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Learning your own ability

Carlos Madeira

Draft

April 2019

Abstract

Families’ human capital investments depend on beliefs about their children’s performance.

I build a dynamic model of expectation formation to show how agents use both observable and

unobservable information to predict their school scores. The model shows parents and students

have substantial knowledge of unobservable factors a¤ecting their performance, especially in

middle and high school. Families are overcon…dent towards expecting higher grades and expectation

formation di¤ers by race. Families’ability to predict future scores improved substantially during

middle school due to several factors: lower bias and variance of the prediction errors, and a better

use of past scores as predictors.

JEL Classi…cation: D83, D84, I20, J00.

Keywords: expectations, learning, academic achievement.

Central Bank of Chile, Agustinas 1180, Santiago, Chile. I would like to express my enormous debt to Orazio

Attanasio, Sergio Urzua, Elie Tamer, Chris Taber, and Chuck Manski for their guidance and comments from seminar

participants at University of Essex, Northwestern University, Universitat Autonoma de Barcelona, Universidad de

Alicante, University of Exeter, Fundacao Getulio Vargas, the Shanghai Econometric Society World Congress, ITAU,

NERA, and the Central Bank of Chile. Financial support from Fundação Calouste Gulbenkian and Fundação para a

Ciência e Tecnologia is gratefully acknowledged. Comments are welcome at carlosmadeira2009@u.northwestern.edu.

All errors are my own. This research used the The Beginning School Study, 1982–2002 (Log# 01293) dataset (made

accessible in 2004, numeric data …les). These data were collected by Karl L. Alexander and Doris R. Entwisle and are

available through the Henry A. Murray Research Archive of the Institute for Quantitative SocialScience at Harvard

University, Cambridge, Massachusetts (Producer and Distributor).

1

1 Introduction

Predicting individual future performance is important in decisions with uncertain outcomes, such as

starting a …rm, choosing a work career, applying to college or saving for retirement (Delavande and

Rohwedder, 2011). Expectations are particularly relevant for academic choices, since many human

capital decisions are made early in life (Cunha et al., 2005). Economic models of education choice

usually assume that agents are able to predict their academic performance and the expected returns

of each option. This is a strong assumption, since a student’s performance may change substantially

when arriving to a new grade or school level. It can be di¢ cult for families to forecast their children’s

academic achievement for several reasons: educators may favor di¤erent teaching’methods, schools

provide di¤erent social environments and curriculum materials may change between years.

This paper uses a publicly-available dataset, the Beginning School Study (BSS), to study

how families use available information to form expectations of their academic achievement. The

BSS elicited point predictions of academic outcomes for a panel of 825 parents and students

from the children’s …rst-grade until their adult years. Previous work with the BSS data shows

that families are systematically overoptimistic about their children’s school performance, although

their predictions improved as the students aged (Madeira, 2018). To evaluate how respondents

use available information I specify a model of student achievement and agents’ expectations,

where respondents forecast their future scores based both on observable information (demographic

characteristics and past performance) and unobservable information (such as the student’s study

habits). The distribution of the unobserved information can be identi…ed from two sources: 1)

the heterogeneity of beliefs among agents with the same observable information, 2) the correlation

of beliefs with the actual outcomes. The model can then be used to examine several sources of

the agents’ prediction errors: 1) overcon…dence, 2) poor use of the information observed in the

student’s previous scores, or 3) noisy use of private information.

Using this model of expectation formation I …nd that in elementary school parents and students

presented both a large bias and variance for their predictions, indicating that respondents are both

overcon…dent and use noisy private information. The results suggest the overcon…dence bias is

smaller for families with higher education levels, older parents, and parents of girls. Respondents in

elementary school react to di¤erences in past academic achievement, but update their expectations

2

more slowly than a rational agent would. Black families are more optimistic than average, but had

prediction errors with a similar variance as white families. Over the years, however, black families

made similar gains in their ability to predict performance as their white counterparts.

The model also shows that the role of parents and students’private information about unobserved

factors a¤ecting school performance increased substantially during middle school and high school.

Students in particular have a much higher degree of private information than their parents during

high school. In middle school families’ bias and prediction variance show a strong decrease, in

particular for students. In addition, I …nd that both parents and students made better use of the

information available in past marks, updating recent information about school performance more

quickly. Therefore, families’ability to predict future scores improved substantially during middle

school and high school due to several factors: lower bias and variance of the prediction errors, and

a better use of past scores as predictors. Economic models of human capital decisions, therefore,

should take into account higher degrees of private information as students grow older and that the

unobserved information known by teenage students increases at quicker rates than for their parents.

Incorrect beliefs about achievement may lead families to make ine¢ cient investments. For

instance, overcon…dent students may put less e¤ort in school. I …nd black parents and students are

more optimistic about their academic scores even after several years, which may explain why they

stay longer as retended students in school (Rivkin, 1995, Lang and Manove, 2011), despite having

similar returns to education (Lang and Rudd, 1986, Wolpin, 1992, Lang and Manove, 2011).

