. $\lambda$ is the multivariate
reliability (or signal-to-total variance) ratio. If $\sigma_{m_{t}}^{2}=0$,
measurement error is absent and the test score delivers a perfect signal on
ability, so that $\lambda=1$. If $\sigma_{m_{t}}^{2}>0$, measurement error is
present and the test score delivers an imperfect signal, so that $\lambda<1$.
As $\sigma_{m_{t}}^{2}$ grows larger, $\lambda$ approaches zero. Hence, the
presence of measurement error implies the estimate $\gamma^{m}$ is attenuated.

Equation 6 clarifies how the attenuation bias in $\gamma^{m}$ contaminates
$\beta^{m}$. If measurement error is absent, $\lambda=1$ and contamination is
also absent, so that $\beta^{m}=\beta$. If measurement error is present,
$\lambda<1$ and $\beta^{m}=\beta+\gamma\delta\big{(}1-\lambda\big{)}$. As
$\lambda$ approaches zero the test does not deliver any signal on ability and
we have that $\beta^{m}=\beta+\gamma\delta$, which is the OLS estimate when
not controlling for test scores. Hence, given the presence of measurement
error, $\beta^{m}$ is larger than $\beta$ when $\gamma$ and $\delta$ are
positive. This bias grows as $\gamma$ and $\delta$ become larger, and
$\lambda$ becomes smaller. See also Pei et al. (2019) for a discussion of the
result in Proposition 1.

###  2.2 The IV strategy

The instrumental variables (IV) strategy uses a second objective measure of
ability in period $t$ as an instrumental variable for the first objective
measure. Intuitively, under 1 the measurement errors of the two objective
measures are uncorrelated, and the second stage IV estimates are consistent
since they only use the variation generated by unobserved ability. In
practice, one may use the scores from a test administered in period $t-1$ as
the second objective measure. This test score, $s_{t-1}^{m}$, also contains
classical measurement error according to 1,

$\displaystyle s_{t-1}^{m}=s_{t-1}+m_{t-1}.$ (7)

Using $s_{t-1}^{m}$ as an instrument for $s_{t}^{m}$ generates the following
first and second stage regression equations respectively,

$\displaystyle s_{t}^{m}$ $\displaystyle=\kappa SES+\pi
s_{t-1}^{m}+e_{t}^{fs},$ (8) $\displaystyle y_{t}$
$\displaystyle=\beta^{iv}SES+\gamma^{iv}\widehat{s}_{t}^{m}+e_{t}^{iv}.$ (9)

To analyze how this IV strategy can address measurement error, it is useful to
also introduce the balancing regression for $t-1$,

$\displaystyle\begin{pmatrix}s_{t-1}\\\
s_{t-1}^{m}\end{pmatrix}=\begin{pmatrix}s_{t-1}\\\
s_{t-1}+m_{t-1}\end{pmatrix}=\begin{pmatrix}\delta_{t-1}\\\
\delta_{t-1}^{m}\end{pmatrix}SES+\begin{pmatrix}u_{t-1}\\\
u^{m}_{t-1}\end{pmatrix},$ (10)

and the variable $\Delta_{s_{t}}=s_{t}-s_{t-1}$, which captures the change in
unobserved ability between time period $t$ and $t-1$. We describe two
additional assumptions that place increasing restrictions on $\Delta_{s_{t}}$.

######  Assumption 2.

The covariance between $\Delta_{s_{t}}$, $SES$ and $s_{t}$ is zero

  1. a.

(SES) $\mathrm{Cov}[\Delta{s_{t}},SES]=0$.

  2. b.

(Ability) $\mathrm{Cov}[\Delta_{s_{t}},s_{t}]=0$.

2(a) requires the change in ability to be the same, on average, across low and
high SES students. 2(b) requires this for low and high ability students in
time $t$.

######  Assumption 3.

Constant $\Delta_{s_{t}}$

  1. (Constant) $\Delta{s_{t}}$ is the same for each student.

3 requires that the change in ability is the same for each student, which is
stronger than 2. The following proposition summarizes what the IV strategy
identifies under only 1 and while combining 1 with either 2 or 3.

######  Proposition 2.

(i) Under 1, it holds that:

$\displaystyle\widetilde{\pi}=$
$\displaystyle\lambda\Bigg{(}\frac{1}{B_{\pi}^{1}}\Bigg{)},$ (11)
$\displaystyle\gamma^{iv}=$ $\displaystyle\gamma B_{\gamma}^{1}B_{\pi}^{1},$
(12) $\displaystyle\beta^{iv}=$
$\displaystyle\beta+\gamma\delta\bigg{(}1-B_{\gamma}^{1}B_{\pi}^{1}\bigg{)}.$
(13)

With
$\widetilde{\pi}=\pi\Bigg{(}\frac{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}$,
$B_{\pi}^{1}=\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\mathrm{Cov}[u_{t},u_{t-1}]}\Bigg{)}$,
and
$B_{\gamma}^{1}=\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}$.

(ii) Under 1 and 2, it holds that:

$\displaystyle\widetilde{\pi}=$ $\displaystyle\lambda,$ (14)
$\displaystyle\gamma^{iv}=$ $\displaystyle\gamma B_{\gamma}^{2},$ (15)
$\displaystyle\beta^{iv}=$
$\displaystyle\beta+\gamma\delta\bigg{(}1-B_{\gamma}^{2}\bigg{)}.$ (16)

With
$\widetilde{\pi}=\pi\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{\Delta_{s_{t}}}^{2}+\sigma_{m_{t-1}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}$
and
$B_{\gamma}^{2}=1-\Bigg{(}\frac{\mathrm{Cov}[e_{t},\Delta_{s_{t}}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}$.

(iii) Under 1 and 3, it holds that:

$\displaystyle\widetilde{\pi}=$ $\displaystyle\lambda,$ (17)
$\displaystyle\gamma^{iv}=$ $\displaystyle\gamma,$ (18)
$\displaystyle\beta^{iv}=$ $\displaystyle\beta.$ (19)

With
$\widetilde{\pi}=\pi\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t-1}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}$.

Proposition 2 formalizes that the IV strategy only addresses measurement error
under 3. This assumption, and the corresponding result, is not new. Gillen et
al. (2019) and Hausman (2001) provide a review of the literature that uses the
IV strategy to address mismeasurement, which dates back to Reiersøl (1941).

Intuitively, a constant $\Delta{s_{t}}$ implies that the test score in period
$t-1$ is truly a second objective measure of ability in period $t$. The first
stage regression of $s_{t}^{m}$ upon $s_{t-1}^{m}$ measures to which extent
this relationship is diluted due to measurement error, so that it reveals the
reliability ratio $\lambda$. This resembles the test-retest method to estimate
test reliability, where the reliability ratio is identified by administering
the same test to the same students within a relatively short period of time
(Guttman, 1945). Furthermore, note that the reduced form equation is similar
to the medium regression equation (3) while replacing $s_{t}^{m}$ with
$s_{t-1}^{m}$,

$\displaystyle y_{t}=\theta_{0}SES+\theta_{1}s_{t-1}^{m}+e_{t}^{iv}.$ (20)

With a constant $\Delta{s_{t}}$ the reduced form coefficient $\theta_{1}$
therefore reveals the estimate $\gamma^{m}=\gamma\lambda$ of the medium
regression, which was attenuated towards zero by the reliability ratio.
Dividing the reduced form by the first stage corrects for this attenuation.
Since $\gamma^{iv}=\gamma$ and $\widehat{s_{t}}^{m}$ still spans the relevant
aspects of $s_{t}^{m}$ in the data, we also have that $\beta^{iv}=\beta$.

###  2.3 The EIV strategy

The errors-in-variables (EIV) strategy uses the results in Proposition 1 to
express the estimates of interest $\gamma$ and $\beta$ as follows,

$\displaystyle\gamma=$ $\displaystyle\frac{\gamma^{m}}{\lambda},$ (21)
$\displaystyle\beta=$
$\displaystyle\beta^{m}-\gamma^{m}\delta\Bigg{(}\frac{1-\lambda}{\lambda}\Bigg{)}.$
(22)

All the elements on the right-hand side are known under 1, except for the
reliability ratio. The EIV strategy then plugs a reliable value for $\lambda$
into (21) and (22) to identify the estimates of interest.

