How much is patience worth?
Discount rates of children and high school completion
Marco Castillo*, Jeffrey L. Jordan**, Ragan Petrie*
*Interdisciplinary Center for Economic Science (ICES), George Mason University, Fairfax, VA 22030 USA
Castillo: [email protected], Petrie: [email protected]
**Agricultural and Applied Economics, University of Georgia, Griffin, GA 30223 USA
Jordan: [email protected]
March 22, 2015
Abstract: We present direct evidence that children with a higher discount rate are less
likely to complete high school on time. The discount rates of 878 children are measured
experimentally at the beginning of 8th grade and compared to graduation outcomes 5 years
later. Controlling for relevant demographic variables our results show that the effect of
the measured discount rate is economically and statistically significant. The difference
in graduation rates between the most patient and least patient student is 10 percentage
points. Controlling for standardized test scores in 7th and 8th grade we find that the
effect of the discount rate on high school completion is not uniform – it matters most for
those whose scholarly performance is poor. Our results are consistent with the existence of
heterogenous non-pecuniary costs to finishing high school and highlight the importance of
taking into account preference diversity and variation in scholarly ability when investigating
the channels of educational attainment.
1
1
Introduction
Recent research on the effects of compulsory education (Oreopoulos, 2006, 2007) shows
large positive effects on income of acquiring one extra year of schooling, with increases
of 10 to 14 percent. Indeed, the returns to high school graduation might even be larger
than previously thought once some of the assumptions in the estimated wage equation
are relaxed (Heckman, Lochner, and Todd, 2003). The positive effects of schooling also
extend to other outcomes. Those forced into one extra year of education are less likely
to report being in poor health, unemployed, and unhappy (Oreopoulos, 2007). Given this
evidence, it is puzzling that a significant proportion of the population do not finish high
school (Heckman and LaFontaine, 2010; Murnane, 2013). Several authors have suggested
that this might be due to the existence of large differences in discount rates or the presence
of non-pecuniary costs of schooling (Heckman et al., 2003; Lang and Ruud, 1986; Murnane,
2013; Oreopoulos, 2007). Evidence consistent with this hypothesis is presented in Eckstein
and Wolpin (1999).1
An individual’s discount rate is the preference parameter that is theoretically related to
future schooling decisions, and the existing literature strongly hints at the importance of
this to high school completion. Nonetheless, there is little direct evidence of this link. The
results that do provide support are indirect.2 For instance, Castillo, Ferraro, Jordan, and
Petrie (2011) show that discount rates predict disciplinary referrals two years in the future,
and disciplinary referrals have been shown to predict high school completion (Alexander,
Entwisle, and Horsey, 1997; Rumberger, 1995). In this paper, we provide evidence of a direct
link between lower discount rates and completing high school by collecting theoretically
motivated (Harrison, Lau, and Williams, 2002) and empirically validated (Castillo et al.,
2011; Sutter, Kocher, Glaetzle-Ruetzler, and Trautmann, 2013) experimental measures of
discount rates of 8th graders and testing if they predict high school completion five years
later.
The experimental discount rate measure is obtained by having children in 8th grade make
choices from a multiple price list that asks them to decide between a fixed amount of money
in 1-month and a larger amount of money in 7-months. This is a front-end delay design
that is simple and has several advantages. First, the procedure is incentive-compatible
and expected to reveal true individual preferences under standard preference assumptions.
Second, by delaying all payments into the future, we eliminate potential confounds due
to mistrust that non-immediate payments will be made. Third, by design, it measures
individual discount rates even if the individual has quasi-hyperbolic preferences.3 These
1
Using data from the Panel Study of Income Dynamics (PSID), these authors estimate a structural model
showing that children who drop out of high school have lower abilities, lower expectations of the returns to
education and higher returns to working while attending high school.
2
(Discuss papers showing that some sort of time-delay question predict several contemporaneous behaviors. Mention also the paper by Golsteyn, Gronqvist, and Lindahl (2014)
3
Consumers with quasi-hyperbolic preferences behave as exponential discounters when comparing streams
1
methodological issues have been shown to influence the measurement of time preferences in
adults (Andreoni and Sprenger, 2012) and are likely to be important with children as well.
Having a theoretically motivated measure of preferences is useful because it allows us
to test additional hypotheses of the human capital accumulation model as it pertains to
high school completion. This not only makes our results easier to interpret, but it also
allows us to relate them to the research on labor economics which suggests preference
heterogeneity is important in determining school outcomes (Eckstein and Wolpin, 1999;
Lang and Ruud, 1986; Oreopoulos, 2007). The theory predicts that discount rates interact
with non-pecuniary costs of schooling and therefore are heterogenous. The intuition is that
discount rates matter most to those facing relatively higher (present) costs of schooling.
Discount rates are less important for those who accumulate human capital at low cost.
Our approach, using economic experiments to test theoretical predictions, can be viewed as
complementary to the research that uses structural models to identify preference parameters
and the costs and benefits of education.4 We directly test theoretical predictions using
experimental and field data, and this methodological approach might be fruitful in other
contexts as well (e.g. risk preferences and behavior).
An important challenge for our study is the inability to observe the educational outcomes
of all the original participants in the experiment. The high school completion status of
about one-third of participants cannot be determined with certainty. Students transferred
to other schools, moved out of the state or country or simply disappeared from the system.5
The inability to track an entire 8th grade cohort to graduation is not uncommon, and
missing data are frequently encountered. Recent research on the measurement of high school
completion in the U.S. (Murnane, 2013) suggests that this attrition is not random. Children
who are more likely to drop out of high school often transfer to other schools (including
home schooling and private schools), and school officials might be lax in following up on
the status of transfers in order to improve graduation statistics.
Given this, we estimate the relationship between discount rates and graduation by allowing for the potential endogeneity of missing outcome data. The basic identification
assumptions of the model are that outcomes affect whether the data are missing or not and
covariates affect these patterns only through their influence on the outcome. To illustrate
how this would apply to our situation, our data shows that missing outcomes are twice
as likely among students with below the median standardized math scores in 7th and 8th
grades than among students with above the median test scores. The assumptions of the
model would say that these data more likely to be missing for those with low grades beof consumption occurring strictly in the future. The front-end delay design therefore identifies discount rates
separate from present-biasedness.
4
This is a two-way street. Structural estimation can help account for unobserved variables that affect
behavior in experiments (see Andersen, Harrison, Lau, and Rutstrom (2008)). It can also help develop
bounds on preferences parameters (e.g. Chetty (2006)).
5
While graduation status is available for all students remaining in the Georgia educational system, we
have access only to the records of those students in the district in which the experiments were implemented.
2
cause they are more likely to dropout, not because having low grades makes it more likely
to have missing dropout data. These assumptions seem sensible in the situation we are
studying. It is easy to observe and verify whether an individual graduated high school.
Because dropping out is socially undesirable and dropouts are more likely to move out of
the school district, then those who dropout would be more likely to have missing data. Test
score data, however, are less likely to be missing. Contrary to graduation outcomes, test
results are more difficult to observe and verify, and moreover, test completion is required
for all students.6 In the methodology section, we discuss how sensible this strategy is for
the data at hand, and in the results section, we provide estimates using alternative methods
and assumptions to test the robustenss of the findings.
We have two main results. First, higher individual discount rates are correlated with a
higher probability of not completing high school in four years. The estimated difference in
graduation rates between the most impatient and least impatient student is 10 percentage
points. This holds controlling for the child’s age, sex, race, and whether the child receives
free or reduced price meals or is in special or gifted education. Second, the effect of the
discount rate is heterogenous. Consistent with the notion that those doing relatively worse
on standardized math tests face larger non-pecuniary costs to educational attainment,7 the
effect of the discount rate on dropping out of high school diminishes as standardized math
scores in 7th and 8th grade increase. The discount rate has a significant effect on dropping
out of high school for those with low test scores in middle school, but no significant effect
for those with average scores. Indeed, controlling for standardized test scores in 7th and
8th grade we find that discount rates do not have a linear effect on graduation rates in some
specifications. This highlights the importance of school performance in the early years of
middle school on future graduation rates (see also Murnane (2013)) and the heterogenous
effect of discount rates on behavior. How well a child has performed in primary and middle
school is likely to be related not only to cognitive ability but also to underlying preferences
(Almlund, Duckworth, Heckman, and Kautz, 2011). Our data are not rich enough as to
distinguish how much of the effect of tests scores is due to preferences or abilities, however,
this is an area that deserves further research.
In sum, our main findings are that children with lower discount rates are more likely
to complete high school on-time and that this relationship is heterogenous, as would be
expected in the presence of differing non-pecuniary costs to educational attainment. Our
results lend support for recent results suggesting that heterogeneity in preferences are important determinants of labor market outcomes (Barsky, Juster, Kimball, and Shapiro, 1997;
Bonin, Dohmen, Falk, Huffman, and Sunde, 2007; Burks, Carpenter, Goette, and Rustichini, 2009; Chabris, Laibson, Morris, Schuldt, and Taubinsky, 2008; Kimball, Sahm, and
Shapiro, 2008; Meier and Sprenger, 2010). Discount rates, i.e. preferences, play an impor6
This approach is developed and described in detail in Ramalho and Smith (2013).
Murnane (2013) shows the importance of math scores in explaining the sex and race gap in graduation
rates.
7
3
tant role in the population at risk of not completing high school. This suggests that uniform
incentives to improve educational performance might have differential impacts across the
population. Optimal policies would need to account for the existence of diverse preferences
and costs of educational attainment. Incentives schemes might not be equally appealing
to students, and perhaps importantly, they might be least appealing to those who would
benefit the most.