This work is related to a large body of literature testing rational expectations which has

produced mixed …ndings (Madeira and Zafar, 2015, Madeira, 2018). Other work …nds that incorrect

beliefs about academic performance may explain ine¢ cient education choices, such as college

dropout decisions (Stinebrickner & Stinebrickner, 2012, 2014). Dominitz (1998) and Das and Soest

(1999) studied how agents revise their income expectations after one year. Many empirical studies

show that agents tend to overestimate their ability and their estimates do not improve signi…cantly

with feedback on the past performance (Hoelzl and Rustichini, 2005). Some laboratory experiments

studied how agents update their beliefs with new information (El-Gamal and Grether, 1995, Houser

et al., 2004). However, lab studies may fail to replicate how agents learn over longer periods or

in less standardized environments. Because few datasets follow the same respondents over many

years, little is known about how agents adjust their beliefs (Manski, 2004). This work …lls some of

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that gap, since the extended time panel of the BSS dataset allows the researcher to observe how

families change their beliefs as they age and learn more information. Finally, the empirical analysis

of the expectations of BSS parents and students helps to shed light on how di¤erent household

members interact to form expectations about the future and make education decisions (Attanasio

and Kaufmann, 2014, Giustinelli, 2016, Giustinelli and Manski, 2016, Oyserman, 2015).

The paper is organized as follows. Section 2 describes the BSS data. Section 3 describes

the structural model of expectation formation. Section 4 presents the estimation results of the

structural model and explains the main changes in parents and students’ predictions of their

academic performance. Finally, Section 5 presents a summary of the main results.

2 Data description

2.1 Sample design of the survey

The Beginning School Study (BSS) consists of 838 children that were randomly selected from

the …rst-grade rosters in 1982 of a set of 20 Baltimore public elementary schools. First, a set of

20 schools strati…ed by racial attendance was selected: 6 schools predominantly black, 6 schools

predominantly white and 8 mixed schools. Afterwards, within each school a random sample of

students were selected from each …rst-grade classroom. Parental permission for participation in

the study was obtained from 825 parents. These families make up the initial sample size in 1982.

The families in the BSS survey come mainly from a disadvantaged background and are quite close

to federal poverty lines. Around 63% of the BSS students participated in the federal program for

subsidized school lunches. Most parents had low education, with more than 30% of both white

and black parents having less than a complete high school information. Also, less than 35% of the

parents went to college and only 10% actually completed a four year college degree. Furthermore,

28% of the mothers in the BSS sample were teenagers at time of the birth of their children.

4

Table 1.1: Nr of past failed grades by academic year

Nr of Failed Grades 1983/84 1984/85 1987/88 1989/90

One grade ahead 0.4% 0.8% 1.1% 0.8%

Never failed a grade 81.8% 72.2% 61.4% 57.0%

Failed one grade 17.5% 24.2% 31.0% 34.2%

Failed two grades or more 0.4% 2.8% 6.3% 8.0%

Missing 7.0% 5.0% 6.6% 9.3%

All statistics are a percentage of the observed sample, except for the

missing sample values which are a percentage of the whole population.

All children attended schools with the same basic curriculum and with teachers on the same

salary scale. Students’ grades were assigned on a letter-mark basis for an academic year with 4

quarters. Public schools marks consist of letters in the same percentage scale: Excellent (90-100),

Good (80-89), Satisfactory (70-79) and Unsatisfactory (0-69). Marks in middle and high school

were also reported on a known interval scale with + and - for each letter. Standardized California

Achievement Tests (CATs) were administered in all Baltimore public schools in October and May

of each year until 1990. The CAT scores were made available not only to parents and children,

but also to teachers and school o¢ cials. Information on grade years, school marks, CAT scores

and subsidized-lunch status of the students were collected directly from the school records. Most

students in the BSS have poor performance, with more than 40% of the students having repeated

one grade or more before starting high school (Table 1.1). Around 60% of the students had marks

of Satisfactory or Unsatisfactory in Math during their elementary and middle school periods (Table

1.2). Therefore a large proportion of the BSS students were at risk of su¤ering grade retention.

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Table 1.2 - Distribution of Students marks:

Letter mark (Math, 1982-89) 1982-89 Q1 1982-89 Q2 1982-89 Q3 1982-89 Q4

1 - Unsatisfactory 19.2% 19.2% 18.5% 16.5%

2 - Satisfactory 47.0% 43.2% 42.5% 42.4%

3 - Good 26.8% 28.9% 29.0% 28.8%

4 - Excellent 7.0% 8.7% 10.0% 12.4%

Missing 30.3% 29.9% 29.1% 30.5%

All statistics are a percentage of the observed sample, except for the

missing sample values which are a percentage of the whole population.

Throughout the paper my results are based on a missing conditionally at random (MCAR)

assumption. The BSS had very low rates of item nonresponse to particular questions (item

nonresponse tends to be inferior to 1% in all survey years) and collected retroactive data on

students’school scores (Alexander and Entwisle, 2004); therefore marks are observed even for years

with missing interviews. Madeira (2018) presents a comprehensive analysis of the missing survey

data of the BSS, showing that in general the MCAR assumption is reasonable. Parents and students

with missing interviews had similar characteristics in terms of parental education and test scores

relative to the families that participated in all the survey years (Madeira, 2018). Furthermore, linear

regressions of parents and students expectations conditional on the children’s past academic scores

have similar coe¢ cients for the updating of Math and English/Reading expectations (Madeira,

2018), con…rming the validity of the MCAR assumption for the BSS sample.

It is also relevant to note that performing all the exercises of this article with just the sample

of households that participated in all the panel surveys does not change the results signi…cantly.

2.2 Questions about the expectations of future marks

The time-line of the survey in each academic year started with the parent questionnaire in the

summer or at the beginning of the fall quarter. CAT scores were implemented in the fall and spring

quarter before the student surveys. Student surveys were then implemented in the fall and spring

quarters before the quarterly marks were given. From 1982 to 1994 the survey asked the following

expectation questions of students and parents:

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Students: What mark do you think you are going to get in (Math / Reading) - Excellent

(90-100), Good (80-89), Satisfactory (70-79), and Unsatisfactory (0-69)?