####  2.3.1 The first stage

Case (ii) of Proposition 2 shows that the first stage can be used as an
estimate for $\lambda$ under 2. The intuition for this result follows more
easily in a univariate setting without $SES$ as an independent variable. The
univariate reliability ratio $\lambda^{u}$ is defined by replacing
$\sigma_{u_{t}}^{2}$ in $\lambda$ with $\sigma_{s_{t}}^{2}$. 2(b) implies that
$\mathrm{Cov}[s_{t},s_{t-1}]=\sigma_{s_{t}}^{2}$, so that the univariate first
stage coefficient $\pi^{u}$ is equal to
$\lambda^{u}\Big{(}\frac{\sigma_{s_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{s_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Big{)}$.
If we define $\widetilde{\pi}^{u}$ as the multiplication of $\pi^{u}$ with the
inverse of the second term, which contains the variances of the test scores
instead of the variances of the error terms from the balancing regressions as
in Proposition 2, we have that $\widetilde{\pi}^{u}=\lambda^{u}$. This result
extends to the multivariate setting since together 2(a)-(b) imply that
$\mathrm{Cov}[u_{t},u_{t-1}]=\sigma_{u_{t}}^{2}$ (instead of only
$\mathrm{Cov}[s_{t},s_{t-1}]=\sigma_{s_{t}}^{2}$).

The EIV first stage (EIV FS) strategy uses the first stage as an estimate for
$\lambda$. The following proposition formalizes what EIV FS identifies under
only 1 and under 1 and 2.

######  Proposition 3.

(i) Under 1, it holds that:

$\displaystyle\gamma^{fs}=$ $\displaystyle\gamma B_{\pi}^{1},$ (23)
$\displaystyle\beta^{fs}=$
$\displaystyle\beta+\gamma\delta\bigg{(}1-B_{\pi}^{1}\bigg{)}.$ (24)

(ii) Under 1 and 2, it holds that:

$\displaystyle\gamma^{fs}=$ $\displaystyle\gamma,$ (25)
$\displaystyle\beta^{fs}=$ $\displaystyle\beta.$ (26)

Where $\gamma^{fs}$ and $\beta^{fs}$ are defined by replacing $\lambda$ with
$\widetilde{\pi}$ in (21) and (22).

The EIV FS strategy only requires the weaker 2 since it directly uses the
medium regression estimate. In contrast, the IV strategy uses the estimate
from the reduced form, which is equal to the medium regression estimate only
if $\mathrm{Cov}[e_{t},\Delta_{s_{t}}]=0$. The latter holds when
$\Delta_{s_{t}}$ is constant. Whereas 3 only seems likely when the two tests
are administered shortly after each other, 2 seems generally more plausible.
In particular, under 2 the teachers may pay attention to $\Delta_{s_{t}}$ in
their subjective evaluations, so that it is part of the error term $e_{t}$,
but $\Delta_{s_{t}}$ cannot correlate with the included variable $SES$ and
unobserved ability $s_{t}$.

####  2.3.2 Test score history

The more novel EIV test score history (EIV TH) strategy recovers the
reliability ratio under 1 only, while making use of data on students’ test
score histories. The intuition of this strategy is to mimic the ideal of the
test-retest method, which uses a test in time $t-\epsilon$ close to time $t$
to estimate the reliability ratio. The first stage equation with a test in
$t-\epsilon$ is,

$\displaystyle
s_{t}^{m}=\kappa_{t-\epsilon}SES+\pi_{t-\epsilon}s_{t-\epsilon}^{m}+e_{t}^{fs_{\epsilon}},$
(27)

where we include a time subscript on the coefficients. Proposition 2 shows
that $\widetilde{\pi}_{t-\epsilon}=\lambda$ under 3. While such a test score
close to time $t$ (i.e., $s_{t-\epsilon}^{m}$) may not be available, we may
observe test scores further away from time $t$ (i.e., $s_{t-\tau}^{m}$ for
$\tau\geq 1$). It seems plausible that all observed test scores contain
classical measurement error according to 1. However, the stronger 2 or 3 may
be less likely to hold.

The EIV TH strategy consists of four steps. First, we repeat the first stage
equation with $s_{t-\tau}^{m}$ for all $\tau\geq 1$,

$\displaystyle
s_{t}^{m}=\kappa_{t-\tau}SES+\pi_{t-\tau}s_{t-\tau}^{m}+e_{t}^{fs_{\tau}},\quad\forall\tau\geq
1.$ (28)

It follows from Proposition 2 that under 1 only, $\widetilde{\pi}_{t-\tau}$
can be expressed as,

$\displaystyle\widetilde{\pi}_{t-\tau}=$
$\displaystyle\lambda\Bigg{(}\frac{1}{B_{\pi_{t-\tau}}^{1}}\Bigg{)},\quad\forall\tau\geq
1,$ (29)

with
$B_{\pi_{t-\tau}}^{1}=\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\mathrm{Cov}[u_{t},u_{t-\tau}]}\Bigg{)}$.
Second, we use $\widetilde{\pi}_{t-\tau}$ as a dependent variable in a
forecasting regression equation of the form,

$\displaystyle\widetilde{\pi}_{t-\tau}=$ $\displaystyle
f(t-\tau)+\varepsilon_{t-\tau}.$ (30)

Third, we make a one-period out-of-sample forecast towards $\tau=\epsilon$.
Without forecasting error this equals,

$\displaystyle\widehat{f}(t-\epsilon)=f(t-\epsilon)=\widetilde{\pi}_{t-\epsilon}=\lambda.$
(31)

Fourth, we define $\gamma^{th}$ and $\beta^{th}$ as the parameters that result
after replacing $\lambda$ with $f(t-\epsilon)$ in (21) and (22). Proposition 3
shows that $\gamma^{th}=\gamma$ and $\beta^{th}=\beta$.

Although not explicitly written down as an assumption, the absence of
forecasting error implies that the chosen polynomial for $f(t-\tau)$
accurately captures the changes in $\widetilde{\pi}_{t-\tau}$ over time. This,
in turn, implies that the underlying factors that drive $B_{\pi_{t-\tau}}^{1}$
away from one, which are $\mathrm{Cov}[\Delta{s_{t}},SES]\neq 0$ and
$\mathrm{Cov}[\Delta{s_{t}},s_{t}]\neq 0$, should close over time in a
predictable way.

##  3 Institutional setting and data

###  3.1 The Dutch education system

In the Netherlands, the transition to tracked secondary education takes place
around the age of twelve when students leave the sixth and final grade of
primary education. The secondary school system consists of three main tracks:
(i) a university track (vwo), which has a duration of six years and gives
students access to university education, (ii) a college track (havo), which
has a duration of five years and prepares students for college education, and
(iii) a vocational track (vmbo), which has a duration of four years and serves
as a preparation for vocational education. The vocational track consists of
four sub-tracks, which can also be ordered from more practice-oriented to more
theory-oriented tracks.

The allocation of students to secondary school tracks is based on the track
recommendation provided by the sixth-grade primary school teacher. The
recommendation process consists of two main steps. First, by March of the
sixth grade the teacher provides an initial track recommendation. Each student
then takes a standardized end-of-primary education test in April or May. This
end-of-primary education test also maps into a track recommendation. If this
test-based recommendation is higher than the teacher recommendation, the
teacher should consider an upwards revision (WPO, 2014). Hence, the final
track recommendation may be higher (but not lower) than the initial one. The
teacher track recommendation is binding in the sense that secondary schools
cannot place students in a lower track than indicated by the track
recommendation (WVO, 2014).

An important characteristic of the Dutch education system is that primary
schools are required to implement a student monitoring system. This includes
the requirement that all primary school students take two standardized tests
per school year, from grade one to six. The first test is administered in the
middle of the year around February and the second at the end of the school
year in June. We will refer to these tests as the midterm and end term test,
respectively. Note that the end term test in sixth grade is the end-of-primary
school test. The biannual tests generally focus on two domains: mathematics
and reading. The mathematics domain contains both abstract problems and
contextual problems that describe a concrete task. The reading domain assesses
the ability to understand written texts, including both factual and literary
content.

There are no strict national rules or procedures on which information to use
when formulating the track recommendation. However, the guidelines from the
Ministry of Education (2023a) state that standardized test results provide a
“good starting point”. Consistent with this guideline, a recent survey among
400 primary school teachers shows that 85% of the respondents attach
“considerable” to “a lot of” value to the standardized test results when
formulating track recommendations (DUO, 2023). Teachers are also expected to
make a holistic assessment and may therefore take into account their
subjective judgements on factors such as motivation, attitude towards school,
and classroom behavior. This discretionary power of primary school teachers in
the Dutch track assignment process has been criticized for creating
inconsistency and bias (OECD, 2016).

In this paper we use the initial, instead of the final, track recommendation
as the teacher’s subjective evaluation. The reason is that the initial track
recommendation can be considered the most important. Many secondary schools
already start allocating students to their first-year classes in March using
the initial track recommendations. Secondary schools may face challenges when
changing their allocation by the time the final track recommendation is
provided in May (Inspectorate of Education, 2019). Moreover, in practice, the
initial and final track recommendation are identical for 9 out of 10
students.444For instance, in 2018-19, 41.50% of all students obtained a test-
based recommendation above the initial track recommendation, and 22.96% of
them received an upwards revision (Inspectorate of Education, 2022). This
implies that the initial and final recommendation were different for 9.53% of
the students in this year.