Recent studies show a contemporaneous correlation between experimental measures of
risk and time preferences and field behavior. These include occupational choice (Bonin et al.,
2007; Burks et al., 2009), credit card borrowing (Meier and Sprenger, 2010) and smoking,
nutrition and exercise (Chabris et al., 2008). Our study is one of the few longitudinal
studies and, to our knowledge, the first to use an experimental measure of the discount
rate to predict future field behavior. Mischel, Shoda, and Rodriguez (1989) show that
a child’s ability to delay immediate consumption at age 4 predicts social and cognitive
competency during adolescence as well as the ability to deal with stress and frustration
(see also Mischel, Shoda, and Peake (1988); Shoda, Mischel, and Peake (1990)). Because
children in our experiments were asked to choose between quantities that were both in the
future, our study is silent with respect to the importance of impulse-control or presentbiasedness on future field behavior. Instead, it points to the separate importance of the
discount rate apart from self-control.
Using panel data from 1,000 New Zealanders, Moffitt, Arseneault, Belsky, Dickson,
Hancox, Harrington, Houts, Poulton, Roberts, Ross, Sears, Thomson, and Caspi (2011)
show that self-control during the first decade of life predicts better health outcomes, more
wealth and lower participation in crime. Self-control is measured by a composite index of
self-reports and reports by parents and teachers on observed behavior. A potential confound
in measures based on observed behavior is that it is unclear how much of the behavior is due
to the preferences or the constraints faced by the child. Impatient children might differ in
the opportunity cost of their time and/or on the strategies they use during interactions with
other children and adults. Our study makes clear that preferences themselves are partly
responsible for observed behavior.
The paper closest to ours is that of Golsteyn et al. (2014). Using panel data for over
11,000 people in Sweden, the authors show that a measure of time preference obtained at age
13 predicts educational attainment, income level and health outcomes. Time preference is
measured from hypothetical questions asking how likely the respondent would be to choose
an amount of money to be received immediately and a larger amount to be received in
5 years. Our study uses a theoretically-motivated measure and incentivized experiments
to construct the discount rate and presents a shorter time horizon over which to make
decisions.8 We find that discount rates matter most for children with higher non-pecuniary
8
Most incentivized experiments measuring time preferences have delays of 0-7 months. One exception is
Harrison et al. (2002) who elicit responses from adults over time periods as long as 3 years.
4
costs to education and therefore are likely to benefit most from developing non-cognitive
skills.9 In Golsteyn et al. (2014), time preferences are most predictive of the behavior of
relatively more cognitively able children.
The paper is organized as follows. Section 2 presents a simple model of the decision to
drop out of high school. Section 3 discusses the estimation method in the presence of missing
outcome data. Section 4 describes the sample selection, the experiment and the definitions
of high school completion. Section 5 presents the results, and Section 6 concludes.
2
A Simple Model
In this section we present a simple model of the decision to finish high school. The model is
an extension of Oreopoulos (2007) in which we make explicit that the cost to finish might
depend on heterogeneous levels of ability. The model will guide our empirical tests of the
effect of discount rates on high school completion.
A student must choose between finishing high school (S = 1) or not (S = 0). For a
given level of ability, A, the lifetime utility over T + 1 periods is:
V (S, A) = u(c0 ) − φ(S, A) +
Pt=T
t=1 (1
+ d)−t u(ct )
The term u(c0 ) − φ(S, A) accounts for the utility of consumption in period 0 and a
non-pecuniary cost associated with choice S. For those pursuing education (S = 1), this
cost is expected to decrease with the ability of the student. That is, we assume that
φ(1, A) > φ(1, A0 ) for A < A0 . This condition expresses the idea that lower ability students
find it more difficult to attain the knowledge required for graduation. For a discount rate, d,
the last term of the equation represents the geometrically discounted utility of consumption
for time separable preferences over consumption. In our study, we measure paramater d
with economic experiments.
The corresponding intertemporal budget constraint for an interest rate r is the following,
where yt (S) is the income level at education level S:
Pt=T
t=0 (1
+ r)−t ct =
Pt=T
t=0 (1
+ r)−t yt (S)
Given that our simple model only allows for two educational outcomes (graduation or
not), we assume that φ(0, A) = 0. We also assume that the non-pecuniary cost of schooling
is separable into two components. In particular, φ(1, A) = φ0 − h(A). This expression
assumes that the cost decreases linearly in the ability of the student. This assumption is
restrictive but amenable to econometric estimation.
9
Recent research (Gertler, Heckman, Pinto, Zanolini, Vermeersch, Walker, Chang, and GranthamZmcgregor, 2013; Heckman, Pinto, and Savelyev, forthcoming; Heckman, Stixrud, and Urzua, 2006) show
promise on early interventions in developing non-cognitive skills.
5
The first order conditions associated with the intertemporal problem outline above can
be used to obtain conditions which determine whether a student drops out of school or not.
Specifically, we have that a student drops out of school if:
[yt (0) − yt (1)] −
Pt=T
t=1 (1
+ r)−t [yt (1) − yt (0)] +
1+d φ1 −h(A)
1+r u0 (ct )
>0
The first term of the expression accounts for the additional income that a high school
student would obtain by dropping out of school.10 This would imply that, for the type of
jobs available to teenagers, [yt (0) − yt (1)] might be negatively correlated with the student’s
ability to succeed at school. The second term is the present value of the income that a
student would obtain if she finished high school. If wages are increasing in the cognitive
and non-cognitive abilities of the student, conditional on the level of education and market
interest rates, this term would depend also on measures of these abilities.
The final term in the expression captures the effect of individual discount rates on the
decision to drop out of school. The term shows that, conditional on the non-pecuniary cost
of school, more impatient students (higher d’s) are more likely to drop out. It also shows
that the magnitude of the effect of individual discount rates on the decision to drop out of
school is increasing in the non-pecuniary cost to school. If this cost depends on the ability
of the student, less able students are expected to react more strongly to changes in discount
rates.11
In sum, the simple model shows that the discount rate is expected to be positively
correlated with dropping out of high school and the effect of the discount rate is likely to
be heterogenous. The effect will be stronger among those who have higher non-pecuniary
costs to attain schooling. This prediction is intuitive. Patience is most important for those
who have a relatively higher marginal utility of consumption earlier in life rather than later.
We will present empirical evidence consistent with this prediction.
3
Estimation Methodology
We assess the relationship between an individual’s discount rate and on-time high school
completion using Probit regression analysis. The dependent variable equals 1 if a student
does not finish high school in 4 years. This is the current definition for a high school
dropout by the state of Georgia (see www.gadoe.gov) and is commonly used in the education
literature (Murnane, 2013). The regressions include a set of covariates together with the
individual discount rate estimated from experimental data.
The first challenge to the validity of this analysis is the potential endogeneity of measured
discount factors. While the experiments were conducted almost 5 full years prior to the
10
Indeed, empirical evidence suggests that this differential income is larger for those that do drop out
relative to those who remain in school (Eckstein and Wolpin (1999)).
11
We do not consider the possibility of uncertainty in the model. When streams of income are uncertain
we should expect that the present value of future income is smaller. This implies that more risk averse
students will discount future returns to education more heavily.
6
final date for an on-time high school graduation, it is still possible that decisions in the
experiment are correlated with the determinants of high school completion. For instance,
those who are likely to dropout because of family issues might perceive the future as more
uncertain and act more impatiently in the experiment (Halevy, 2008; Saito, 2011). At the
moment, we do not have a way to address the issue of endogeneity of preferences. However,
we will include controls that are contemporenous to the experimental data and are likely
to be correlated with the probability of a timely graduation to control for differences in
backgrounds.
The second challenge is the lack of information on graduation outcomes for all the
students in the original economic experiment. As we will show later, there are about 32
percent of students for whom it is not possible to determine with certainty if they graduated
from high school or not. Restricting the analysis to the subset of complete cases not only
affects the efficiency of the estimates but also requires making the untestable assumption
that outcome data is missing at random (Manski, 2003). If attempts to hide failure to
complete high school are as likely by individuals as bureacrats (Murnane, 2013), estimates
based on complete cases only are likely to be biased.
In the results section we will explore several approaches to deal with missing outcome
data. First, we considered probit and logit regressions of the likelihood of not finishing
high school in 4 years on complete cases only. Estimates on this subsample are consistent,
yet inefficient, if outcome data are missing at random. The estimates of the logit model,
except for the intercept, on the subsample of complete cases remain unbiased under the
weaker assumption that the probability of response conditional on the dependent variable
is independent of the covariates (Allison, 2001; Ramalho and Smith, 2013). These estimates
are therefore potentially informative on the relationship between discount rates and timely
high school completion.
As a second approach, we use the method proposed by Ramalho and Smith (2013) to
deal with non-response in discrete choice models. The problem of missing data is recast as
one of choice-based sampling (Manski and Lerman, 1977; Manski and McFadden, 1981). In
the case of endogenous missing outcome data, observations are neither randomly drawn from
the population nor always observed. The basic identification assumption is that patterns
of missingness of data are independent of covariates conditional on the actual outcome.
That is, covariates affect missingness only through their effect on outcomes. In our data,
we observe that students with low scores on standarized math tests in the 7th and 8th are
more likely to have missing graduation data than students with high scores. The model
assumes that this is because those with low scores are more likely to dropout out of school,
and therefore outcomes may be missing, and not because their scores are low per se. The
assumption would fail if those with low scores are more likely to transfer to other schools
regardless of whether they are more likely to dropout of school or not.