Parents: Please guess the marks your child will get in Reading and Mathematics on the …rst

report card this fall: Excellent (90-100), Good (80-89), Satisfactory (70-79), Unsatisfactory (0-69).

The wording of the questions changes slightly from year to year, but remains consistent in

asking for the respondents’best guess. During the whole time period of the survey, school grades

in Baltimore were awarded in an Excelent, Good, Satisfactory, and Unsatisfactory category scale

(Entwisle, Alexander, and Olson, 1997), which translated into an equivalent numerical scale of

90-100, 80-89, 70-79, and 0-69. This educational scale system of the Baltimore school system

was di¤erent from the one applied in the majority of the United States, which is based on A-F

letters. However, it was a system that was universally applied in Baltimore during the whole time

period of the BSS survey (1982-1994). Families also answered questions about their predictions of

the education level and jobs they would have as adults. Parents indicated their age, education,

occupation, employment status and weekly hours of work if employed. Other questions included

whether they had stress, problems at work or within the family. Teacher surveys were also

implemented at the end of the school year, eliciting their best guesses of the students’ marks

in the following fall quarter and their future educational attainment.

Table 1.3 - Respondents’expectations Parents Students

Best guess for the Math (Fall) mark 82 82-90 82 82-90

0 - Not taking

1 - Unsatisfactory 3,2% 2,5% 2,5% 2,6%

2 - Satisfactory 35,6% 30,2% 12,5% 16,8%

3 - Good 48,7% 47,2% 37,4% 44,7%

4 - Excellent 12,5% 20,2% 47,6% 35,9%

Nr of observations 786 3379 824 3335

Some work on expectations has been done before with the BSS data. Alexander, Entwisle and

Thompson (1988) show that parents’expectations are signi…cant for predicting …rst grade outcomes,

even after conditioning on previous academic results and family background. The authors also show

that parental expectations are a¤ected by race and education. However, no previous work studied

7

how expectations were revised over time based on observable and private information signals.

In Table 1.3 I show that around 60% of the parents believed their children would attain a

Good or Excellent mark at Math in the next fall quarter, although only 33% of students actually

obtained those marks (Table 1.2). Madeira (2018) presents a set of nonparametric tests that

rejected the rationality of both parents and students’expectations, even after controlling for a wide

range of assumptions about the agents’information sets, although their predictions improved as

the children aged. Both parents and students were found to have optimistic expectations, with

guesses statistically being higher than the obtained marks. There is a strong persistence of families

with correct predictions over time, but the BSS data does show that respondents with incorrect

guesses did have a signi…cant probability of providing accurate predictions in future years, which

is evidence that respondents take their predictions seriously (Madeira, 2018).

3 A joint model of student achievement and expectations

3.1 A simple model of achievement and expectations

Families can depart from rationality in several ways: 1) over-optimism, 2) poor use of the observable

information available (families may not adjust their expectations when they receive new information),

and 3) agents give predictions that are too random. Here I specify a model of student achievement

and agents’expectation formation that measures the sources of respondents’forecast errors.

Suppose that at the end of each period tteachers observe student achievement, si;t in the

continuous scale of [0 100] and teachers assign a mark, Yi;t =j2 f1; :::; J gto students who fall

in the interval si;t 2(vj1; vj). Assume that there is an increasing monotone transformation

1) y

i;t =f(si;t) = m(xi;t ) + "i;t,

which can be expressed as the sum of a predictable component m(xi;t)and a term "i;t unknown

to the econometrician. The probability of student ireceiving a mark below jis:

2) Pr(Yi;t jjx) = Pr(y

i;t =m(xi;t) + "i;t f(vj)jx) = F"jx(f(vj)m(x)),

where I assume F"jx(:)is a known distribution up to a parameter vector.

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It is not easy to establish a link between the expected interval of the agent and his expectation

of the latent variable. For example, if agents express their mode interval, then their mode interval

does not necessarily contain the mode of the latent variable. However, a set of interval forecasts

allows to evaluate the direction of the prediction losses, therefore a median or quantile interval

is a plausible interpretation for the agents’interval expectations. Note that the agents’categorical

prediction, Pi;t = arg minp(Pr(Yi;t p)), is also the interval that contains their subjective

quantile for the latent variable, y

i;t, therefore the -quantile(y

i;t)belongs to the interval [f(vP1); f(vP)].

Assuming agents know the cuto¤s that teachers use to assign marks and use an absolute loss

criterion with = 0:5to form discrete predictions, Pi;t, then Pi;t corresponds to the interval that

contains their subjective median for the continuous outcome. This result is a consequence of the

property of invariance of quantiles in relation to monotone transformations.

Now I specify the agents’predictions process

3) p

i;t =mp(xi;t) + pii;t ,

as the sum of a systematic component, mp(xi;t), and a private information factor, pii;t, known

to the agent but not to the econometrician. mp(:)denotes the mean prediction made by each family

based on observable information and pii;t denotes the private information possessed by each agent

and not observed by the econometrician. Therefore pii;t may be correlated with "i;t, although

it does not happen necessarily that families know all the factors a¤ecting their achievement,

which implies that the correlation of both terms is less than one. An example of pii;t could be

information that parents and students know about their homework or how much the teacher likes

the student. Parents and students may have some knowledge of the unobservable factors a¤ecting

student achievement. All families in Baltimore knew the numerical intervals assigned to each grade,

therefore I assume the cuto¤s for each grade, vj, were known by the agents. This implies that the

probability of the agent igiving a discrete prediction Pi;t below value jis:

4) Pr(Pi;t jjx) = Pr(p

i;t f(vj)jx) = Fpijx(f(vj)mp(x)),

where I assume that Fpijx(:)is a known distribution up to a parameter vector.