We use the fifth-grade end term test as a measure of ability because it can be
considered the final test score that the teacher observes before formulating
the initial track recommendation. Alternatively, one could use the final track
recommendation as the teacher’s subjective evaluation and the end-of-primary
school test result as the ability measure. This approach is problematic since
the final track recommendation can only be adjusted upwards, introducing
additional sources of measurement error for the end-of-primary school test:
the test is low (high) stakes for students who are (un)satisfied with the
initial recommendation.

###  3.2 Data

We use proprietary administrative data from Statistics Netherlands on primary
school students in the Netherlands. The data on the biannual standardized
tests is gathered by the Netherlands Cohort Study on Education (NCO) and
subsequently made available to Statistics Netherlands. The test score data can
be linked to other administrative data using an anonymized personal ID,
allowing us to also observe the initial teacher track recommendation, parental
income and education, and several other student background characteristics.

The biannual standardized test scores are available for roughly 50% of the
primary schools in the Netherlands. The NCO collected this data from 2018-19
onwards by individually contacting primary schools. Each school had to provide
permission for the data collection, where parents were given the possibility
to raise objections. The NCO only collected the data of schools that made use
of standardized tests from the test developer Cito. With roughly 80% of the
primary schools using Cito across years, this is the largest test provider in
the Netherlands. See Haelermans et al. (2020) for more details on the NCO data
collection.

The complete standardized test score history, from grade one to six, is
available for the students who were enrolled in sixth grade from 2018-19
onwards. Our sample therefore covers four cohorts: students who are in sixth
grade in the school years 2018-19, 2019-20, 2020-21, and 2021-22. For these
four cohorts, the biannual test score data contains information on 318,630
students in 5,887 schools. After merging the biannual test score data with the
data on track recommendations and student background characteristics, we
observe 276,841 students in 5,802 schools. After dropping the students with
missing values for any variable used in our analyses, and with extreme values
for household income (discussed below), our estimation sample includes 148,019
students in 3,295 schools.

###  3.3 Descriptive statistics

Table 1: Descriptive statistics

| (1) | (2) | (3) | (4)
---|---|---|---|---
| Mean | SD | Min | Max
Panel A: Teacher recommendation
$\mathbbm{1}$(recom. $\geq$ upper voc. tr.) | 0.770 | 0.421 | 0.000 | 1.000
$\mathbbm{1}$(recom. $\geq$ college tr.) | 0.497 | 0.500 | 0.000 | 1.000
$\mathbbm{1}$(recom. $\geq$ university tr.) | 0.211 | 0.408 | 0.000 | 1.000
Panel B: SES measures
SES income | 6.439 | 3.772 |  $<$-0.410 |  $>$55.500
$\mathbbm{1}$(SES inc. $\geq$ median) | 0.517 | 0.500 | 0.000 | 1.000
SES years of schooling | 15.975 | 2.334 | 8.000 | 18.000
$\mathbbm{1}$(SES edu. $\geq$ college) | 0.559 | 0.497 | 0.000 | 1.000
Panel C: Test scores
Reading midterm grade 5 | 191.618 | 26.766 |  $<$81.000 |  $>$340.000
Reading end term grade 5 | 197.279 | 26.983 |  $<$84.000 |  $>$354.000
Math midterm grade 5 | 254.142 | 27.142 |  $<$118.000 |  $>$379.000
Math end term grade 5 | 261.959 | 26.085 |  $<$118.000 |  $>$394.000
Avg. midterm grade 5 | 0.005 | 0.900 | -5.177 | 5.077
Avg. end term grade 5 | 0.004 | 0.905 | -5.438 | 5.130
Panel D: Control variables
$\mathbbm{1}$(female) | 0.499 | 0.500 | 0.000 | 1.000
$\mathbbm{1}$(Western imm.) | 0.076 | 0.265 | 0.000 | 1.000
$\mathbbm{1}$(non-Western imm.) | 0.200 | 0.400 | 0.000 | 1.000
Age | 11.590 | 0.647 |  $<$9.250 |  $>$14.160
Observations | 148019

  * Notes: this table shows the descriptive statistics for our estimation sample. Income is measured in €10,000. We translated the schooling levels in the Netherlands to years of schooling as follows: primary school $=$ 8, mbo 1 $=$ 13 , havo or vwo $=$ 14, mbo 2 or 3 $=$ 15, mbo 4 $=$ 16, college or university bachelor $=$ 17, and university master $=$ 18 years. Statistics Netherlands makes a distinction between individuals with a Western and non-Western migration background. Non-Western immigrants have a background from all countries in Africa, Latin America or Asia (including Turkey, excluding Indonesia and Japan), and Western immigrants from the remaining countries. Due to confidentiality reasons we cannot show minima and maxima that are based upon less than 10 observations. Therefore, the minimum (maximum) may contain the smallest (largest) number, such that at least ten observations have values lower (higher) than that number.

Table 1 presents the descriptive statistics for our estimation sample. Panel A
focuses on several measures for the initial track recommendation. Our baseline
outcome variable $y_{t}$ is a dummy variable that equals one if the track
recommendation is at least equal to the college track. Enrollment in the
college or university track is most relevant from a policy perspective, since
it implies the student is tracked towards higher education. Panel A shows that
49.7% of the students receive a track recommendation equal to, or higher than,
the college track.

Panel B shows the statistics for four different measures of SES. Our baseline
measure of SES will be gross annual household income, which is €64,390 on
average. Household income is right-skewed with a maximum (minimum) of €550,000
(€-4,100). We dropped observations with a yearly household income above
(below) the 99.9th (0.1st) percentile. To facilitate interpretation, for our
analysis below we standardize household income per cohort. In our robustness
checks we use three alternative SES measures. First, we use parental years of
schooling of the highest educated parent, which is 16 years on average.
Moreover, both the income and education measure are binarized. The binary
income measure equals one if household income is above the median of a
student’s cohort, while the binary education measure equals one if the highest
educated parent holds at least a college degree.

Panel C shows the statistics for the midterm and end term test in fifth grade,
per domain and the average across domains. The latter is calculated by first
standardizing the domain-specific test scores per cohort and subsequently
taking the average. We use this average score on the fifth-grade end term test
as our main measure of ability $s_{t}$. We address measurement error, among
others, by using the average score on the fifth-grade midterm test as our
measure for $s_{t-1}$. Table 5 in Appendix D shows the descriptive statistics
on the midterm and end term tests from grades two to four, which form our
measures for $s_{t-\tau}$ with $\tau>1$. We do not use the data from grade one
since the biannual reading test is not observed in that grade.

While our baseline analysis does not include any control variables, in
robustness tests we control for gender, migration background, age, and cohort
dummies. Panel D shows that 49.9% of the students are female, 7.6% (20%) of
the students have a Western (non-Western) immigration background, and students
are on average 11.6 years old in March of the sixth grade.

##  4 Results

###  4.1 The OLS strategy

The baseline OLS estimates are presented in Table 2. Column (1) shows the OLS
estimates of the medium regression equation (3). Consistent with previous
evidence from the Netherlands (Inspectorate of Education, 2016; Timmermans et
al., 2018; Zumbuehl et al., 2022), the findings show that students with higher
income parents are more likely to receive at least a college track
recommendation, conditional on the fifth-grade end term test. The estimate for
$\beta^{m}$ indicates that a one standard deviation increase in parental
income is associated with a 2.8 percentage point increase in receiving at
least a college track recommendation.

As expected, column (1) further shows that test scores are positively
associated with teacher track recommendations. This estimate is attenuated in
the presence of measurement error. Whether and how this contaminates the
estimate for the conditional SES gap depends upon the results of the balancing
regression equation (4), which is presented in column (2). The OLS estimate
for $\delta$ is positive, meaning that parental income is positively
associated with the fifth-grade end term test score. We can therefore conclude
that our estimate for $\gamma\delta$ is positive and that, in the presence of
measurement error, OLS overestimates the conditional SES gap.

Table 2: OLS estimates

| (1) | (2)
---|---|---
|  $\mathbbm{1}$(recom. $\geq$ | End term
| college tr.) | grade 5
SES income | 0.028*** | 0.229***
| (0.001) | (0.004)
End term grade 5 | 0.381*** |
| (0.001) |
Observations | 148019 | 148019
$R^{2}$ | 0.498 | 0.064

  * Notes: ***, **, * refers to statistical significance at the 1, 5, and 10% level. The numbers represent the estimated coefficients and the numbers in parentheses the corresponding standard errors. Coefficients are estimated with OLS. Standard errors are clustered on the school id.