We first introduce some necessary notation. The data are characterized by a vector
7
(y, x, z) where y is a variable that equals 1 if a student does not finish high school in 4 years
and 0 if she graduates. z is a variable that equals 1 if outcome y is observed and 0 if the
outcome is not observed. Finally, x is a set of covariates. Ramalho and Smith (2013) show
that the unconditional likelihood function of vector (y, x, z) is:
lnL = yzln(ω1 P{y = 1|x, θ}fX (x)) + (1 − y)zln(ω0 P{y = 0|x, θ}fX (x))
+(1 − z)ln((1 − ω1 P{y = 1|x, θ} − ω0 P{y = 0|x, θ})fX (x))
In the expression above, the term P{y|x, θ} is a parametric model of the conditional
probability of observing outcome y given covariates x and parameters θ. The term fX (x)
is the marginal probability density of vector x. Note that in the absence of missing data
(z = 1) the model collapses to a weighted likelihood function as encountered in choice-based
sampling. In those models, as well as ours, the term ωy equals
unconditional probability of observing outcome y and
Hy1
Hy1
Q(y) ,
where Q(y) is the
is the probability of observing
(z = 1) outcome y. The authors shows that function fX can be factored out from the
estimation. Also, the problem can be simplified if there is knowledge of the value of Q(y),
the unconditional probability of completing high school in 4 years or not. In the results
section we will present the corresponding maximum-likelihood estimation of this model when
P{y|x, θ} follows a probit model and Q(y = 0) and Q(y = 1) are observed (e.g. countylevel graduation data). In this context, the model requires estimating two additional free
parameters H11 and H01 .
Another approach to deal with missing outcome data is to combine estimations based
on the complete cases of 8th graders with information on graduation at the county level
from official records. Since our sample of 8th graders is a representative sample of the population of high school students in the county, we would expect that if the estimates on the
subsample of complete cases are unbiased, then the extrapolated graduation rates on the
whole sample (complete and incomplete cases) would reproduce the patterns of graduation
at the district level. In other words, we can impose additional moment restrictions on the
estimations using the incomplete data set to reflect the knowledge we have of graduation
outcomes by subgroup at the county level. For instance, records show that the graduation
rate of economically disadvantaged children is 5 percentage points smaller than the average
(38 percent versus 33 percent). We follow Imbens and Lancaster (1994) who derive general method of moment estimators that combine different sources of information. In our
estimations we impose the additional restriction that official records are measured without
error. This is an assumption made out of convenience since we do not have an independent
random sample that would allow us to determine the error in these estimates.
8
4
Sample selection
The setting for our study is a suburban/rural county school district in Georgia.12 The
district is typical of suburban/rural school districts in the U.S. in that income and education
levels are lower compared to urban areas. For example, 2011 per capita income in the district
was $28,305 ($36,979 in Georgia). According to the Georgia Department of Education, 66.7
percent of students in the district graduated in 4 years as of 2013.13 Our experiment was
conducted at all four public middle schools in the district and our sample represents 82%
of the entire student population. The students in our sample come from a broad range of
socio-economic backgrounds (sample statistics are presented in Table 1). At the time of the
experiment, 96% of our participants were 13 or 14 years old (mean=13.80, SD=0.56), while
3% were 15 years old. In Georgia, students can make the decision to drop out of school
at the age of 16. Thus, we elicit discount rates in the period prior to when this important
decision would be made.
4.1
Experimental design
We measure time preferences by eliciting discount rates with the front-end delay design used
by Harrison et al. (2002). Instead of allowing an option of payment immediately after the
experiment, both payments are delayed. This design mitigates the potential for confounding
trust and patience in the experiment and makes the transaction costs of receiving payment
across options the same. In our experiment, subjects are asked, orally and in writing, to
make twenty decisions in total. For each decision, subjects must choose whether they would
prefer $49 one month from now or $49+$X seven months from now. The amount of money,
$X, is strictly positive and increases over the twenty decisions. Table 2 shows the decision
sheet the subject sees.14 For example, in the first decision, a subject is asked if she would
prefer $49 one month from now or $50.83 seven months from now. In the ninth decision, a
subject is asked if she would prefer $49 one month from now or $67.61 seven months from
now. Subjects are asked to make one choice for each of the twenty decisions on the decision
sheet. Based on discussions with teachers and students at other schools, we determined
that the range of $50 to $99 would be considered by adolescents to be “large” payoffs, but
not so large as to potentially cause problems with their parents.
Coller and Williams (1999) and Harrison et al. (2002) argue that one should elicit the
market rates of interest that subjects face so that one can control for arbitrage opportunities
(field censoring) in the econometric analysis. However, our discussions with teachers at the
study site and with similar aged students at other schools led us to believe that students
12
The next two sections draw heavily from Castillo et al. (2011).
See
http://www.gadoe.org/External-Affairs-and-Policy/communications/
Pages/PressReleaseDetails.aspx.
14
Subjects did not see the last two columns indicating the implied annual interest rate and effective interest
rate.
13
9
do not price field investments in terms of interest rates. Thus information and questions
on rates would simply confuse decision making. If subjects were to have access to credit
markets, and these interest rates were binding in the experiment, our estimates would be
lower bounds on the true discount rates.
Economic theories of discounting predict that an individual faced with the decision
sheet in Table 2 would either choose (a) $49 for all decisions, (b) the higher payment for all
decisions, or (c) $49 for a number of decisions starting with Decision 1 and then switch to
the higher payment for the remaining decisions. In other words, if an individual chose to
receive $Y in seven months rather than $49 in one month, then the individual will prefer any
amount $Z > $Y in seven months rather than $49 in one month. Following Harrison et al.
(2002), we call these individuals “consistent” decision-makers. However, in experiments
using decision sheets like the one in Table 2, some individuals are “inconsistent” decisionmakers: they choose $Y in seven months rather than $49 in one month, but then choose
$49 in one month rather than $Z > $Y in seven months. Harrison et al. (2002) and Meier
and Sprenger (2010) found that 4% and 11%, respectively, of their adult subjects were
inconsistent in their choices. Bettinger and Slonim (2007), whose subjects were between
5 and 16 years old, found that 34% of their sample were inconsistent. The proportion of
inconsistent decision-makers in our sample (31%) is closer to that of Bettinger and Slonim
(2007).
In each session, subjects are assigned a unique identification code. This code is private,
and subjects do not know the identification codes of other subjects. Subjects make their
decisions by circling one amount, either $49 or $49+$X, on their decision sheet for each of
the twenty decisions. After subjects make their decisions, each subject puts her decision
sheet in an envelope and the envelopes are collected.
One decision out of the twenty decisions is randomly chosen for payment by taking 20
index cards with the numbers 1-20 written on them, shuffling them in front of the subjects,
presenting them “face down,” and asking a subject to choose one card. The number on the
card is the decision number to be paid for each of the three subjects in each session who are
chosen to receive payment. So, for example, if decision 15 is chosen for payment and one of
the winning subjects circled $83.03, the subject would receive $83.03 in seven months. If
another winning subject circled $49, that subject would receive $49 in one month.
After determining the decision to be paid, all the envelopes are shuffled in front of the
subjects, and three envelopes per session are chosen for payment. The identification codes of
those chosen to receive payment are written on the blackboard. Because identification codes
are kept private by each subject, no other subject knows which subjects have been chosen
to receive payment. Subjects who are chosen to receive payment are paid with a Wal-Mart
gift card by the school principal on the specific date for the decision chosen. We chose to
pay with a Wal-Mart gift card for two reasons. It minimizes potential problems associated
with giving children cash and it can be transformed into many goods that children desire,
10
so it very similar to cash. We chose to have the school administration store and distribute
the cards to assure the children that they would be paid in the future. In all schools,
the principal is regarded as a permanent fixture and interacts regularly with the children.
Within a week of the experiment, the winning subjects stop by the principal’s office to
verify the gift card. On or within a week of the payment date, the subjects go privately to
the principal’s office to pick up their gift cards. Their names and payment are kept private.
Subjects know all of these procedures before making their decisions.
All experiments were conducted by the authors, and 878 8th grade students participated
(ages 13 to 15). One hundred and twenty students were paid an average of $62.88 (std dev
= $18.04), with a total payout of $7,546.17. One month after the experiment, 66 students
received gift cards of $49. Seven months after the experiment, 54 students received gift
cards ranging from $52.71 to $98.02. The experiments were conducted in three sets and
encompass all four middle schools in the school district. The first set was on September 19,
2006. The second was on August 31, 2007, and the third was on August 26, 2008.
4.2
Defining dropouts
Table 4 summarizes the graduation status information for the sample in our study. This
information was provide to us by the Georgia Department of Education. According to Table
4, 57.1 percent of students graduated in four years. Students in the following categories
are know to not have graduated in four years: Death, Expulsion, Financial hardship/Job,
Low grades/school failure, Adult/Post secondary, Lack of attendance, and Still enrolled in
high school. These categories account for 12.5 percent of the students. For the remaining
30.4 percent of students it is not clear whether they graduated in four years or not. For the
analysis, we treat the graduation status of these students as unknown.15
For the subsample for which we know the graduation status, the graduation rate is 82
percent (501 graduates, 110 dropouts). This graduation rate is high in comparison with
the state level graduation rate of 71 percent. Given that the county is known to have a
somewhat lower graduation rate than the state, this suggests that data are not missing at
random. Those less likely to finish high school in four years are more likely to have an
unknown graduation status.
5
Results
Table 5 previews the main patterns in the data of discount rates, math and reading scores
for standarized test in 7th and 8th grades and disciplinary referrals in 8th and 9th grades.16
15
The school system classifies the category Unknown as a dropout as well. We do not follow this practice
since, by definition, we do not know what happen to these students.
16
For test scores, we use the average of 7th and 8th grades. If one of the two scores is missing, we use
whichever one is present in the data. The scores are rescaled to reflect a change in the scoring of the test
between 2006 and 2007. For the number of disciplinary referrals per year, we use the average number from
11
The data are disaggregated by those who we know have not finished high school, those we
know have finished high school and those whose graduation status cannot be verified.
The table shows that those who do not graduate high school, relative to those that do
graduate, are more impatient (higher discount rates, difference in means p-value = 0.0010),
have lower math (p-value = 0.0000) and reading scores (p-value = 0.0001) and have a
higher number of disciplinary referrals (p-value = 0.0000). It also shows that the outcomes
of those whose graduation status is unknown lie between those of graduates and dropouts.