The process of joint expectations and student achievement can then be summarized as:

5) Pr(Yi=l; Pi=jjx) = Zf(vl)

f(vl1)Zf(vj)

f(vj1)

f";pijx(t1m(x); t2mp(x))@t2@t1.

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Now I face the problem of choosing a suitable parametric family for F";pijx(:)that can be

identi…ed from expressions 2), 4) and 5). I make the assumption that (y

i;t; p

i;t)are bivariate-normal

distributed, with means (m(xi;t); mp(xi;t)), standard-deviations (c(xi;t); cp(xi;t )) and correlation-coe¢ cient

(xi;t). The di¤erence mp(xi;t)m(xi;t)can be interpreted as respondents’average bias, while

cp(xi;t)denotes the heterogeneity of agents’private information. Note that cp(xi;t)is a parameter

for intra group heterogeneity of infomation and not a parameter of each agent’s subjective uncertainty.

cp(xi;t)represents how much agents in the same group xdi¤er between themselves. The correlation

coe¢ cient can be interpreted as a measure of the quality of respondents’private information.

Under the bivariate-normal assumption, the prediction error of the respondents, p

i;t y

i;t,

is completely described by its bias and variance components, being distributed as: N(mp(x)

m(x); c(x)2+cp(x)22(x)c(x)cp(x)). A rational agent with no private information would have

a prediction variance of c(x)2, therefore the term cp(x)22(x)c(x)cp(x)measures the variance

component in families’predictions that a rational agent would not have and can be interpreted as

a measure of the quality of their private information. Since the error terms "and pi are speci…ed

to be uncorrelated with X, agents of the group xcan only be rational if mp(x)m(x)=0and

cp(x)22(x)c(x)cp(x)0. The term mp(x)m(x)examines how respondents’ bias varies

across demographic groups. Rational agents will have a 0 prediction bias, while overcon…dent

students will have positive values of mp(x)m(x). This model can also show if families are using

observable information in an optimal way. For instance, families could put too much focus on their

older scores or on their more recent ones. The estimated coe¢ cients about how people use the

information contained in previous school grades will say whether people are updating observable

information too slowly or too quickly by comparing the weights given to recent school performance

by the parents and students to the optimal weights a rational agent would have.

3.2 Identi…cation proof

The bivariate-normal model is identi…ed by imposing location and scale normalizations: A.1)

f(v0) = 1; f(v1)=0,f(vJ)=+1; A.2) c(x0)=1for some x0with positive density. In

this case the probability of a student getting a mark lower than jis given by:

6) Pr(Yjjx)=f(vj)m(x)

c(x)=)m(x) = f(vj)c(x)1(Pr(Yjjx)).

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Using expression 6) for categories jand 1evaluated at x0identi…es the cut-o¤ points f(vj):

7) f(vj)=1(Pr(Yjjx)) 1(Pr(Y1jx)).

Using expression 6) evaluated at any xfor category jand category k < j identi…es c(x):

8) c(x) = f(vj)f(vk)

1(Pr(Yjjx)) 1(Pr(Ykjx)).

m(x)is then identi…ed from expression 6). The identi…cation of mp(x)and cp(x)follows the

same argument, except that the scale normalization of the variance is unnecessary since the cuto¤s

are already identi…ed. The reason why this model allows for heteroscedasticity across groups is that

the variance is standardized for only one group of agents and cuto¤s are assumed to be the same

for all groups. Also, the correlation of private information with the unobserved factors a¤ecting

student achievement, (x), is identi…ed from equation 5) (Zellner and Lee, 1965)

3.3 Dynamic structure of achievement and expectation updating

Parents and students report their beliefs in several periods and it is reasonable to assume that

expectations are correlated over time. Therefore I estimate a parsimonious structure that accounts

for the initial private information each agent has and how this information is updated every period.

Student achievement has both a permanent component, i, and a transitory component, vi;t,

that evolves over time. In the same way there is a permanent component of the expectations of each

agent, p

i, and a component, vp

i;t, that is updated in each period as the agent gets more information.

Note that in this dynamic model the superscript pdenotes the coe¢ cients that correspond to the

parents and students’s observable private information to di¤erentiate them from the coe¢ cients

a¤ecting the distribution of unobserved factors that determine student achievement:

9) "i;t =i+vi;t,vi;t =ui;t +(xi;t )vi;t1,"i;1=i+ui;1,

10) pii;t =p

i+vp

i;t,vp

i;t =up

i;t +p(xi;t)vp

i;t1,"p

i;1=p

i+up

i;1,

where the random e¤ects (i; p

i)N(0;[(xi); p

(xi); (xi)]) and the time-varying unobservables

are normally distributed, (ui;t; up

i;t)N(0;[u(xi;t); p

u(xi;t); u(xi;t)]). I apply the standardizations

11

c0=1,c1=r,cJ= +1and V ar("i;1jx0

i)=1for some x0

iwith positive density. The AR

coe¢ cients 1(xi;t)1and 1p(xi;t )1are restricted to be within the unit-root circle.