Our baseline models do not include any controls other than the fifth-grade end
term test score. Table 6 in Appendix D shows OLS estimates of the conditional
SES gap while including additional control variables: gender, migration
dummies, age, and cohort fixed effects. The results show that the additional
controls hardly affect the conditional SES gap: the estimate changes from
0.028 to 0.029. This may be expected as many of the controls are not
associated with SES. However, this is not the case for migration background,
which is strongly correlated with SES.555Note that the results in Table 6
indicate that students with a migration background are more likely to receive
at least a college track recommendation, conditional on SES and test scores.
This is not inconsistent with the literature, which reports mixed evidence on
teacher bias by migration status (Batruch et al., 2023). In general, these
results suggest that our main estimates are rather robust to controlling for
additional variables typically observed in administrative data.

###  4.2 The IV strategy

Table 3 shows the results of the IV strategy, which uses the fifth-grade
midterm test as an instrument for the fifth-grade end term test. The first
stage in column (1) shows that the coefficient estimate of the midterm test is
equal to $0.887$. This estimate measures the reliability ratio after
multiplication with the fraction containing the variances of the error terms
from the balancing regression.666As shown in Proposition 2, under 2 this
fraction can be written as
$(\sigma_{u_{t}}^{2}+\sigma_{\Delta_{s_{t}}}^{2}+\sigma_{m_{t-1}}^{2})\big{/}(\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2})$.
Hence, whether this fraction is bigger or smaller than one depends on the
variance of the change in ability and the change in the variance of
measurement error over time. Hence, the first stage estimate for the
reliability ratio is $0.887\times 0.992=0.880$. The presence of measurement
error implies that the OLS estimate of $0.028$ overestimates the conditional
SES gap.

Table 3: IV estimates

| (1) | (2) | (3)
---|---|---|---
| End term |  $\mathbbm{1}$(recom. $\geq$ |  $\mathbbm{1}$(recom. $\geq$
| grade 5 | college tr.) | college tr.)
Midterm grade 5 | 0.887*** | 0.383*** |
| (0.002) | (0.001) |
SES income | 0.031*** | 0.030*** | 0.016***
| (0.001) | (0.001) | (0.001)
End term grade 5 |  |  | 0.432***
|  |  | (0.002)
Observations | 148019 | 148019 | 148019
$R^{2}$ | 0.794 | 0.500 | 0.490
$(\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2})\big{/}(\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2})$ | 0.992 |  |

  * Notes: ***, **, * refers to statistical significance at the 1, 5, and 10% level. The numbers represent the estimated coefficients and the numbers in parentheses the corresponding standard errors. Coefficients in column (1) and (2) are estimated with OLS and in column (3) with 2SLS. Standard errors are clustered on the school id.

The IV strategy addresses measurement error by using the first stage to
correct the reduced form estimate on the midterm term test shown in column
(2). In particular, the second stage estimate on the end term test in column
(3) can be calculated by dividing the reduced form by the first stage,
$\frac{0.383}{0.887}=0.432$. If the second stage estimate $\gamma^{iv}$ is
equal to $\gamma$, there cannot be any contamination bias, and the second
stage estimate $\beta^{iv}$ identifies the conditional SES gap $\beta$. The IV
estimate for the conditional SES gap is equal to 0.016, which implies that a
one standard deviation increase in parental income is associated with a 1.6
percentage point increase in receiving at least a college track
recommendation.

###  4.3 The EIV strategy

The EIV strategy requires a value for the reliability ratio of the end term
test to address measurement error. From the IV strategy we know that the first
stage estimate for the reliability ratio is 0.880. Reliability ratios near
0.90 are considered high (Feld and Zölitz, 2017). Depending on the topic, the
Graduate Management Admission Test (GMAT) is advertised to have a reliability
ratio between 0.89 and 0.90 (Graduate Management Admission Council, 2023), the
Graduate Record Examination (GRE) between 0.87 and 0.95 (Educational Testing
Service, 2023), and the Scholastic Assessment Test (SAT) between 0.89 and 0.93
(College Board, 2013).

The EIV FS strategy uses the first stage estimate for the reliability ratio to
directly estimate $\gamma$ and $\beta$. In particular, with the reliability
ratio equal to $0.880$ the OLS estimate on the fifth-grade end term test
$\gamma^{m}$ is attenuated by a factor of $0.880$. Hence, the EIV FS estimate
$\gamma^{fs}$ is equal to $\frac{0.381}{0.880}=0.433$. This attenuation bias
contaminated the OLS estimate for $\beta^{m}$ from Table 2. Accordingly, the
conditional SES gap is overestimated by a value of
$\gamma\delta(1-\lambda)=0.433\times 0.229\times(1-0.880)=0.012$. The EIV
strategy subtracts this value from $\beta^{m}$, and so the estimate for
$\beta^{fs}$ is equal to $0.028-0.012=0.016$.

Figure 1: Estimate for the reliability ratio using students’ test score
histories Notes: this figure plots the estimates for
$\widetilde{\pi}_{t-\tau}$ and the forecasting regression equation to estimate
$\lambda=f(t-\epsilon)$. We use OLS to estimate $\pi_{t-\tau}$ from (28), and
the variance of the error term from (4), to obtain $\widetilde{\pi}_{t-\tau}$
for each $\tau$. The forecasting regression equation (30) is specified as a
polynomial of degree one, estimated with OLS, and then used to obtain the one-
period out-of-sample forecast $f(t-\epsilon)$.

An alternative estimate for the reliability ratio can be obtained by using the
biannual test score data throughout grade two to five. Figure 1 presents the
estimates from the first stage regressions that separately use the midterm and
end term tests from grade two to five as independent variable. Instead of
using the rightmost estimate on the fifth-grade midterm test as the
reliability ratio, the EIV TH approach relies on the first stage estimates of
all previous test scores to predict the first stage estimate of a test score
very close to the fifth-grade end term test. We use a linear polynomial for
the forecasting regression equation (30). With an $R^{2}$ of $0.981$ this fits
the development of the first stage estimates relatively well. The estimate for
the reliability ratio is the one-period out-of-sample forecast and is equal to
$0.899$. The EIV TH strategy produces an estimate for the fifth-grade end term
test of 0.424 and an estimate for the conditional SES gap of $0.018$.

###  4.4 Comparing the results

Figure 2: OLS, IV, and EIV estimates Notes: this figure shows the estimated
coefficients for the OLS, IV, and EIV strategy. The lines are 95% confidence
intervals. The estimates and standard errors for the OLS and IV strategy can
be found in Table 2 and 3 respectively. The estimates and standard errors for
the EIV FS are obtained via a stacking approach. We triplicate the data and
estimate (3), (4), and (8) with OLS via a single regression equation. We
cluster on the school id that is similarly triplicated. The EIV TH strategy
uses the same stacking approach to estimate (3) and (4). See Figure 1 for
details on the estimation of the one-period out-of-sample forecast
$f(t-\epsilon)$. To obtain standard errors, we assume that the estimate for
$f(t-\epsilon)$ is fixed.

Figure 2 shows the estimates with 95% confidence intervals for the OLS
strategy together with all three strategies that aim to address measurement
error. The left figure presents the estimates for the end term test and
visualizes the attenuation bias of OLS. The IV and EIV FS strategy inflate the
test score estimate by a comparable amount, since the reduced form estimate on
the midterm test scores ($0.383\times 0.992=0.380$) is similar to the medium
regression estimate on the end term test scores ($0.381$). The right figure
presents the estimates for the conditional SES gap. Similar attenuation bias
translates into a similar contamination bias, and so the IV and EIV FS
strategy also produce essentially the same estimate for the conditional SES
gap. Since the estimate for the reliability ratio is somewhat higher when
using the one-period out-of-sample forecast instead of the first stage
($0.899$ versus $0.880$), the attenuation and contamination bias are somewhat
smaller for the EIV TH strategy.

Table 4 summarizes how much of the conditional SES gap in track
recommendations can be explained by measurement error in test scores across
the three strategies. Column (1) shows that the IV strategy estimates that
42.14% of the SES gap is due to measurement error. Column (2) and (3) show
that the EIV strategies estimate this percentage between 35.13 and 42.78,
depending on the method used to estimate the reliability ratio. In general,
the results show that mitigating contamination bias matters substantially and
that the three strategies addressing measurement yield similar results.

Table 4: Percentage change in the conditional SES gap

| (1) | (2) | (3)
---|---|---|---
| $\beta^{\bullet}=\beta^{iv}$ | $\beta^{\bullet}=\beta^{fs}$ | $\beta^{\bullet}=\beta^{th}$
$\Bigg{(}\frac{\beta^{m}-\beta^{\bullet}}{\beta^{m}}\Bigg{)}\times 100$ |  42.14% |  42.78% |  35.13%

  * Notes: the numbers represent the estimated values for$\big{(}\frac{\beta^{m}-\beta^{\bullet}}{\beta^{m}}\big{)}\times 100$. The estimate for $\beta^{m}$ is shown in Table 2, for $\beta^{iv}$ in Table 3, and for $\beta^{fs}$ and $\beta^{th}$ in Figure 2.