Those who have an unknown graduation status are on average different from those known
to not have graduated (math scores, p-value = 0.0006; reading scores, p-value = 0.1693;
disciplinary referrals, p-value = 0.0018; discount rate, p-value = 0.0057) and from those
known to have graduated (math scores, p-value = 0.0000; reading scores, p-value = 0.0000;
disciplinary referrals, p-value = 0.0000; discount rate, p-value = 0.9616).
Table 6 presents logit and probit regressions for the probability of not finishing high
school in 4 years based on the subsample with complete data. It should be noted that given
the level of missing data in our sample (about 30 percent), a completely non-parametric
approach (e.g. Horowitz and Manski (2001); Manski (2003)) would not be informative. Appendix 2 discusses how to construct outer bounds for the parameters of the model specified
in Table 6 and presents evidence that non-parametric bounds are two wide to determine
the significance of covariates of the probit regression.
All specifications of the model in Table 6 show that age, sex, race, and receiving free
or reduced price meals are significant in predicting failing to complete high school in 4
years. In all tables, we mark parameters with a
+
if they are significant in a one-side test
(p-value < 0.20). We do this because most of the variables have clear theoretical predicted
directions of how they affect high school graduation (e.g. higher discount rates should
lower the probability of graduating) or there is empirical evidence of the direction of the
relationship (e.g. boys graduate at lower rates and those with higher math scores graduate
at higher rates).
Looking at the effect of the discount rate, column 1 in Table 6 presents the estimates
of the logit regression on the failure to graduate in 4 years. The regression shows that a
one standard deviation increase in the discount rate is equivalent to about two-fifths the
effect that being male has on dropping out of high school ( 0.367×0.495
∼ 25 ). This means that
0.394
the difference in the graduation rate of the least and most patient student is 10 percentage
points. As mentioned above, these estimates are likely to be biased due to the fact that
patterns of missing data are not random. Indeed, the predicted probability of dropping out
of school from the model is 20 percent while the reported percent in the county is 33.4.
Nonetheless, Allison (2001) and Ramalho and Smith (2013) show that for the logit model,
only the constant, and not the slope parameters, will be biased if data is missing only due
to the outcome and not the covariates. That is, the slope parameters of the logit model will
8th and 9th grades. Again, if one of the two numbers is missing, we use the one that is present.
12
be unbiased if the data is incomplete because those not graduating are more likely to be
absent in the sample regardless of their personal characteristics. Therefore, the estimates
in column 1 can provide a useful benchmark of the relationship between the discount rate
and graduation.
Column 2 of Table 6 presents a probit regression and shows that similar patterns are
obtained as in column 1 (using a logit model). Column 3 presents estimates accounting
for the effect of standardized math scores in the 7th and 8th grade. It shows that, even
controlling for math performance in middle school, a one standard deviation increase in
the discount rate has an effect similar to one half of being male ( 0.252×0.495
∼ 12 ). Given
0.367
that the difference in graduation of boys and girls is about 9 percentage points, these
estimates suggest that, on average, a child choosing most patiently in the experiment in
comparison with a kid choosing most impatiently (∼1.5 difference) will have a 9 percentage
points difference in the rate of graduation in 4 years. Similar results are obtained using a
probit model. To reiterate, discount rates are predictive of graduation rates even controlling
for test scores in the 7th and 8th grade. These results suggest that discount rates might
be informative of future behavior and that the effect might be large. Columns 5 and 6
investigate if the effect of the discount rate depends on math scores in middle school as we
would expect if kids have heterogenous non-pecuniary cost to schooling that manifest in
standardized test scores. The estimated parameters suggest that the effect of the discount
rate is nonlinear yet imprecisely estimated. This is not entirely surprising given that drop
out rates are severely underestimated in our sample of complete cases.
5.1
Accounting for missing outcome data
We now turn to the estimates that account for the missing outcome data. Table 7 presents
probit regressions using the approach proposed by Ramalho and Smith (2013). Columns 1,
2 and 3 reproduce the regressions in Table 6, and columns 4, 5 and 6 present the estimates
for the subsample of Black children, White children and children receiving free and reduced
price meals. These last three regressions only include the model that allows for a nonlinear
effect of the discount rate on the probability of failing to finish high school in 4 years.
Column 1 in Table 7 confirms the relationship found in previous tables between the
discount rate and failure to complete high school in 4 years. Column 2 suggests that the
effect of the discount rate on graduation disappears once performance on math tests is
included. Column 2 also shows that both the discount rate and sex have no significant
effects in this specification. This result suggests that accounting for missing outcome data
is important. The last four columns of the table allow for a nonlinear effect of the discount
rate by math scores and show a consistent effect. For those with lower match scores,
being more impatient increases the likelihood of not completing high school on time. The
parameter estimates are similar across all the subsamples for which we have county-level
graduation data. The consistency of the estimates across subgroups lends support for the
13
assumption that data missingness depends on outcomes and not covariates.
Table A1 in Appendix 1 reproduces the results in Table 7 for all subgroups in our
sample. We observe that while the nonlinear relationship between failure to graduate in 4
years and discount rates is stable across subgroups, other specifications are not. The direct
effect of discount rate on graduation is not significant for white children, but it is for black
children and those on free and reduced price lunch. Indeed, children on free and reduced
lunch with a higher discount rate are less likely to graduate in all specifications. Table A2
shows logit regressions on complete cases only for each one of these groups and confirms
this heterogeneity.17 Recall that the slope parameters of these regressions are not biased
if missing data is independent of covariates conditional on outcomes. Table A2 shows that
discount rates are significant for Black children and children receiving free and reduced price
lunch even controlling for standardized test scores in 7th and 8th grade. In these groups,
the estimated difference in graduation between the least and most patient student is 18.5%
and 17.3% respectively. These differences are 13.4% and 10.3% once test scores in 7th and
8th grades are accounted for.
The bottom panel of Table 7 reports the predicted rate of failure to complete high
school in 4 years for each of the models and each of the subpopulations in the sample for
which we have graduation data at the county level. The observed county-level graduation
rate is also listed by subgroup. Looking at the third column in the bottom panel, we see
that the model does not adequately capture the variance in behavior of Black and White
children. It underestimates the graduation rate of Black children (31.5% versus 37.8%)
and overestimates that of White children (31.3% versus 28.5%). However, the estimated
graduation rate of children receiving free and reduced price meals is closer to the actual
rate (39.4% versus 38.5%). Not surprinsingly, columns 4, 5 and 6 show that the model
performs better when each of these subpopulations are estimated separately. Importantly,
these estimations show that the measured relationship between the discount rate and failure
to graduate in 4 years is robust across populations.
5.2
Robustness checks
As a robustness check on the results in Table 7, we estimate model (3) under alternative
assumptions of the rate of failure to graduate. These results are presented in the Appendix
(Table A4). These estimates test the sensitivity of the results to assumptions on graduation rates for the county as a whole and are an important check given that county-level
graduation rates also could be measured with error. Table A4 shows that the nonlinear
relationship between the discount rate and the failure to graduate in 4 years remains significant for county-level rates that range from 25 percent to 37.5 percent. The table also
shows that the significance of the discount rate diminishes as those with missing data are
17
The same pattern is observed in Table 3 which presents estimation using probit regression on complete
cases instead.
14
less likely to have graduated in 4 years. Note that given the implied observed graduation
rate in 4 years in Table 4 (501 graduates out of 611 complete cases) and the proportion of
complete cases (69.6 percent) all the estimated models in Table A4 imply that the failure to
graduate among those with incomplete data is very high. For instance, an assumed rate of
failure to graduate in 4 years of 25 percent implies that 54.8 percent of those with missing
data did not graduate in 4 years ( 0.25−0.696×0.12
). An assumed rate of failure to graduate in
1−0.696
4 years of 37.5 percent implies that 95.9 percent of those with missing data did not graduate
in 4 years ( 0.375−0.696×0.12
). The table therefore shows that the found relation between the
1−0.696
discount rate and failure to graduate in 4 years is robust.
Next, we test the robustness of the results by using an alternative estimation method
using complete cases only and additional moment conditions. The estimation assumes that
the probability of failing to graduate in 4 years follows a probit model and is estimated
using the generalized method of moments. We add 3 additional moment conditions to the
model following Imbens and Lancaster (1994)’s approach which combines micro data with
aggregate data. In our case, the micro data is the sample of children in the study and
the aggreggate data is the official rate of graduation for different subpopulations at the
county level. The first condition is that the expected probability of failure to graduate in
4 years for White children is equal to the observed county-level graduation rate of White
children. This moment condition uses all the data in our sample since it is based on what
the model predicts for the whole population not just the complete cases. The second and
third conditions are similarly defined for Black children and children receiving free and
reduced price meals.
The estimates of this model are presented in Table 8 and show similar results to those
in Table 7. The first column of Table 8 shows that a one standard deviation increase in
the discount rate is equivalent to about two-fifths the effect that being male has on failing
to graduate in 4 years ( 0.459×0.495
∼ 52 ). The second column shows that the effect of the
0.523
discount rate on graduation disappers when performance in standardized math exams in the
7th and 8th grade are account for. The last column shows that the effect of the discount rate
on graduation is nonlinear. The bottom panel of Table 8 shows the predicted probability
of not graduating in 4 years by subpopulation. As in Table 7, the model underpredicts
the probability that a Black student will not finish high school in 4 years. The Hansen
overidentification tests show that the additional moment conditions are violated. The likely
reason for this is that the county-level estimates are biased. This is possible due to the fact
that students move out of the district or transfer to other schools. The second reason is
that the underlying distribution of characteristics in our sample does not coincide exactly
with that used to calculate official graduation rates. While this is reason to be cautious
in interpreting the results, we find that they are similar to those in Table 7 which uses an
alternative identification strategy.