This bivariate random-e¤ects and AR(1) structure can be interpreted in a simple way. The

Random-e¤ects represent the e¤ects of unobserved variables that are …xed for each family and

the knowledge each agent has of those permanent factors. Examples of persistent unobserved

factors a¤ecting student performance could be, for example, the permanent income earned by the

family, the conditions of the house and neighborhood where they live, or whether grandparents

help with the children. The AR(1) process represents the e¤ect of unobservable variables with

a transitory e¤ect in student achievement and the knowledge agents have of these components.

Transitory components a¤ecting achievement could be a change of residence or lack of knowledge

about teachers’abilities. This is important because the model allows to estimate whether transitory

unobservable factors are more di¢ cult for families to forecast in relation to temporary ones.

3.4 Estimation

Let

Yi;t (^

Yi;1; ::; ^

Yi;t) = (j1; :::; jt) = gt,~

Yi;t = (Yi;1; ::; Yi;t) = (l1; :::; lt) = qt,~xi;t (xi;1; ::; xi;t), be

the vectors of agent i’s guesses, academic scores and covariates until time t. The agents’bivariate

outcome probability and the overall likelihood function are given by:

11) Pr(

Yi;T =gT;~

Yi;T =qTjxi;T ) =

T

Y

t=1

Pr( ^

Yi;t =jt; Yi;t =ltjxi;t;

Yi;t1)

12) L=

N

P

i=1

log(Pr(

Yi;T =gT;~

Yi;T =qTjxi))

This model is estimated by Simulated Maximum Likelihood, using the GHK procedure (Geweke

et al., 1994) with 30 draws. The draws are obtained using the Modi…ed Latin Hypercube Sampling

(MLHS) method, which has been shown to strongly outperform other numerical simulators in many

applications (Hess, Polak and Train, 2004). The MLHS method to obtain Rmultivariate draws

basically starts with an equal spaced sequence of values, '(j) = j1

Rfor j= 1; :::; R, in each

dimension. Then a pseudo-uniform number xis added to the draws of each dimension to get

~'(j) = '(j) + x

Rfor j= 1; :::; R. This allows the econometrician to get an equal-spaced coverage of

each dimension. To get the pseudo-uniform number xI use scrambled Halton draws option, which

have been shown to outperform standard uniform numbers (Hess et al., 2004).

12

Finally, for the purposes of estimation I assume the following functional forms:

13) m(xi;t) = xi;t ,V ar("i;1) = exp(~xi;1)2, where ~xi;1has no constant (therefore V ar("i;1) = 1

is satis…ed for ~xi;1(0; ::; 0)),

(xi)=( exp(xi;1)

1 + exp(xi;1)) exp(~xi;1),u(xi;t) = exp(xi;tu)for t > 1,

u(xi;1) = (1 (exp(xi;1)

1 + exp(xi;1))2)1=2exp(~xi;1),

(xi) = 2( exp(xi;1)

1 + exp(xi;1))1,

u(xi;t) = 2( exp(xi;tu)

1 + exp(xi;tu))1,

and (xi;t) = 2( exp(xi;t)

1 + exp(xi;t))1;

14) mp(xi;t) = xi;t p,V ar(pii;1) = exp(xi;1p),

p

(xi)=( exp(xi;1p

)

1 + exp(xi;1p

)) exp(~xi;1p),p

u(xi;t) = exp(xi;tp

u)for t > 1,

p

u(xi;1) = (1 (exp(xi;1p

)

1 + exp(xi;1p

))2)1=2exp(~xi;1p),

and p(xi;t) = 2( exp(xi;tp

)

1 + exp(xi;tp

))1.

I estimate the model with di¤erent coe¢ cients for each of the main school levels: First grade,

Elementary, Middle, and High school. The coe¢ cient vectors m(x),c(x),mp(x),cp(x),(x)also

control for time-dummies for both races. The parameters of the autocorrelation, dispersion and

correlation of unobservables use race, gender and maternal education as explanatory variables.

This dynamic model involves a large set of parameters, therefore I focus only on the most

representative results of the regressions for Math. The results of the respondents’expectations for

English are quite similar and available from the author upon request.

13

4 Estimation results

4.1 Bias of respondents’expectations across demographic groups

Tables 2.1 to 2.4 show the results of the Bivariate-normal regressions for Math1. Table 2.1 shows

the coe¢ cients a¤ecting the mean achievement and mean expectations of parents and students

during elementary school. The mean expectations across demographic groups in …rst grade, middle

and high school are qualitatively similar to the elementary school, therefore for brevity those results

are not reported but they are available from the author upon request.

Table 2.1: Bivariate Regression of Math marks in Elementary school - Q1, 83/84-87/88

Marks (w/ students) Parents Students

Mean of latent marks/ expectations:

Constant 0.42 (0.14)* 1.84 (0.15)* 2.73 (0.3)*

black race -0.24 (0.06)* 0.11 (0.05)* -0.01 (0.09)

female 0.09 (0.05)** -0.03 (0.05) 0.01 (0.09)

Student’s month of birth -0.01 (0.01) 0 (0.01) 0.01 (0.01)

age of mother at birth (10) 0.01 (0.05) -0.14 (0.05)* -0.06 (0.08)

mother’s years of education 0.05 (0.01)* 0.02 (0.01) 0.01 (0.02)

nr of failed grades 0.25 (0.06)* -0.04 (0.05) 0.05 (0.08)

CAT score 0.44 (0.08)* 0.24 (0.06)* -0.12 (0.09)

Average of all past marks 3.78 (0.82)* 2.2 (0.67)* 0.45 (1.18)

Average of last 4 quarters 0.91 (0.6) 2.05 (0.53)* 2.3 (1)*

(controls for year-dummies) yes yes yes

Nr of observations 2359 1493 1307

Standard-deviations in (), * signi…cant at the 5% level, ** signi…cant at the 10% level

Table 2.2 hows the standard-deviation of the heterogeneity in the temporary unobservable

private information for each group and how the private information of parents and students is

correlated with the actual unobserved factors a¤ecting student achievement. Table 2.3 shows the

1Standard-errors and t-statistics were also estimated with 100 bootstrap sample replications or with the outer

product of the gradient, but the results are similar to the inverse Hessian and are not reported.