###  4.5 Testing the assumptions

Since measurement error and ability are unobserved, we are unaware of testable
implications for 1 and 2(b). However, 2(a) and 3 do have immediate testable
implications. Define $\Delta_{m_{t}}=m_{t}-m_{t-1}$, so that
$s_{t}^{m}-s_{t-1}^{m}=\Delta_{s_{t}}+\Delta_{m_{t}}=\Delta_{s^{m}_{t}}$. Let
$\alpha_{x}=\big{(}\frac{\mathrm{Cov}[\Delta{s^{m}_{t}},x]}{\sigma_{x}^{2}}\big{)}$
be the OLS estimator of a model that regresses $\Delta_{s^{m}_{t}}$ upon a
single student characteristic $x$. Classical measurement error on the left-
hand side does not affect the OLS estimate, and so 2(a) requires that
$\alpha_{SES}$ is zero. In turn, 3 requires that $\alpha_{x}$ is zero for
every student characteristic $x$, including $SES$.777Using test scores
$s^{m}_{t}$ as student characteristic $x$, instead of unobserved ability
$s_{t}$, may produce an overestimate for $\alpha_{s_{t}}$ and thus a
misleading test for 2(b). To see this, we specify the following equations,
$\displaystyle\begin{pmatrix}\Delta_{s^{m}_{t}}\\\
\Delta_{s^{m}_{t}}\end{pmatrix}=\begin{pmatrix}\alpha_{s_{t}}\\\
\alpha^{m}_{s_{t}}\end{pmatrix}\begin{pmatrix}s_{t}\\\
s_{t}^{m}\end{pmatrix}+\begin{pmatrix}v_{t}\\\ v_{t}^{m}\end{pmatrix}.$ It can
be shown that the OLS estimate for
$\alpha^{m}_{s_{t}}=1+\bigg{(}\alpha_{s_{t}}-1\bigg{)}\lambda^{u}$, where
$\lambda^{u}=\Bigg{(}\frac{\sigma_{s_{t}}^{2}}{\sigma_{s_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}$
is the univariate reliability ratio. Given $\lambda^{u}<1$, we have that
$\alpha^{m}_{s_{t}}>0$ even if $\alpha_{s_{t}}=0$. The intuition for this
positive bias is that the same measurement error $m_{t}$ enters both the left-
hand and right-hand side of the equation.

Table 7 in Appendix D shows OLS estimates of the change in test scores on SES
and all additional control variables. Column (1) shows that the estimate of
SES is statistically significant at the 1% level. Although we can reject 2(a),
the SES estimate of 0.005 seems economically small. In particular, the
estimate is reduced by 97.82% compared to the SES estimate from the regression
of the fifth-grade end term test score on SES in Table 2. Column (2) to (5)
further show that the gender dummy, the immigrant dummies, and age also
correlate with the change in test scores at the 1% level. The immigrant
dummies seem of non-negligible size. These results cast strong doubt on 3. In
general, 3 seems unlikely in settings where the difference between time period
$t$ and $t-1$ is large, especially if the mismeasured variable is highly
dynamic.

Though it is difficult to empirically test 2(b), one may discuss its
implications for the change in ability over time. In particular, if
$\Delta_{s_{t}}$ is uncorrelated with $s_{t}$, the following two equalities on
the process $s_{t}=s_{t-1}+\Delta_{s_{t}}$ directly follow:

$\displaystyle-\sigma_{\Delta{s_{t}}}^{2}=$
$\displaystyle\mathrm{Cov}[\Delta_{s_{t}},s_{t-1}],$ (32)
$\displaystyle\sigma_{\Delta{s_{t}}}^{2}=$
$\displaystyle\sigma_{s_{t-1}}^{2}-\sigma_{s_{t}}^{2}.$ (33)

Low ability students in $t-1$ must have had larger positive changes in ability
than high ability students in $t-1$ (from (32)) and the variance of ability
must be larger in $t-1$ than in $t$ (from (33)). Whether these equalities are
likely depends on the setting at hand. In an education context, (32) and (33)
would be consistent with diminishing marginal returns to ability.

Although not explicitly written down as an assumption, using students’ test
score histories to estimate the reliability ratio requires that we can
accurately capture the changes in the first stage estimate over time. Figure 1
shows this is the case, as the linear polynomial captures 98.1% of the
variance in the first stage estimate over time. This also suggests that the
underlying factors that drive the first stage away from the reliability ratio,
which are $\mathrm{Cov}[\Delta{s_{t}},SES]\neq 0$ and
$\mathrm{Cov}[\Delta{s_{t}},s_{t}]\neq 0$, close over time in a predictable
way. Figure 3 in Appendix D plots OLS estimates of $\Delta{s^{m}_{t}}$ on
$SES$ over time, and shows that $\mathrm{Cov}[\Delta{s^{m}_{t}},SES]$ indeed
approaches zero in a near linear fashion.

###  4.6 Explaining the difference in results

The results above suggest that both 2 and 3 are violated. Can such violations
explain the relatively small difference in results across the various
strategies? To analyze this question, recall that the EIV TH strategy
identifies the true estimates under 1 only. Our empirical results can be
explained by violations of 2 and 3 with $B_{\gamma}^{1}\approx 1$ and
$B_{\pi}^{1}>1$, as this generates:

$\displaystyle\gamma^{m}<\gamma^{th}=\gamma<\gamma^{iv}\approx\gamma^{fs},$
(34) $\displaystyle\beta^{fs}\approx\beta^{iv}<\beta^{th}=\beta<\beta^{m}.$
(35)

First, Proposition 1 shows that OLS underestimates $\gamma$, and in turn
overestimates $\beta$, given the presence of classical measurement error.
Second, case (i) of Proposition 2 shows that the IV strategy overestimates
$\gamma$, and in turn underestimates $\beta$, if $B_{\gamma}^{1}\approx 1$ and
$B_{\pi}^{1}>1$. The reason is that the reduced form will be equal to the
medium regression estimate $\gamma^{m}=\gamma\lambda$ if
$B_{\gamma}^{1}\approx 1$, but the first stage will be an underestimate for
the reliability ratio $\lambda$ if $B_{\pi}^{1}>1$. Note that
$B_{\gamma}^{1}\approx 1$ if
$\mathrm{Cov}[y_{t},u_{t}]\approx\mathrm{Cov}[y_{t},u_{t-1}]$ and
$B_{\pi}^{1}>1$ if $\sigma_{u_{t}}^{2}>\mathrm{Cov}[u_{t},u_{t-1}]$. The
latter may happen if for instance 2(b) is violated, so that
$\sigma_{u_{t}}^{2}=\mathrm{Cov}[u_{t},u_{t-1}]+\mathrm{Cov}[\Delta_{s_{t}},s_{t}]$,
and $\mathrm{Cov}[\Delta_{s_{t}},s_{t}]>0$. This implies that high ability
students in $t$ must have had larger positive changes in ability than low
ability students in $t$.

Third, case (i) of Proposition 3 shows that the EIV FS strategy also
overestimates $\gamma$, and in turn underestimates $\beta$, if
$B_{\pi}^{1}>1$. A comparison between case (i) of Proposition 2 and 3 shows
that the bias to the EIV FS and IV strategy are similar if
$B_{\gamma}^{1}\approx 1$. The reason is that in this case the medium
regression estimate, which is used by the EIV FS strategy, is similar to the
reduced form estimate, which is used by the IV strategy.

Our empirical results can be interpreted in this light. Both the IV and EIV FS
strategy use the first stage estimate of $0.880$ for the reliability ratio.
The IV strategy uses this to correct the reduced form estimate of $0.380$, and
produces $\widehat{\gamma}^{iv}=\frac{0.380}{0.880}=0.432$. In contrast, the
EIV FS strategy uses this to correct the medium regression estimate of
$0.381$, and produces $\widehat{\gamma}^{fs}=\frac{0.381}{0.880}=0.433$. As
the medium and reduced form regression estimates are similar, the results for
the IV and EIV FS are similar. However, the correction for the attenuation
bias by both strategies is slightly too large, since with data on students’
test score histories we estimate a somewhat larger reliability ratio of
$0.892$. Using this to correct the medium regression estimate produces
$\widehat{\gamma}^{th}=\frac{0.381}{0.899}=0.424$.