Table 9 presents an alternative estimation strategy to measure the effect of the discount
15
rate on graduation status by using all the data. The dependent variable takes the value of
0 if the student graduates in 4 years, 1 if the status of graduation is unknown and 2 if the
student did not finish high school in 4 years. In this model, we assume that those with an
unknown graduation status have a probability of not finishing high school that is strictly
between 0 and 1. This table estimates ordinal probit and ordinal logit regressions using the
same specifications in Tables 6, 7 and 8. This table confirms that the effect of the discount
rate on graduation follows a similar pattern as those reported previously.
Table 10 presents estimates under the assumption that either all students with missing
outcome data graduated or all students with missing outcome data failed to graduate in 4
years. Both these assumptions are extreme and most likely erroneous. This table shows
that discount rates have a significant effect in graduation rates under both assumptions.
However, alternative assumptions imply a different relationship between discount rates and
graduation.
We should also remark that our estimates of the effect of the discount rate on high
school completion are likely affected by measurement error. We only have one measure of
the discount rate per student and, as mentioned in Section 4.1, about one-third of children
make an inconsistent (non-monotonic) choice. To put this in perspective, the scale reliability
coefficient (Cronbach alpha) for the math standardized tests in the 7th and 8th grade is
0.9009. That is, standarized math scores are a reliable measure of acquired knowledge.
Regarding experimental measures of preferences, we have evidence that for risk preferences
in this population (Castillo, Jordan, and Petrie, 2014) measurement error is important and
not accounting for it would lead to accepting the null hypothesis of no effect of preferences
on field behavior too often.18 We suspect some measurement error may hold for discount
rates. It is therefore remarkable that the discount rate measures are able to predict behavior
5 years in the future. Finally, the estimations presented in this section could be affected
by the degree of risk aversion of the subjects (Andersen et al., 2008). Unfortunately, we do
not have measures of risk attitudes for all the students in our sample, so we cannot account
for risk attitudes.
In sum, our estimates show that the discount rate predicts high school completion and
that this effect is heterogenous as would be expected if the costs of schooling are different
across the population. The effects can be very large among students who are ex-ante more
likely to not finish high school in 4 years.
5.3
Magnitude of effects
The results so far point to a robust and signficant nonlinear relationship between the discount rate and the probability of graduating on time. We next ask, what is the magnitude
of the effect of the discount rate on the failure to graduate in 4 years? Table 11 presents
some results to address this question. The table shows the probability of dropping out
18
We found that the scale reliability coefficient between the answers to five lottery questions was 0.3117.
16
for different values of math scores in 7th and 8th grade and discount rates based on the
estimates of column 3 in Table 7. The estimates presented in the table correspond to a
black, 13 year-old male who receives free and reduced price meals. The first column shows
the value of the discount rate and the average math score for 7th and 8th grades. The
range of discount rates and math scores we consider is from 50 to 150 and from 0.45 to 0.55.
Fifty-six percent of discount rates and 44 percent of math scores fall within these ranges.
Looking at the first row of Table 11, the estimates predict that a black, 13 year-old male
who receives free and reduced price meal, has a discount rate of 50 and an average math
score of 0.45 has a probability of dropping out of school of 37.4 percent. Similarly, row 5
shows that if this student had a discount rate of 150 instead his probability of dropping out
would be 56.2. The bottom panel tests whether the change in the probability of dropping
out is significant across these two scenarios. Row 16 shows that indeed the difference in
the probability of dropping out for a 100 point increase in the discount rate for a test score
of 0.45 (18.8 percent change) is significant. Row 17 of shows that a smaller difference in
discount rates (100 versus 150 or about one standard deviation of the discount rate) would
also predict a significant reduction (9.3 percent) in the probability of dropping out of high
school. This shows the effects of the discount rate on dropping out are very large.
Table 11 also allows us to examine the heterogeneous costs to schooling on dropping out.
For instance, the probability of dropping out of high school for a student with a discount
factor of 50 and an average math scores of 0.55 is 6.3 percent (row 11). This is 31 percentage
points less than the probability of dropping out for a student with an average math score of
0.45. This difference is highly significant (row 22). Indeed, Table 11 shows that math scores
in 7th and 8th grade are good predictors of dropping out of school. It also confirms that the
effect of the discount rate is heterogenous. Discount rates have a negative relationship with
graduation rates among those with poorer performance in middle school. The relationship
is of the opposite sign for those doing better academically. The data reveals that 13 percent
of those with math scores above 0.55 drop out of high school while 62 percent of those with
scores below 0.45 do.19 For those doing well academically in middle school, preferences are
not predictive of graduation, however, they are informative for those doing poorly. Table
12 repeats the analysis but using the estimates from column 3 in Table 8. We observe the
same pattern as in Table 1.
The effect of a child’s discount rate on completing high school on time is large. A 50
point increase in the discount rate (equivalent to one standard deviation) of a student with
a median math score reduces the probability of graduating by 9.3 percentage points. Also,
the effect is larger for those who are doing poorly in school. For a given discount rate, a
child whose math score is in the lower quintile of the distribution is almost five times more
likely to drop out of school than a child whose math score is in the upper quartile.
19
The calculation is based on the subsample with known graduation status.
17
6
Conclusions
We set out to investigate if the individual discount rate of children predicts on-time graduation rates. To do this, we collected an experimental measure of the individual discount rate
of children in 8th grade and tested if it predicted completing high school five years later.
We find that an individual with a higher discount rate is less likely to complete high school
on time, even controlling for demographics and test scores, and that this effect is heterogeneous by academic performance. The effect of a one standard deviation increase in the
discount rate is comparable to two-fifths of the effect of being a male in the probability of
dropping out of school. Importantly, the discount rate is a significant predictor of behavior
for those with relatively lower standardized math scores in 7th and 8th grades. Children
who perform well in middle school are largely unaffected by their time preferences, and this
is likely due to the fact that they are unlikely to drop out of high school anyway. That is,
impatience matters for those children who are relatively more at risk of not graduating.
Our research is a direct test of the hypothesis that unobservable differences in preferences and abilities are important determinants of labor market outcomes. Previous evidence
on the importance of preferences has been mainly indirect. For instance, structural estimations of the decision to drop out of high school point to the importance of unaccounted
preferences, beliefs, non-pecuniary costs of schooling and productivity. Our experiments
show that measured preferences can predict behavior. Given that experimental measures
are likely to suffer from significant measurement error, our results can be encouraging. Theory and experiments can be combined to derive precise testable hypotheses, and collecting
theoretically motivated measures is important in order to derive policy implications.
Not finishing high school is associated with a large set of labor marekt outcomes. We
therefore expect that the ability of experimental measures to predict other life outcomes
will remain. This suggests that focus on interventions aimed at altering preferences or
interventions that take into consideration the heterogeneity is granted.
18
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22
Table 1. Descriptive Statistics
Variable
mean
sd
N
Age
13.8
0.56
866
Male
48.4
50.0
847
Black
46.6
49.9
847
Hispanic & Multiracial
4.3
20.2
847
Free & reduced price meal
63.5
48.2
846
Math score (7th & 8th grade)
0.53
0.09
846
Reading score (7th & 8th grade)
821.5
25.3
845
Special education
24.8
43.2
878
Gifted
8.8
28.3
878
Disciplinary referrals (7th & 8th grade)
2.0
2.9
863
Table 2. Decision Sheet
Paid one
Decision
month from
or
now
Paid Seven
Implied annual
Implied annual
months from
interest rate
effective interest
now
rate
1
$49
or
$50.83
7.35
7.60
2
$49
or
$52.71
14.70
15.73
3
$49
or
$54.66
22.05
24.42
4
$49
or
$56.66
29.40
33.70
5
$49
or
$58.72
36.75
43.62
6
$49
or
$60.85
44.10
54.20
7
$49
or
$63.04
51.45
65.50
8
$49
or
$65.29
58.80
77.54
9
$49
or
$67.61
66.15
90.39
10
$49
or
$70.00
73.50
104.09
11
$49
or
$72.46
80.25
118.68
12
$49
or
$74.99
88.20
134.22
13
$49
or
$77.59
95.55
150.77
14
$49
or
$80.27
102.90
168.38
15
$49
or
$83.03
110.25
187.13
16
$49
or
$85.86
117.60
207.06
17
$49
or
$88.78
124.95
228.26
18
$49
or
$91.77
132.30
250.79
19
$49
or
$94.85
139.65
274.73
20
$49
or
$98.02
147.00
300.16
Note: that subjects did not see the last two columns in this table. These columns are included
to show the implied annual interest rate and effective interest rate as sociated with each choice.
23
Table 3. Distribution of Discount Rates
Discount rate (d.r.)
Frequency
Percent
d.r. ≤ 20
122
13.9
20 < d.r. ≤ 40
44
5.0
40 < d.r. ≤ 60
129
14.7
60 < d.r. ≤ 80
120
13.7
80 < d.r. ≤ 100
103
11.7
100 < d.r. ≤ 120
50
5.7
120 < d.r. ≤ 140
102
11.6
d.r. ≥ 140
208
23.7
Total
878
100.0
Table 4. Distribution of high school outcomes
for the cohort in the experiment
Outcome
Cases
Percent
Court (D)
1
0.11
Death (U)
2
0.23
Expelled (D)
11
1.25
Financial hardship/Job (D)
1
0.11
Graduated (G)
501
57.06
Home study (U)
16
1.82
Incarcerated (D)
2
0.23
Private school (U)
49
5.58
Low grades/school failure (D)
4
0.46
Adult/Post secondary (D)
28
3.19
Lack of attendance (D)
37
4.21
Transfer to other public school (U)
68
7.74
Unknown (U)
34
3.87
Transfer to another school in the state (U)
14
1.59
Out of the state (U)
63
7.18
Transfer to private school (U)
3
0.34
Juvenile justice (U)
7
0.80
Still in school (D)
26
2.96
Unclassified (U)
11
1.25
Total
878
100.0
U = status not verified, D = dropout, G = graduated in four years.