14

standard-deviation of the private information for the persistent factors for parents and students

and how it correlates with the actual unobserved persistent factors a¤ecting students’ school

performance. Finally, Table 2.4 shows how parents and students are using information from recent

school scores versus older performance in order to update their expectations.

The results of Table 2.1 show that black parents expect signi…cantly higher marks but their

students are actually worse than average, implying black parents have a higher optimism bias.

Children that are retained in lower grades have substantially higher marks than others, but the

expectations of parents and students are similar to their peers, which implies these respondents

are less biased than average. Female students are better at Math while their parents have similar

expectations to others, which implies a lower overcon…dence bias of parents in relation to girls.

This result is consistent with a higher parental preference for boys despite the greater academic

success of the girls. Children of parents with more age and education have higher marks but their

parents have similar expectations as others, which again indicates a lower bias.

Parents use CAT scores to form their Math expectations, but students do not appear to do so,

therefore in elementary school children adjust less to new information than parents.

Similar results were found for Math at the other school levels (…rst grade, middle school

and elementary school), which are available from the author upon request. The only signi…cant

di¤erence is that while repeating students have higher scores in elementary school, retention does

not seem to help their grades during middle and high school. The same results are also valid for

the regressions with the Reading/ English marks.

4.2 Knowledge of persistent versus temporary unobserved factors

Table 2.2 shows the standard-deviation of the heterogeneity of temporary factors a¤ecting students’

elementary school performance and how the private infomation of families is correlated with this

term. The estimates for the dispersion of the unobserved factors a¤ecting current performance

show that female students and children of higher educated mothers depart less from the mean,

therefore it is easier to predict their marks. Parents with higher education have a smaller deviation

of unobserved private knowledge, which indicates that these parents follow less noisy prediction

rules. Black parents, however, appear to have a greater dispersion in the knowledge about their

15

children. Students of highly educated parents also have private information that is more correlated

with actual achievement (Table 2.2), which shows their better use of private information.

Table 2.2: Unobserved heterogeneity of Math marks and

expectations in Elementary school - Q1, 83/84-87/88

Standard-deviation of unobservables:

Constant -0.14 (0.12) -0.38 (0.12)* 0.13 (0.13)

black race 0.07 (0.05) 0.12 (0.07)** 0.04 (0.08)

female -0.08 (0.05) -0.09 (0.07) -0.03 (0.08)

mother’s years of education -0.03 (0.01)* -0.06 (0.02)* -0.02 (0.02)

(controls for year-dummies)

Correlation of unobservables with marks:

Constant 0.15 (0.15) -0.03 (0.18)

black race -0.2 (0.13) -0.08 (0.17)

female 0.04 (0.12) -0.05 (0.15)

mother’s years of education -0.01 (0.03) 0.07 (0.04)**

Controls for year-dummies are included.

Standard-deviations in (), * and ** denote 5% and 10% statistical signi…cance.

I now compare families’ private information about new factors a¤ecting school performance

(Table 2.2) with parents and students’private information about the persistent factors a¤ecting

their performance (Table 2.3). The constants for the correlation of parents and students’private

information with persistent performance factors (Table 2.3) are larger than for the correlation of

private information with temporary factors (Table 2.2). This indicates that parents and students

know more about persistent factors a¤ecting their performance than about new factors previously

unknown. However, the standard-errors of the estimation are also large, therefore the model cannot

reject the hypothesis that parents and students’ know little or nothing about the unobserved

elements a¤ecting elementary school performance. The next section will show that the correlation

of parents and students’private information with the actual factors a¤ecting school performance

increases substantially during middle school and high school.

16

Table 2.3: Heterogeneity of families’persistent expectations and Math marks

Coe¢ cients of random e¤ects (Parents)

Std of Exps Std of marks Correlation of Exps and Marks

Constant -0.4 (0.29) -1.7 (0.4)* 1.25 (0.76)

black race 0.06 (0.27) -0.22 (0.32) -1.17 (0.57)*

female -0.15 (0.26) -0.32 (0.32) -0.56 (0.52)

mother’s years of education 0.11 (0.06)** 0.14 (0.06)* 0.1 (0.11)

Coe¢ cients of random e¤ects (Students)

Std of Exps Std of marks Correlation of Exps and Marks

Constant -1.41 (0.26)* -3.95 (2.08)** 2.01 (1.23)

black race -0.3 (0.25) -1.27 (1.04) 0.5 (0.94)

female 0.08 (0.24) -2.49 (5.67) 0.66 (1.58)

mother’s years of education 0.01 (0.07) 0.43 (0.3) 0.12 (1.65)

Standard-deviations in (), * signi…cant at the 5% level, ** signi…cant at the 10% level

Parents of black race have a random-e¤ect information that is less correlated with the actual

random-e¤ect in student achievement (table 2.3), which indicates that they have less access to good

private information relative to white families.