###  4.7 Robustness checks

We test the robustness of our results in four ways. First, Figure 4 in
Appendix D shows the results when using two alternative measures for teacher
recommendation: a dummy that equals one if the teacher recommendation is equal
to or higher than the theory-oriented vocational track (Figure 4(a)) or the
university track (Figure 4(b)). Second, Figure 5 in Appendix D shows the
results when using three alternative SES measures: a dummy that equals one if
parental income is above the median of a student’s cohort (Figure 5(a)), years
of schooling of the highest educated parent (Figure 5(b)), and a dummy that
equals one if the highest educated parent completed at least college education
(Figure 5(c)). Third, Figure 6 in Appendix D shows the results when separately
estimating the models per cohort (Figure 6(a)-(d)). All results are similar to
our baseline results: the pattern of the estimates is identical in each figure
and measurement error explains a sizeable share of the conditional SES gap.

Fourth, as SES is strongly correlated with migration background, a potential
concern is that the conditional SES gap at least partly reflects a gap by
migration background. To test whether this is the case, we estimated our
baseline models for immigrants and natives separately. Figure 7 in Appendix D
shows that the SES estimates are similar in both subgroups. This is consistent
with our OLS estimates in Table 6 of Appendix D that show the conditional SES
gap is hardly affected by controlling for migration background.

##  5 Conclusion

This paper examines the role of measurement error in explaining conditional
gaps in teacher evaluations. Teacher bias is generally measured by estimating
gaps in subjective teacher evaluations, while conditioning on objective
measures of student abilities. A major challenge in the identification of
teacher bias is the presence of measurement error in test scores that are
supposed to reflect student abilities. We discuss three different strategies
to address measurement error and explain under which assumptions these
strategies are valid. Whereas an instrumental variable approach is more
commonly used in applied econometric studies, we show that an errors-in-
variables approach requires weaker assumptions to address measurement error
and contamination bias.

The empirical analysis focuses on conditional SES gaps in teacher track
recommendations in the Netherlands. This is a setting where potential teacher
bias is highly consequential and findings on conditional gaps with respect to
parental SES are well established in the literature. Using recent
administrative data covering about half of the Dutch primary schools, we find
that measurement error in test scores can explain a substantial portion of the
conditional SES gap, between 35 and 43%.

It may seem striking that such a high share of the conditional SES gap can be
explained by measurement error, especially since the test scores we use in our
analyses are highly reliable according to conventional norms. However, when
student abilities are strongly correlated with SES and matter for teacher
track recommendations, the magnitude of contamination bias may be nontrivial -
even when tests are highly reliable. While our findings show that teacher bias
may be smaller than previous studies report, the strong correlation between
SES and ability does suggest large inequalities of opportunity before and
during primary education.

Our findings also have important implications for the larger literature
studying bias and discrimination. Studies often suggest that systematic
differences in outcomes across groups (e.g., student grades, job promotion,
wages) conditional on control variables (e.g., test scores, education,
experience) indicate bias. However, this claim is invalid in the presence of
measurement error in the controls. It is therefore key that studies testing
the presence of systematic differences in outcomes properly address
measurement error in control variables.

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##  Appendix A Proof Proposition 1

Using 1, we can write the balancing regression equation (4) as

$\displaystyle\begin{pmatrix}s_{t}\\\
s_{t}^{m}\end{pmatrix}=\begin{pmatrix}s_{t}\\\ s_{t}+m_{t}\end{pmatrix}=\delta
SES+\begin{pmatrix}u_{t}\\\ u_{t}+m_{t}\end{pmatrix}.$ (36)

Applying the Frisch-Waugh logic, we can now derive the expression for
$\gamma^{m}$ as follows,

$\displaystyle\gamma^{m}=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[y_{t},u^{m}_{t}]}{\mathrm{Var}[u^{m}_{t}]}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t}+m_{t}]}{\mathrm{Var}[u_{t}+m_{t}]}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t}]}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}$
(37) $\displaystyle=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t}]}{\sigma_{u_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}$
$\displaystyle=$ $\displaystyle\gamma\lambda,$

where
$\lambda=\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}$.

To derive the expression for $\beta^{m}$, we first introduce the short
regression equation:

$\displaystyle y_{t}=\beta^{s}SES+e_{t}^{s}.$ (38)

Next, we use the omitted variable bias formula for $\beta^{s}$ to write

$\displaystyle\beta^{m}+\gamma^{m}\delta^{m}=\beta+\gamma\delta.$ (39)

Using that $\delta^{m}=\delta$ and substituting for
$\gamma^{m}=\gamma\lambda$, results in

$\displaystyle\beta^{m}=$
$\displaystyle\beta+\gamma\delta\bigg{(}1-\lambda\bigg{)}.$ (40)

##  Appendix B Proof Proposition 2

###  B.1 Case (i)

We start by rewriting the second stage equation (9) to obtain the reduced form
regression equation,

$\displaystyle y_{t}=\theta_{0}SES+\theta_{1}s_{t-1}^{m}+e_{t}^{iv},$ (41)

where $\frac{\theta_{1}}{\pi}=\frac{\gamma^{iv}\pi}{\pi}=\gamma^{iv}$.

Using 1, we can write the balancing regression equation (10) as

$\displaystyle\begin{pmatrix}s_{t-1}\\\
s_{t-1}^{m}\end{pmatrix}=\begin{pmatrix}s_{t-1}\\\
s_{t-1}+m_{t-1}\end{pmatrix}=\delta_{t-1}SES+\begin{pmatrix}u_{t-1}\\\
u_{t-1}+m_{t-1}\end{pmatrix},$ (42)

Next we derive the expression for the first stage coefficient $\pi$. Applying
the Frisch-Waugh logic, results in

$\displaystyle\pi=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[s_{t}^{m},u^{m}_{t-1}]}{\mathrm{Var}[u^{m}_{t-1}]}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[s_{t}+m_{t},u_{t-1}+m_{t-1}]}{\mathrm{Var}[u_{t-1}+m_{t-1}]}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[s_{t},u_{t-1}]}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$
(43) $\displaystyle=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[s_{t},u_{t-1}]}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$
$\displaystyle=$
$\displaystyle\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\mathrm{Cov}[s_{t},u_{t-1}]}{\sigma_{u_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$
$\displaystyle=$
$\displaystyle\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\mathrm{Cov}[\delta
SES+u_{t},u_{t-1}]}{\sigma_{u_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$
$\displaystyle=$
$\displaystyle\lambda\Bigg{(}\frac{\mathrm{Cov}[u_{t},u_{t-1}]}{\sigma_{u_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}.$

We continue with the reduced form coefficient $\theta_{1}$. Applying the
Frisch-Waugh logic, results in

$\displaystyle\theta_{1}=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[y_{t},u^{m}_{t-1}]}{\mathrm{Var}[u^{m}_{t-1}]}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}+m_{t-1}]}{\mathrm{Var}[u_{t-1}+m_{t-1}]}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$
(44) $\displaystyle=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\sigma_{u_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$
$\displaystyle=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t}]}{\sigma_{u_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$
$\displaystyle=$
$\displaystyle\gamma\lambda\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}.$

We finish with the second stage estimates $\gamma^{iv}$ and $\beta^{iv}$.
Combining (43) and (44), results in

$\displaystyle\gamma^{iv}=$
$\displaystyle\frac{\theta_{1}}{\pi}=\gamma\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\mathrm{Cov}[u_{t},u_{t-1}]}\Bigg{)}.$
(45)

To derive an expression for $\beta^{iv}$ we first introduce the balancing
regression equation for $\widehat{s}_{t}^{m}$,

$\displaystyle\begin{pmatrix}s_{t}^{m}\\\
\widehat{s}_{t}^{m}\end{pmatrix}=\begin{pmatrix}s_{t}^{m}\\\
s_{t}^{m}-e_{t}^{fs}\end{pmatrix}=\begin{pmatrix}\delta^{m}\\\
\delta^{iv}\end{pmatrix}SES+\begin{pmatrix}u^{m}_{t}\\\
u^{iv}_{t}\end{pmatrix}.$ (46)

We have that $\delta^{iv}=\delta^{m}$,

$\displaystyle\delta^{iv}=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[\widehat{s}_{t}^{m},SES]}{\mathrm{Var}[SES]}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[s_{t}^{m}-e_{t}^{fs},SES]}{\sigma_{SES}^{2}}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[s_{t}^{m},SES]}{\sigma_{SES}^{2}}\Bigg{)}$
(47) $\displaystyle=$ $\displaystyle\delta^{m},$

and under 1 we have that $\delta^{iv}=\delta^{m}=\delta$. We use the omitted
variable bias formula for $\beta^{s}$ from (38) to write

$\displaystyle\beta^{iv}+\gamma^{iv}\delta^{iv}=$
$\displaystyle\beta+\gamma\delta.$ (48)

Using $\delta^{iv}=\delta$ and (45) it follows that,

$\displaystyle\beta^{iv}=$
$\displaystyle\beta+\gamma\delta\Bigg{(}1-\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}}{\mathrm{Cov}[u_{t},u_{t-1}]}\Bigg{)}\Bigg{)}.$
(49)