24
Table 5. Behavior by graduation outcome
Discount
Math
Reading
Disciplinary
Group
rate
score
score
referrals
Graduated in 4 years
82.8
0.564
828.0
1.25
s.e.
2.14
0.004
0.826
0.091
N
501
496
496
501
Unverified+
85.6
0.502
814.1
2.71
s.e.
3.07
0.005
1.07
0.205
N
268
247
246
254
Did not graduate−
101.0
0.474
807.3
4.07
s.e.
4.71
0.006
4.84
0.379
N
109
103
103
108
Total
85.8
0.535
821.5
2.04
s.e.
1.67
0.003
0.869
0.099
N
878
846
845
863
+
Home schooling, move out of the country, private school, military, pregnant, serious ill-
ness/accident, transferred to another public school, unknown, advanced to another school system, transferred to another school in the system, transferred out of state, transfer to private
school, transferred to public school, school choice, USCO, transferred under jurisdiction of Department of Juvenile Justice, not subject to compulsory education.
incarcerated, low grades/school failure, lack of attendance.
25
−
Court or legal, expelled,
Table 6. Failure to complete high school in 4 years
Estimations based on complete cases only
Logit
Probit
Logit
Probit
Variable
(1)
(2)
(3)
(4)
Discount rate (d.r.)
0.367∗∗
0.677∗∗∗
0.252+
0.459+
[0.145]
[0.263]
[0.157]
[0.281]
(0.012)
(0.010)
(0.109)
(0.103)
Math score (7th & 8th grade)
-9.843∗∗∗
-18.172∗∗∗
[1.437]
[2.724]
(0.000)
(0.000)
Math score (7th & 8th grade)×d.r.
Age
Male
Black
Hispanic & multiracial
Free & reduced meal
Special education
Inconsistent
Constant
N
log-likelihood
0.572∗∗∗
[0.124]
(0.000)
0.394∗∗∗
[0.141]
(0.005)
-0.572∗∗∗
[0.163]
(0.000)
-0.487
[0.380]
(0.200)
1.042∗∗∗
[0.189]
(0.000)
0.557∗∗∗
[0.150]
(0.000)
0.032
[0.152]
(0.832)
-10.005∗∗∗
[1.703]
(0.000)
1.000∗∗∗
[0.220]
(0.000)
0.653∗∗∗
[0.253]
(0.010)
-0.954∗∗∗
[0.285]
(0.001)
-0.822
[0.682]
(0.228)
1.891∗∗∗
[0.355]
(0.000)
0.973∗∗∗
[0.260]
(0.000)
0.040
[0.272]
(0.884)
-17.560∗∗∗
[3.036]
(0.000)
0.443∗∗∗
[0.132]
(0.001)
0.367∗∗
[0.155]
(0.018)
-0.668∗∗∗
[0.177]
(0.000)
-0.223
[0.405]
(0.581)
0.891∗∗∗
[0.207]
(0.000)
0.027
[0.172]
(0.875)
-0.096
[0.166]
(0.564)
-2.710+
[2.014]
(0.179)
0.760∗∗∗
[0.230]
(0.001)
0.583∗∗
[0.278]
(0.036)
-1.144∗∗∗
[0.315]
(0.000)
-0.349
[0.720]
(0.628)
1.618∗∗∗
[0.384]
(0.000)
0.011
[0.301]
(0.970)
-0.138
[0.298]
(0.643)
-4.092
[3.531]
(0.247)
596
596
588
588
-218.43976
-219.04879
-182.69886
-183.05296
Standard errors in brackets, p-values in parentheses.
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
26
Logit
(5)
1.570
[1.394]
(0.260)
-7.575∗∗∗
[2.715]
(0.005)
-2.655
[2.787]
(0.341)
0.446∗∗∗
[0.132]
(0.001)
0.359∗∗
[0.155]
(0.021)
-0.658∗∗∗
[0.177]
(0.000)
-0.196
[0.402]
(0.626)
0.889∗∗∗
[0.207]
(0.000)
0.027
[0.172]
(0.876)
-0.097
[0.167]
(0.561)
-3.886∗
[2.351]
(0.098)
Probit
(6)
2.410
[2.539]
(0.343)
-14.732∗∗∗
[5.081]
(0.004)
-3.982
[5.148]
(0.439)
0.768∗∗∗
[0.230]
(0.001)
0.570∗∗
[0.279]
(0.041)
-1.130∗∗∗
[0.316]
(0.000)
-0.322
[0.719]
(0.655)
1.618∗∗∗
[0.385]
(0.000)
0.007
[0.302]
(0.982)
-0.140
[0.300]
(0.642)
-5.901+
[4.206]
(0.161)
588
-182.25021
588
-182.75742
Table 7. Failure to complete high school in 4 years
Maximum likelihood estimations accounting for missing outcome data1
Full sample
33.4%
(2)
0.113
[0.162]
(0.485)
-15.405∗∗∗
[2.719]
(0.000)
Rate of failure to graduate in 4 years
Discount rate (d.r.)
Math score (7th & 8th grade)
Math score (7th & 8th grade)×d.r.
Age
Male
Black
Hispanic & multiracial
Free & reduced price meal
Special education
Inconsistent
Constant
Probability of observing a dropout
Probability of observing a graduation
N
Group (observed rate)
(1)
0.253∗
[0.150]
(0.093)
Blacks
37.8%
(4)
5.131∗∗
[2.443]
(0.036)
-8.444∗∗
[3.515]
(0.016)
-10.062∗∗
[5.100]
(0.049)
0.578∗∗∗
[0.174]
(0.001)
0.121
[0.207]
(0.557)
(3)
5.121∗∗∗
[1.798]
(0.004)
-6.943∗∗∗
[2.248]
(0.002)
-10.317∗∗∗
[3.715]
(0.006)
0.708∗∗∗
0.494∗∗∗
0.588∗∗∗
[0.126]
[0.124]
[0.127]
(0.000)
(0.000)
(0.000)
0.275∗∗
0.150
0.119
[0.136]
[0.152]
[0.158]
(0.042)
(0.325)
(0.452)
-0.666∗∗∗
-0.814∗∗∗
-0.794∗∗∗
[0.145]
[0.183]
[0.185]
(0.000)
(0.000)
(0.000)
-0.239
0.088
0.085
[0.355]
[0.377]
[0.372]
(0.500)
(0.816)
(0.820)
0.976∗∗∗
0.653∗∗∗
0.679∗∗∗
0.751+
[0.225]
[0.218]
[0.220]
[0.542]
(0.000)
(0.003)
(0.002)
(0.166)
0.590∗∗∗
-0.174
-0.180
0.012
[0.141]
[0.182]
[0.184]
[0.226]
(0.000)
(0.339)
(0.328)
(0.959)
0.107
-0.057
-0.040
-0.013
[0.144]
[0.158]
[0.165]
[0.209]
(0.457)
(0.719)
(0.809)
(0.949)
-11.289∗∗∗
0.373
-5.079∗∗
-5.167+
[1.716]
[2.141]
[2.127]
[3.221]
(0.000)
(0.862)
(0.017)
(0.109)
0.143∗∗∗
0.129∗∗∗
0.130∗∗∗
0.148∗∗∗
[0.019]
[0.014]
[0.013]
[0.021]
(0.000)
(0.000)
(0.000)
(0.000)
0.551∗∗∗
0.572∗∗∗
0.571∗∗∗
0.582∗∗∗
[0.025]
[0.022]
[0.020]
[0.027]
(0.000)
(0.000)
(0.000)
(0.000)
840
828
828
385
Predicted probability of not graduating in 4
Full sample (0.334)
0.389
0.316
0.314
Blacks (0.378)
0.480
0.314
0.315
0.342
Whites (0.285)
0.305
0.315
0.313
Free & reduced price meal (0.385)
0.525
0.392
0.394
1 Estimations are based on Ramalho and Smith (2013)
Standard errors in brackets, p-values in parentheses.
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
27
Whites
28.5%
(5)
4.949∗
[2.584]
(0.056)
-7.594∗∗
[3.371]
(0.024)
-10.901∗∗
[5.305]
(0.040)
0.606∗∗∗
[0.201]
(0.003)
0.106
[0.256]
(0.679)
Free & red.
price meal
38.5%
(6)
4.977∗∗
[1.969]
(0.012)
-6.523∗∗∗
[2.487]
(0.009)
-9.768∗∗
[4.086]
(0.017)
0.576∗∗∗
[0.149]
(0.000)
0.066
[0.182]
(0.717)
-0.721∗∗∗
[0.207]
(0.001)
0.237
[0.411]
(0.565)
0.723∗∗∗
[0.255]
(0.005)
-0.709∗∗
-0.036
[0.333]
[0.204]
(0.033)
(0.861)
-0.052
-0.012
[0.309]
[0.188]
(0.866)
(0.950)
-4.674+
-4.669∗
[3.186]
[2.474]
(0.142)
(0.059)
0.119∗∗∗
0.162∗∗∗
[0.018]
[0.016]
(0.000)
(0.000)
0.561∗∗∗
0.529∗∗∗
[0.029]
[0.024]
(0.000)
(0.000)
394
524
years by group
0.285
0.390
Table 8. Failure to complete high school in 4 years
The regressions estimate a probit model on the complete cases with added moment conditions to
match the expected graduation rates at the district level following Imbens and Lancaster (1994)
District level failure to graduate in 4 years rates (2013 5-year cohort)
Blacks = 37.8%, White = 28.5%, Free & reduced price meal = 43.0%
Variable
Discount rate (d.r.)
(1)
0.459∗∗∗
[0.137]
(0.001)
(2)
0.032
[0.165]
(0.848)
-14.863∗∗∗
[2.003]
(0.000)
(3)
5.737∗∗∗
[1.731]
(0.001)
Math score (7th & 8th grade)
-5.104∗∗∗
[1.801]
(0.005)
Math score (7th & 8th grade)×d.r.