4.3 How do respondents use past information to form their expectations?

Table 2.4 presents the value of parents and students’coe¢ cients for the past marks and compare

them with the coe¢ cients that are given by the actual process of students’marks. In elementary

school both parents and students use recent academic performance (the last 4 marks) to update

their expectations. However, the coe¢ cients for predicting student achievement actually show

that parents and students in elementary school should rely much more on the entire past school

perfomance of the student, rather than just on last year’s performance. Also, in the spring quarter

of each academic year students do not rely as much as they should on their four most recent grades.

17

Table 2.4: Families’expectations use of the average marks

in the last year and all past performance

Elementary school Middle school High school

Q1 Q4 Q1 Q4 Q1 Q4

Students’expectations:

All past marks 0.45 (1.18) -2.54 (1.17)* 0.82 (1.04) -0.57 (0.83) 1.47 (1.35)

Last 4 quarters 2.3 (1)* 6.84 (1.19)* 3.01 (0.62)* 7.27 (0.88)* 6.13 (0.92)*

Parents’expectations:

All past marks 2.2 (0.67)* 4.96 (0.92)* 3.93 (1.09)*

Last 4 quarters 2.05 (0.53)* 2.7 (0.44)* 4.64 (0.62)*

Mean Achievement:

All past marks 3.78 (0.82)* -1.02 (0.63) 5.28 (1.18)* -0.05 (0.84) 3.81 (1.16)* 2.82 (1.17)*

Last 4 quarters 0.91 (0.6) 12.72 (1.38)* 3.78 (0.66)* 10.55 (1.22)* 7.41 (0.86)* 10 (1.19)*

Standard-deviations in (), * signi…cant at the 5% level, ** signi…cant at the 10% level

In middle school and high school parents and students rely more on both the last year’s school

performance and their entire past school performance. This updating procedure is actually very

close to the actual e¤ect of past scores estimated for the mean process of student achievement.

Parents and students, therefore, improved substantially their use of the observable information

given by past marks from elementary school to middle school and high school.

The main di¤erence between parents and students’predictions and the real process of student

achievement in middle school and high school is that parents and students did not rely as much

on the information in the last four academic quarters as it would be e¢ cient. This means that

parents and students are slower than optimal to update their expectations during each academic

year. Similar conclusions can be drawn from the families’expectations for English marks.

4.4 Bias and variance of prediction errors: Overview of results

This section evaluates the mean values of bias and the variance of respondents’prediction errors

adjusted for the rational variance term (p2

i;t 2i;ti;t p

i;t), across 4 demographic groups (the whole

population, black students, repeating students and families with mothers that have completed high

18

school). I then report how these mean values evolved over elementary (Table 3.1), middle (Table

3.2), and high school (Tables 3.3 and 3.4). Since these average values are complex functions of

several parameters I report bootstrap standard-errors of the values of each group.

Table 3.1: Bias and variance of expectations (Math, Fall - Elementary school, 82-87)

Variables / Demographic group All Black Repeaters Mothers w/

high school

correlation of parents and marks 0.13 (0.05) 0.10 (0.05) 0.08 (0.06) 0.13 (0.06)

mean bias of parental expectations 0.75 (0.15) 0.86 (0.17)* 0.71 (0.18) 0.72 (0.14)

variance of parental expectations’errors 0.83 (0.23) 0.89 (0.26) 1.04 (0.32)** 0.76 (0.23)*

variance of parents - "rational" agent 0.26 (0.09) 0.29 (0.10) 0.36 (0.13)** 0.24 (0.12)

correlation of students and marks 0.14 (0.03) 0.14 (0.04) 0.13 (0.04) 0.16 (0.04)

mean bias of students’expectations 1.15 (0.14) 1.27 (0.15)* 1.30 (0.19)* 1.05 (0.13)

variance of students expectations’errors 1.15 (0.28) 1.18 (0.31) 1.29 (0.38)* 1.05 (0.27)

variance of students - "rational" agent 0.59 (0.16) 0.59 (0.18) 0.67 (0.21)* 0.53 (0.15)

Bootstrap standard-errors in () - 100 sample replications.

* signi…cantly di¤erent from the "All" group at the 5% level, **signi…cant at the 10% level.

The results con…rm that black parents and students are signi…cantly more biased than white

families at all school levels, but they have a similar prediction variance as white families. Parents

and students of families where the mother has a high school degree are better predictors than

average, since both their bias and prediction variance are lower than their peers. Parents and

students decreased both their bias and prediction variance substantially from elementary to middle

and high school. This implies that families improved their predictions as children aged, due to

an improved use of both the observable information given by past marks (Table 2.4) and a better

knowledge of unobservable factors a¤ecting their school performance (Tables 3.1 and 3.2).

19

Table 3.2: Bias and variance of expectations (Math, Fall - Middle school)

Variables / Demographic group All Black Repeaters Mothers w/

high school

correlation of parents and marks 0.22 (0.04) 0.20 (0.05) 0.18 (0.05)** 0.24 (0.05)

mean bias of parental expectations 0.56 (0.13) 0.61 (0.14)* 0.66 (0.16)* 0.50 (0.12)*

variance of parental expectations’errors 0.62 (0.29) 0.60 (0.28) 0.72 (0.33)** 0.56 (0.28)*

variance of parents - "rational" agent 0.13 (0.06) 0.15 (0.07) 0.18 (0.09) 0.11 (0.06)

correlation of students and marks 0.35 (0.04) 0.35 (0.05) 0.32 (0.05) 0.36 (0.05)

mean bias of students’expectations 0.58 (0.11) 0.62 (0.13)* 0.67 (0.13)* 0.52 (0.11)*

variance of students expectations’errors 0.64 (0.22) 0.63 (0.22) 0.80 (0.24)* 0.59 (0.22)*

variance of students - "rational" agent 0.11 (0.05) 0.13 (0.06) 0.23 (0.10)* 0.08 (0.06)**

Bootstrap standard-errors in () - 100 sample replications.