###  B.2 Case (ii)

First we use 2 to find a link between $u_{t-1}$ and $u_{t}$. We can write,

$\displaystyle\delta=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[s_{t},SES]}{\mathrm{Var}[SES]}\Bigg{)}=\Bigg{(}\frac{\mathrm{Cov}[s_{t-1}+\Delta_{s_{t}},SES]}{\sigma_{SES}^{2}}\Bigg{)}$
(50) $\displaystyle=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[s_{t-1},SES]}{\sigma_{SES}^{2}}\Bigg{)}+\Bigg{(}\frac{\mathrm{Cov}[\Delta_{s_{t}},SES]}{\sigma_{SES}^{2}}\Bigg{)},$
$\displaystyle=$ $\displaystyle\delta_{t-1}+\alpha,$

with
$\alpha=\Bigg{(}\frac{\mathrm{Cov}[\Delta_{s_{t}},SES]}{\sigma_{SES}^{2}}\Bigg{)}$.
2(a) guarantees that $\delta=\delta_{t-1}$. This allows us to write (4) and
(10) as

$\displaystyle\begin{pmatrix}s_{t}\\\
s_{t-1}\end{pmatrix}=\begin{pmatrix}s_{t}\\\
s_{t}-\Delta{s_{t}}\end{pmatrix}=\delta SES+\begin{pmatrix}u_{t}\\\
u_{t}-\big{(}\Delta_{s_{t}}-\overline{\Delta}_{s_{t}}\big{)}\end{pmatrix}.$
(51)

The mean of $\Delta_{s_{t}}$, denoted by $\overline{\Delta}_{s_{t}}$, will be
absorbed by the intercept of (10), so that
$u_{t-1}=u_{t}-\big{(}\Delta_{s_{t}}-\overline{\Delta}_{s_{t}}\big{)}$. We use
this to write
$\mathrm{Cov}[u_{t},u_{t-1}]=\sigma_{u_{t}}^{2}-\mathrm{Cov}[u_{t},\Delta_{s_{t}}]$.
Subsequently, 2(a)-(b) together guarantee that
$\mathrm{Cov}[u_{t},\Delta_{s_{t}}]=\mathrm{Cov}[s_{t}-\delta
SES,\Delta_{s_{t}}]=0$. Thus 2 allows us to write
$\mathrm{Cov}[u_{t},u_{t-1}]=\sigma_{u_{t}}^{2}$.

Case (i) assumes 1 and case (ii) assumes both 1 and 2. Hence, we use the
results above derived under 2 to simplify the expressions in case (i).

We start with the first stage coefficient $\pi$ derived in (43). We note that
the term
$\Bigg{(}\frac{\mathrm{Cov}[u_{t},u_{t-1}]}{\sigma_{u_{t}}^{2}}\Bigg{)}$
equals 1 since $\mathrm{Cov}[u_{t},u_{t-1}]=\sigma_{u_{t}}^{2}$. 2 further
allows to substitute for
$\sigma_{u_{t-1}}^{2}=\mathrm{Var}[u_{t}-\big{(}\Delta_{s_{t}}-\overline{\Delta}_{s_{t}}\big{)}]=\mathrm{Var}[u_{t}-\Delta_{s_{t}}]=\sigma_{u_{t}}^{2}+\sigma_{\Delta_{s_{t}}}^{2}-2\times\mathrm{cov}[u_{t},\Delta_{s_{t}}]=\sigma_{u_{t}}^{2}+\sigma_{\Delta_{s_{t}}}^{2}-2\times\mathrm{cov}[s_{t}-\delta
SES,\Delta_{s_{t}}]=\sigma_{u_{t}}^{2}+\sigma_{\Delta_{s_{t}}}^{2}$ in the
term
$\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$.
This allow us to write (43) as follows,

$\displaystyle\pi=$
$\displaystyle\lambda\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{\Delta_{s_{t}}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}.$
(52)

We continue with the reduced form coefficient $\theta_{1}$ derived in (44). We
substitute for
$u_{t-1}=u_{t}-\big{(}\Delta_{s_{t}}-\overline{\Delta}_{s_{t}}\big{)}$ in the
term
$\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}$.
It follows that,

$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}=$
$\displaystyle\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}-\Bigg{(}\frac{\mathrm{Cov}[y_{t},\Delta_{s_{t}}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}$
(53) $\displaystyle=$ $\displaystyle 1-\Bigg{(}\frac{\mathrm{Cov}[\beta
SES+\gamma s_{t}+e_{t},\Delta_{s_{t}}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}$
$\displaystyle=$ $\displaystyle
1-\Bigg{(}\frac{\mathrm{Cov}[e_{t},\Delta_{s_{t}}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}.$

The results above allow us to write (44) as follows,

$\displaystyle\theta_{1}=$
$\displaystyle\gamma\lambda\Bigg{(}1-\Bigg{(}\frac{\mathrm{Cov}[e_{t},\Delta_{s_{t}}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}\Bigg{)}\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{\Delta_{s_{t}}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}.$
(54)

We finish with the second stage estimates $\gamma^{iv}$ and $\beta^{iv}$.
Combining (52) and (54) results in

$\displaystyle\gamma^{iv}=$
$\displaystyle\frac{\theta_{1}}{\pi}=\gamma\Bigg{(}1-\Bigg{(}\frac{\mathrm{Cov}[e_{t},\Delta_{s_{t}}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}\Bigg{)}.$
(55)

Using the results above, it follows from (49) that,

$\displaystyle\beta^{iv}=$
$\displaystyle\beta+\gamma\delta\Bigg{(}\frac{\mathrm{Cov}[e_{t},\Delta_{s_{t}}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}.$
(56)

###  B.3 Case (iii)

First we use 3 to find a link between $u_{t-1}$ and $u_{t}$. Similar to (50),
we write

$\displaystyle\delta=$ $\displaystyle\delta_{t-1}+\alpha,$ (57)

with
$\alpha=\Bigg{(}\frac{\mathrm{Cov}[\Delta_{s_{t}},SES]}{\sigma_{SES}^{2}}\Bigg{)}$.
3 guarantees again that $\delta=\delta_{t-1}$. Moreover, we can write (4) and
(10) as

$\displaystyle\begin{pmatrix}s_{t}\\\
s_{t-1}\end{pmatrix}=\begin{pmatrix}s_{t}\\\
s_{t}-\Delta{s_{t}}\end{pmatrix}=\delta SES+\begin{pmatrix}u_{t}\\\
u_{t}\end{pmatrix},$ (58)

where the constant $\Delta{s_{t}}$ is completely absorbed by the intercept of
(10), so that $u_{t-1}=u_{t}$.

Case (i) assumes 1 and case (iii) assumes both 1 and 3. Hence, we use the
results above derived under 3 to simplify the expressions in case (i).

We start with the first stage coefficient $\pi$ derived in (43). We substitute
for $u_{t-1}=u_{t}$ in
$\Bigg{(}\frac{\mathrm{Cov}[u_{t},u_{t-1}]}{\sigma_{u_{t}}^{2}}\Bigg{)}$, so
that this term equals 1. We further substitute for
$\sigma_{u_{t-1}}^{2}=\sigma_{u_{t}}^{2}$ in the term
$\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t-1}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}$.
This results in

$\displaystyle\pi=$
$\displaystyle\lambda\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}.$
(59)

We continue with the reduced form coefficient $\theta_{1}$ derived in (44). We
substitute for $u_{t-1}=u_{t}$ in the term
$\Bigg{(}\frac{\mathrm{Cov}[y_{t},u_{t-1}]}{\mathrm{Cov}[y_{t},u_{t}]}\Bigg{)}$,
so that this term equals 1. This results in,

$\displaystyle\theta_{1}=$
$\displaystyle\gamma\lambda\Bigg{(}\frac{\sigma_{u_{t}}^{2}+\sigma_{m_{t}}^{2}}{\sigma_{u_{t}}^{2}+\sigma_{m_{t-1}}^{2}}\Bigg{)}.$
(60)

We finish with the second stage estimates $\gamma^{iv}$ and $\beta^{iv}$.
Combining (59) and (60) results in

$\displaystyle\gamma^{iv}=$ $\displaystyle\frac{\theta_{1}}{\pi}=\gamma.$ (61)

Using the results above, it follows from (49) that

$\displaystyle\beta^{iv}=$ $\displaystyle\beta.$ (62)

##  Appendix C Proof Proposition 3

###  C.1 Case (i)

Case (i) of Proposition 2 shows that under 1 we have that
$\widetilde{\pi}=\lambda\Bigg{(}\frac{1}{B_{\pi}^{1}}\Bigg{)}$. We substitute
$\widetilde{\pi}$ for $\lambda$ in (21) to arrive at,