-11.610∗∗∗
[3.524]
(0.001)
Age
0.845∗∗∗
0.589∗∗∗
0.556∗∗∗
[0.143]
[0.137]
[0.132]
(0.000)
(0.000)
(0.000)
Male
0.523∗∗∗
0.423∗∗∗
0.505∗∗∗
[0.126]
[0.156]
[0.155]
(0.000)
(0.007)
(0.001)
Black
-0.310∗∗∗
-0.631∗∗∗
-0.576∗∗∗
[0.077]
[0.118]
[0.112]
(0.000)
(0.000)
(0.000)
Hispanic & multiracial
-2.994+
-3.072+
-2.893
[2.015]
[2.234]
[2.400]
(0.137)
(0.169)
(0.228)
Free & reduced price meal
0.528∗∗∗
0.572∗∗∗
0.551∗∗∗
[0.087]
[0.124]
[0.121]
(0.000)
(0.000)
(0.000)
Special education
0.706∗∗∗
-0.489∗∗
-0.469∗∗
[0.152]
[0.223]
[0.235]
(0.000)
(0.029)
(0.046)
Inconsistent
0.084
0.275∗
0.185
[0.141]
[0.161]
[0.162]
(0.551)
(0.087)
(0.255)
Constant
-13.301∗∗∗
-1.365
-5.771∗∗∗
[1.975]
[2.124]
[2.085]
(0.000)
(0.520)
(0.006)
N
846
832
832
Hansen’s J (χ2 (3)) (p-value)
74.2 (0.000)
77.5 (0.000)
75.2 (0.000)
Group (observed)
Predicted probability of not graduating in 4 years
Full sample (0.334)
0.304
0.292
0.291
Blacks (0.378)
0.335
0.317
0.316
Whites (0.285)
0.305
0.296
0.296
Free & reduced price meal (0.385)
0.371
0.357
0.356
Standard errors in brackets, p-values in parentheses.
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
28
Table 9. Failure to graduate high school in 4 years
Dependent variable: y = 0 if graduated, 1 if unknown, 2 if failed to graduate in 4 years
Multinomial probit
Multinomial logit
(1)
(2)
(3)
(4)
(5)
DR
0.161∗∗
[0.076]
(0.035)
0.067
[0.088]
(0.442)
-6.411∗∗∗
[0.744]
(0.000)
0.498∗∗∗
[0.075]
(0.000)
0.211∗∗∗
[0.071]
(0.003)
-0.417∗∗∗
[0.116]
(0.000)
-0.109
[0.179]
(0.543)
0.551∗∗∗
[0.103]
(0.000)
0.443∗∗∗
[0.103]
(0.000)
0.063
[0.087]
(0.467)
0.356∗∗∗
[0.081]
(0.000)
0.204∗∗
[0.089]
(0.022)
-0.589∗∗∗
[0.101]
(0.000)
-0.128
[0.199]
(0.520)
0.483∗∗∗
[0.103]
(0.000)
0.069
[0.108]
(0.520)
-0.073
[0.093]
(0.436)
Math score (7th & 8th grade)
Math score (7th & 8th grade)×d.r.
Age
Male
Black
Hispanic & multiracial
Free & reduced price meal
Special education
Inconsistent
—
c ons
0.999∗
[0.581]
(0.086)
-4.937∗∗∗
[1.192]
(0.000)
-1.841+
[1.144]
(0.108)
0.358∗∗∗
[0.081]
(0.000)
0.202∗∗
[0.089]
(0.024)
-0.590∗∗∗
[0.102]
(0.000)
-0.120
[0.200]
(0.547)
0.486∗∗∗
[0.103]
(0.000)
0.062
[0.108]
(0.569)
-0.070
[0.094]
(0.458)
0.265∗
[0.145]
(0.068)
0.102
[0.149]
(0.496)
-11.242∗∗∗
[1.296]
(0.000)
0.870∗∗∗
[0.137]
(0.000)
0.316∗∗
[0.146]
(0.030)
-0.675∗∗∗
[0.165]
(0.000)
-0.133
[0.329]
(0.686)
0.861∗∗∗
[0.173]
(0.000)
0.752∗∗∗
[0.166]
(0.000)
0.092
[0.156]
(0.555)
0.612∗∗∗
[0.140]
(0.000)
0.294∗
[0.153]
(0.056)
-0.990∗∗∗
[0.178]
(0.000)
-0.169
[0.338]
(0.617)
0.783∗∗∗
[0.180]
(0.000)
0.091
[0.185]
(0.622)
-0.116
[0.160]
(0.466)
1.567+
[1.004]
(0.118)
-8.862∗∗∗
[2.077]
(0.000)
-2.921+
[1.996]
(0.143)
0.617∗∗∗
[0.140]
(0.000)
0.292∗
[0.154]
(0.057)
-0.991∗∗∗
[0.178]
(0.000)
-0.165
[0.338]
(0.627)
0.783∗∗∗
[0.180]
(0.000)
0.079
[0.186]
(0.671)
-0.109
[0.161]
(0.496)
832
-653.7
832
-652.8
846
832
832
846
-717.5
-653.1
-652.0
-718.5
Robust standard erros in brackets, p-values in parentheses
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
29
(6)
Table 10. Failure to graduate high school in 4 years (logistic regressions)
Dependent variable: y = 0 if graduated, X if unknown, 1 if failed to graduate in 4 years
Missing = graduated
Missing = Not graduated
X=0
X=1
(1)
(2)
(3)
(4)
(5)
(6)
Discount rate (d.r.)
0.627***
[0.232]
(0.007)
0.421*
[0.237]
(0.075)
-11.241***
[2.105]
(0.000)
0.638***
[0.187]
(0.001)
0.632***
[0.232]
(0.006)
-0.538**
[0.247]
(0.030)
-0.743
[0.642]
(0.247)
1.438***
[0.324]
(0.000)
0.688***
[0.232]
(0.003)
-0.181
[0.244]
(0.459)
-12.715***
[2.581]
(0.000)
0.395**
[0.196]
(0.043)
0.652***
[0.246]
(0.008)
-0.674***
[0.259]
(0.009)
-0.495
[0.654]
(0.450)
1.243***
[0.333]
(0.000)
0.054
[0.260]
(0.836)
-0.342
[0.257]
(0.183)
-3.040
[3.081]
(0.324)
Math score (7th & 8th grade)
Math score (7th & 8th grade)×d.r.
Age
Male
Black
Hispanic & multiracial
Free & reduced price meal
Special education
Inconsistent
Constant
Observations
-0.495
[1.868]
(0.791)
-13.012***
[4.204]
(0.002)
1.916
[3.879]
(0.621)
0.390**
[0.196]
(0.046)
0.655***
[0.246]
(0.008)
-0.677***
[0.260]
(0.009)
-0.501
[0.655]
(0.444)
1.243***
[0.333]
(0.000)
0.056
[0.260]
(0.829)
-0.344
[0.257]
(0.181)
-2.122
[3.605]
(0.556)
0.165
[0.158]
(0.295)
0.004
[0.168]
(0.979)
-11.866***
[1.372]
(0.000)
0.945***
[0.151]
(0.000)
0.239
[0.155]
(0.123)
-0.791***
[0.183]
(0.000)
-0.049
[0.373]
(0.895)
0.835***
[0.188]
(0.000)
0.771***
[0.178]
(0.000)
0.221
[0.168]
(0.187)
-14.085***
[2.062]
(0.000)
0.714***
[0.158]
(0.000)
0.173
[0.166]
(0.299)
-1.125***
[0.199]
(0.000)
-0.077
[0.398]
(0.847)
0.742***
[0.202]
(0.000)
0.092
[0.199]
(0.644)
0.008
[0.179]
(0.966)
-4.036*
[2.361]
(0.087)
2.140*
[1.277]
(0.094)
-8.565***
[2.316]
(0.000)
-4.172*
[2.474]
(0.092)
0.719***
[0.158]
(0.000)
0.169
[0.167]
(0.311)
-1.128***
[0.200]
(0.000)
-0.068
[0.397]
(0.864)
0.742***
[0.202]
(0.000)
0.075
[0.200]
(0.709)
0.016
[0.180]
(0.931)
-5.811**
[2.577]
(0.024)
832
832
840
828
828
846
Robust standard erros in brackets, p-values in parentheses
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
30
Table 11. Probability of failing to graduate in 4 years
(estimates based on model (3) in Table 7
Assumptions: Male, Black, Age = 13, Free & reduced price meal
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
(Discount rate, math score in 7th & 8th grade)
(50, 0.45)
(75, 0.45)
(100, 0.45)
(125, 0.45)
(150, 0.45)
(50, 0.50)
(75, 0.50)
(100, 0.50)
(125, 0.50)
(150, 0.50)
(50, 0.55)
(75, 0.55)
(100, 0.55)
(125, 0.55)
(150, 0.55)
Probability of
dropping out
0.374∗∗∗
0.420∗∗∗
0.467∗∗∗
0.515∗∗∗
0.562∗∗∗
0.177∗∗∗
0.175∗∗∗
0.172∗∗∗
0.170∗∗∗
0.168∗∗∗
0.063∗∗
0.047∗
0.035+
0.025
0.019
Change in probability
Comparison
of dropping out
(150, 0.45) v. (50, 0.45)
0.188∗∗
(150, 0.45) v. (100, 0.45)
0.093∗∗
(150, 0.50) v. (50, 0.50)
-0.009
(150, 0.50) v. (100, 0.50)
-0.004
(150, 0.55) v. (50, 0.55)
-0.044∗∗
(150, 0.55) v. (100, 0.55)
-0.026∗
(50, 0.55) v. (50, 0.45)
-0.311∗∗∗
(100, 0.55) v. (100, 0.45)
-0.432∗∗∗
(150, 0.55) v. (150, 0.45)
-0.544∗∗∗
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
31
s.e.