* signi…cantly di¤erent from the "All" group at the 5% level, **signi…cant at the 10% level.

Table 3.3: Bias and variance of expectations (Parents - Math, Fall, High school)

Variables / Demographic group All Black Repeaters Mothers w/

high school

correlation of parents and marks 0.22 (0.12) 0.05 (0.17)** 0.07 (0.15)* 0.28 (0.12)

mean bias of parents’expectations 0.44 (0.10) 0.51 (0.11)* 0.49 (0.15) 0.40 (0.09)**

variance of parents expectations’errors 0.32 (0.15) 0.35 (0.19) 0.41 (0.19)** 0.25 (0.14)*

variance of parents - "rational" agent 0.08 (0.07) 0.14 (0.11) 0.15 (0.10)** 0.04 (0.05)

Bootstrap standard-errors in () - 100 sample replications.

* signi…cantly di¤erent from the "All" group at the 5% level, **signi…cant at the 10% level.

20

Table 3.4: Bias and variance of students’expectations (Math - Spring, High school)

Variables / Demographic group All Black Repeaters Mothers w/

high school

correlation of students and marks 0.31 (0.06) 0.38 (0.06)** 0.35 (0.09) 0.28 (0.07)

mean bias of students’expectations 0.56 (0.18) 0.61 (0.20) 0.55 (0.19) 0.53 (0.18)*

variance of students expectations’errors 0.29 (0.24) 0.30 (0.26) 0.29 (0.25) 0.26 (0.26)

variance of students - "rational" agent 0.00 (0.03) -0.02 (0.05) -0.01 (0.06) 0.02 (0.03)

Bootstrap standard-errors in () - 100 sample replications.

* signi…cantly di¤erent from the "All" group at the 5% level, **signi…cant at the 10% level.

Note that the correlation of parents and students’ private information with actual school

performance increased substantially from elementary school (Table 3.1) to middle school (Table 3.2)

and high school (Tables 3.3 and 3.4), especially among students. In elementary school, parents and

students’private information only had a 14% correlation with actual school performance. However,

during middle and high school the correlation of private information with actual unobserved

factors a¤ecting academic grades was 22% for parents and 30 to 35% for students. Parents are

as knowledgeable as children at an early age, but teenagers have a higher assessment of both

observable and unobservable factors a¤ecting achievement than their parents. This result shows

that the role of private knowledge of unobserved factors increases substantially as children grow

older. Therefore economic models of education and human capital that ignore private information

held by teenagers are making unrealistic assumptions.

Similar results are found for the parents and students’English/Reading forecasts and for the

students’spring quarter expectations, which are available from the author upon request.

5 Conclusions

This paper studies how families learn to predict their academic performance. The prediction errors

of the agents can be explained by several factors: 1) excessive optimism; 2) failure to adjust to

new information on the students’ academic performance; and 3) families rely on noisy private

information. To evaluate the role of these factors, I specify and estimate a model of student

21

achievement and expectation formation. I then use the results of this model to compare the bias

and variance of predictions across demographic groups and how respondents use the information

of past marks in relation to their true predictive value.

I show that respondents are overcon…dent and use noisy private information, with overcon…dence

being smaller for families with higher education. Female students, children of parents with more

age and education have higher marks but their parents have similar expectations as others, which

indicates a lower overcon…dence of the parents in these groups.

There were also racial di¤erences in how families form expectations. Black families are more

optimistically biased than average, but over the years they made similar gains as white families in

their ability to forecast. Parents’predictions in middle school and high school were too focused

on older school scores and adjusted too conservatively to new information. After controlling for

parental education, black students are more likely to stay more years in retention before dropping

out of high school (Lang and Manove, 2011), even if they have similar returns to education as white

families (Carneiro et al., 2005), a higher distaste for schooling (Austen-Smith and Fryer, 2005), and

similar discount rates as white families (Lang and Rudd, 1986). I …nd black parents and students

are more optimistic about their academic scores even after several years of schooling, which may

explain the puzzle of why they stay longer in school.

In elementary school, students had a larger bias and prediction variance than their parents.

Families also updated their expectations more slowly than would be e¢ cient. In middle school,

families’bias and prediction variance decreased a lot, particularly among students who became as

good predictors as their parents. In addition, both parents and students made better use of the

information available in past marks. The accuracy of families’ predictions was worse during high

school. Perhaps high school transition represents a curriculum shock for the BSS students, who

often became high school drop-outs. It is also possible that teenage behavior is hard to predict.

The model also shows that parents and students possess signi…cant private information about

unobservable factors. The accuracy of this private information - as measured by its correlation

with actual academic scores - increased for both parents and students during middle school and

high school. Students in particular have a much higher degree of private information than their

parents during high school. This result shows that the role of private knowledge of unobserved

factors increases substantially as children grow older. Economic models of human capital decisions,

22

therefore, should take into account higher degrees of private information as students grow older.

The results of this paper also suggest a new view about the causes of why families fail to make

more early childhood investments, despite their high return (Cunha, Heckman, Lochner, Masterov,

2005). I show that a large proportion of parents at the start of elementary school expected their

children to receive high marks and reach high levels of education. Overcon…dent parents could

therefore fail to foresee that these investments are essential to their children’s academic success.

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