$\displaystyle\gamma^{fs}=$
$\displaystyle\frac{\gamma^{m}}{\widetilde{\pi}}=\Bigg{(}\frac{\gamma\lambda}{\lambda}\Bigg{)}B_{\pi}^{1}$
(63) $\displaystyle=$ $\displaystyle\gamma B_{\pi}^{1}.$

Next we substitute $\widetilde{\pi}$ for $\lambda$ in (22) to arrive at,

$\displaystyle\beta^{fs}=$
$\displaystyle\beta^{m}-\gamma^{m}\delta\bigg{(}\frac{1-\widetilde{\pi}}{\widetilde{\pi}}\bigg{)}$
(64) $\displaystyle=$
$\displaystyle\beta^{m}-\gamma^{m}\delta\bigg{(}\frac{1-\lambda}{\lambda}\bigg{)}+\gamma^{m}\delta\bigg{(}\frac{1-\lambda}{\lambda}\bigg{)}-\gamma^{m}\delta\bigg{(}\frac{1-\widetilde{\pi}}{\widetilde{\pi}}\bigg{)}$
$\displaystyle=$
$\displaystyle\beta+\gamma^{m}\delta\bigg{(}\frac{1-\lambda}{\lambda}-\frac{1-\widetilde{\pi}}{\widetilde{\pi}}\bigg{)}$
$\displaystyle=$
$\displaystyle\beta+\gamma\delta\bigg{(}1-B_{\pi}^{1}\bigg{)}.$

###  C.2 Case (ii)

Case (ii) of Proposition 2 shows that under 1 and 2 we have that
$B_{\pi}^{1}=1$, so that $\widetilde{\pi}=\lambda$. It follows from case (i)
of Proposition 3 that

$\displaystyle\gamma^{fs}=$ $\displaystyle\gamma,$ (65)
$\displaystyle\beta^{fs}=$ $\displaystyle\beta.$ (66)

##  Appendix D Additional empirical results

Table 5: Descriptive statistics on the test scores across all grades

| (1) | (2) | (3) | (4) | (5)
---|---|---|---|---|---
| Obs | Mean | SD | Min | Max
Panel A: Reading tests
Midterm grade 2 | 114743 | 139.354 | 27.933 |  $<$42.000 |  $>$264.000
End term grade 2 | 126688 | 146.532 | 26.752 |  $<$55.000 |  $>$282.000
Midterm grade 3 | 137499 | 157.415 | 26.149 |  $<$66.000 |  $>$279.000
End term grade 3 | 121392 | 161.807 | 27.028 |  $<$71.000 |  $>$280.000
Midterm grade 4 | 142069 | 173.447 | 25.339 |  $<$73.000 |  $>$309.000
End term grade 4 | 111460 | 182.413 | 27.147 |  $<$79.000 |  $>$337.000
Midterm grade 5 | 148019 | 191.618 | 26.766 |  $<$81.000 |  $>$340.000
End term grade 5 | 148019 | 197.279 | 26.983 |  $<$84.000 |  $>$354.000
Panel B: Math tests
Midterm grade 2 | 132855 | 167.815 | 30.385 |  $<$30.000 |  $>$278.000
End term grade 2 | 132976 | 189.235 | 29.063 |  $<$53.000 |  $>$289.000
Midterm grade 3 | 137502 | 206.446 | 28.042 |  $<$50.000 |  $>$313.000
End term grade 3 | 137643 | 218.366 | 27.299 |  $<$64.000 |  $>$334.000
Midterm grade 4 | 141986 | 230.064 | 26.684 |  $<$89.000 |  $>$356.000
End term grade 4 | 123702 | 241.663 | 26.635 |  $<$100.000 |  $>$361.000
Midterm grade 5 | 148019 | 254.142 | 27.142 |  $<$118.000 |  $>$379.000
End term grade 5 | 148019 | 261.959 | 26.085 |  $<$118.000 |  $>$394.000

  * Notes: this table shows the descriptive statistics for the test scores across all grades. Due to confidentiality reasons we cannot show minima and maxima that are based upon less than 10 observations. Therefore, the minimum (maximum) may contain the smallest (largest) number, such that at least ten observations have values lower (higher) than that number.

Table 6: OLS estimates with control variables

| (1) | (2) | (3)
---|---|---|---
|  $\mathbbm{1}$(recom. $\geq$ college tr.)
SES income | 0.115*** | 0.028*** | 0.029***
| (0.002) | (0.001) | (0.001)
End term grade 5 |  | 0.381*** | 0.375***
|  | (0.001) | (0.001)
$\mathbbm{1}$(female) |  |  | 0.011***
|  |  | (0.002)
$\mathbbm{1}$(Western imm.) |  |  | 0.036***
|  |  | (0.004)
$\mathbbm{1}$(non-Western imm.) |  |  | 0.036***
|  |  | (0.003)
Age |  |  | -0.072***
|  |  | (0.002)
$\mathbbm{1}$(Cohort$=$2019-20) |  |  | -0.042***
|  |  | (0.005)
$\mathbbm{1}$(Cohort$=$2020-21) |  |  | 0.030***
|  |  | (0.005)
$\mathbbm{1}$(Cohort$=$2021-22) |  |  | -0.072***
|  |  | (0.005)
Observations | 148019 | 148019 | 148019
$R^{2}$ | 0.053 | 0.498 | 0.506

  * Notes: ***, **, * refers to statistical significance at the 1, 5, and 10% level. The numbers represent the estimated coefficients and the numbers in parentheses the corresponding standard errors. Coefficients are estimated with OLS. Standard errors are clustered on the school id.

Table 7: $\mathrm{Cov}[\Delta{s^{m}_{t}},x]$ for observable characteristics
$x$

| (1) | (2) | (3) | (4) | (5)
---|---|---|---|---|---
| $\Delta_{s^{m}_{t}}=s^{m}_{t}-s^{m}_{t-1}$
SES income | 0.005*** |  |  |  |
| (0.001) |  |  |  |
$\mathbbm{1}$(female) |  | 0.007*** |  |  |
|  | (0.002) |  |  |
$\mathbbm{1}$(Western imm.) |  |  | 0.018*** |  |
|  |  | (0.004) |  |
$\mathbbm{1}$(non-Western imm.) |  |  |  | 0.012*** |
|  |  |  | (0.004) |
Age |  |  |  |  | -0.005**
|  |  |  |  | (0.002)
Observations | 148019 | 148019 | 148019 | 148019 | 148019
$R^{2}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000

  * Notes: ***, **, * refers to statistical significance at the 1, 5, and 10% level. The numbers represent the estimated coefficients and the numbers in parentheses the corresponding standard errors. Coefficients are estimated with OLS. Standard errors are clustered on the school id.

Figure 3: $\mathrm{Cov}[\Delta{s^{m}_{t}},SES]$ over time Notes: this figure
plots the estimates for the coefficients of SES from regressions of the change
in test scores $(s^{m}_{t}-s^{m}_{t-\tau})$ upon SES over time $\tau$.
Coefficients are estimated with OLS. Standard errors are clustered on the
school id.  Figure 4: OLS, IV, and EIV estimates with two alternative measures
of teacher recommendation

(a)

(b)

Notes: this figure shows the estimated coefficients for the OLS, IV, and EIV
strategy with two alternative measures of teacher recommendation. The lines
are 95% confidence intervals. The two alternative measures are a dummy
variable that is equal to one if the recommendation is equal to or higher than
the theory-oriented vocational track (Figure 4(a)) or the university track
(Figure 4(b)). See Figure 2 for details on the estimation.  Figure 5: OLS, IV,
and EIV estimates with three alternative SES measures

(a)

(b)

(c)

Notes: this figure shows the estimated coefficients for the OLS, IV, and EIV
strategy with three alternative SES measures. The lines are 95% confidence
intervals. The three alternative measures are a dummy that equals one if
parental income is above the median (Figure 5(a)), years of schooling of the
highest educated parent (Figure 5(b)), and a dummy that equals one if the
highest educated parent completed at least college education (Figure 5(c)).
See Figure 2 for details on the estimation.  Figure 6: OLS, IV, and EIV
estimates per cohort

(a)

(b)

(c)

(d)

Notes: this figure shows the estimated coefficients for the OLS, IV, and EIV
strategy, separately for each cohort (Figure 6(a)-(d)). The lines are 95%
confidence intervals. See Figure 2 for details on the estimation.  Figure 7:
OLS, IV, and EIV estimates separately for immigrants and natives

(a)

(b)

Notes: this figure shows the estimated coefficients for the OLS, IV, and EIV
strategy, separately for immigrants (Figure 7(a)) and natives (Figure 7(b)).
The lines are 95% confidence intervals. See Figure 2 for details on the
estimation.