0.085
0.083
0.085
0.089
0.096
0.057
0.055
0.056
0.059
0.063
0.031
0.028
0.025
0.022
0.019
p-value
0.000
0.000
0.000
0.000
0.000
0.002
0.002
0.002
0.004
0.008
0.045
0.086
0.156
0.246
0.340
s.e.
0.077
0.038
0.043
0.022
0.022
0.015
0.069
0.080
0.096
p-value
0.014
0.014
0.824
0.826
0.040
0.059
0.000
0.000
0.000
Table 12. Probability of failing to graduate in 4 years
(estimates based on model (3) in Table 8
Assumptions: Male, Black, Age = 13, Free & reduced price meal
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
(Discount rate, math score in 7th & 8th grade)
(50, 0.45)
(75, 0.45)
(100, 0.45)
(125, 0.45)
(150, 0.45)
(50, 0.50)
(75, 0.50)
(100, 0.50)
(125, 0.50)
(150, 0.50)
(50, 0.55)
(75, 0.55)
(100, 0.55)
(125, 0.55)
(150, 0.55)
Probability of
dropping out
0.456∗∗∗
0.507∗∗∗
0.558∗∗∗
0.607∗∗∗
0.656∗∗∗
0.256∗∗∗
0.250∗∗∗
0.245∗∗∗
0.239∗∗∗
0.234∗∗∗
0.115∗∗
0.086∗∗∗
0.063∗∗
0.046∗
0.032+
Change in probability
Comparison
of dropping out
(150, 0.45) v. (50, 0.45)
0.200∗∗
(150, 0.45) v. (100, 0.45)
0.102∗∗
(150, 0.50) v. (50, 0.50)
-0.022
(150, 0.50) v. (100, 0.50)
-0.011
(150, 0.55) v. (50, 0.55)
-0.083∗∗
(150, 0.55) v. (100, 0.55)
-0.051∗∗
(50, 0.55) v. (50, 0.45)
-0.341∗∗∗
(100, 0.55) v. (100, 0.45)
-0.494∗∗∗
(150, 0.55) v. (150, 0.45)
-0.624∗∗∗
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
32
s.e.
0.093
0.091
0.093
0.097
0.103
0.064
0.059
0.057
0.058
0.062
0.038
0.031
0.028
0.025
0.023
p-value
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.002
0.006
0.022
0.071
0.156
s.e.
0.081
0.042
0.053
0.027
0.011
0.025
0.070
0.090
0.108
p-value
0.013
0.016
0.687
0.690
0.126
0.101
0.000
0.000
0.000
7
Appendix 1. Additional results
Table A1. Failure to complete high school in 4 years
Maximum likelihood estimations accounting for missing outcome data1
Blacks
37.8%
(2)
0.234
[0.229]
(0.307)
-15.967∗∗∗
[2.850]
(0.000)
Rate of failure to graduate in 4 years
Discount rate (d.r.)
(1)
0.376∗
[0.211]
(0.074)
0.700∗∗∗
[0.169]
(0.000)
0.304∗
[0.180]
(0.092)
0.478∗∗∗
[0.153]
(0.002)
0.195
[0.188]
(0.298)
(3)
5.131∗∗
[2.443]
(0.036)
-8.444∗∗
[3.515]
(0.016)
-10.062∗∗
[5.100]
(0.049)
0.578∗∗∗
[0.174]
(0.001)
0.121
[0.207]
(0.557)
0.625∗
[0.332]
(0.060)
0.909∗∗∗
[0.183]
(0.000)
0.162
[0.190]
(0.393)
-11.734∗∗∗
[2.329]
(0.000)
0.172∗∗∗
[0.025]
(0.000)
0.544∗∗∗
[0.025]
(0.000)
0.498+
[0.388]
(0.199)
0.080
[0.208]
(0.700)
-0.008
[0.191]
(0.969)
0.087
[2.509]
(0.973)
0.145∗∗∗
[0.020]
(0.000)
0.588∗∗∗
[0.024]
(0.000)
0.751+
[0.542]
(0.166)
0.012
[0.226]
(0.959)
-0.013
[0.209]
(0.949)
-5.167+
[3.221]
(0.109)
0.148∗∗∗
[0.021]
(0.000)
0.582∗∗∗
[0.027]
(0.000)
Math score (7th & 8th grade)
Math score (7th & 8th grade)×d.r.
Age
Male
(4)
0.070
[0.217]
(0.745)
Whites
28.5%
(5)
-0.432∗
[0.260]
(0.096)
-15.459∗∗∗
[3.977]
(0.000)
0.703∗∗∗
[0.185]
(0.000)
0.246
[0.205]
(0.231)
0.502∗∗
[0.202]
(0.013)
0.158
[0.244]
(0.518)
(6)
4.949∗
[2.584]
(0.056)
-7.594∗∗
[3.371]
(0.024)
-10.901∗∗
[5.305]
(0.040)
0.606∗∗∗
[0.201]
(0.003)
0.106
[0.256]
(0.679)
0.966∗∗∗
[0.229]
(0.000)
0.189
[0.230]
(0.411)
0.050
[0.235]
(0.832)
-10.927∗∗∗
[2.461]
(0.000)
0.115∗∗∗
[0.022]
(0.000)
0.576∗∗∗
[0.042]
(0.000)
0.758∗∗∗
[0.246]
(0.002)
-0.717∗∗
[0.337]
(0.034)
0.027
[0.272]
(0.922)
0.696
[3.449]
(0.840)
0.117∗∗∗
[0.018]
(0.000)
0.566∗∗∗
[0.031]
(0.000)
0.723∗∗∗
[0.255]
(0.005)
-0.709∗∗
[0.333]
(0.033)
-0.052
[0.309]
(0.866)
-4.674+
[3.186]
(0.142)
0.119∗∗∗
[0.018]
(0.000)
0.561∗∗∗
[0.029]
(0.000)
0.699∗∗∗
[0.152]
(0.000)
0.119
[0.153]
(0.435)
-10.639∗∗∗
[2.007]
(0.000)
0.162∗∗∗
[0.018]
(0.000)
0.536∗∗∗
[0.031]
(0.000)
-0.039
[0.206]
(0.852)
0.017
[0.187]
(0.928)
0.514
[2.561]
(0.841)
0.165∗∗∗
[0.017]
(0.000)
0.524∗∗∗
[0.026]
(0.000)
-0.036
[0.204]
(0.861)
-0.012
[0.188]
(0.950)
-4.669∗
[2.474]
(0.059)
0.162∗∗∗
[0.016]
(0.000)
0.529∗∗∗
[0.024]
(0.000)
394
534
524
524
Black
Hispanic & multiracial
Free & reduced price meal
Special education
Inconsistent
Constant
Probability of observing a dropout
Probability of observing a graduate
Observations
392
1
Economically disadvantaged
38.5%
(7)
(8)
(9)
0.337∗∗
0.250+
4.977∗∗
[0.160]
[0.185]
[1.969]
(0.036)
(0.177)
(0.012)
-15.148∗∗∗
-6.523∗∗∗
[2.788]
[2.487]
(0.000)
(0.009)
-9.768∗∗
[4.086]
(0.017)
0.730∗∗∗
0.502∗∗∗
0.576∗∗∗
[0.142]
[0.151]
[0.149]
(0.000)
(0.001)
(0.000)
0.190
0.155
0.066
[0.152]
[0.181]
[0.182]
(0.210)
(0.392)
(0.717)
-0.662∗∗∗
-0.753∗∗∗
-0.721∗∗∗
[0.154]
[0.212]
[0.207]
(0.000)
(0.000)
(0.001)
-0.132
0.288
0.237
[0.327]
[0.442]
[0.411]
(0.687)
(0.515)
(0.565)
385
385
398
394
Estimations are based on Ramalho and Smith (2013)
Standard errors in brackets, p-values in parentheses.
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
33
Table A2. Failure to complete high school in 4 years
Logit regressions on complete cases only
Rate of failure to graduate in 4 years
(1)
Discount rate (d.r.)
Blacks
37.8%
(2)
0.930**
[0.384]
(0.015)
1.021**
[0.428]
(0.017)
-23.051***
[4.917]
(0.000)
1.110***
[0.310]
(0.000)
0.720*
[0.369]
(0.051)
0.684**
[0.331]
(0.039)
0.723*
[0.412]
(0.079)
Math score (7th & 8th grade)
Math score (7th & 8th grade)×d.r.
Age
Male
Whites
28.5%
(5)
(3)
(4)
(6)
4.082
[4.283]
(0.341)
-17.026*
[9.321]
(0.068)
-6.444
[8.946]
(0.471)
0.722**
[0.336]
(0.032)
0.695*
[0.414]
(0.093)
0.283
[0.385]
(0.462)
-0.287
[0.428]
(0.502)
-17.478***
[3.581]
(0.000)
0.961***
[0.334]
(0.004)
0.637*
[0.369]
(0.084)
0.858**
[0.352]
(0.015)
0.504
[0.406]
(0.214)
-1.083
[3.666]
(0.768)
-18.805***
[7.106]
(0.008)
1.586
[7.251]
(0.827)
0.862**
[0.353]
(0.015)
0.505
[0.406]
(0.213)
0.285
[0.427]
(0.504)
-0.016
[0.424]
(0.970)
-4.358
[7.035]
(0.536)
1.720***
[0.367]
(0.000)
0.356
[0.420]
(0.396)
-0.203
[0.431]
(0.637)
-16.282***
[4.595]
(0.000)
1.420***
[0.402]
(0.000)
-0.706
[0.488]
(0.148)
-0.311
[0.472]
(0.510)
-4.682
[5.123]
(0.361)
0.705**
[0.285]
(0.013)
0.537*
[0.307]
(0.080)
-18.083***
[3.158]
(0.000)
0.944***
[0.247]
(0.000)
0.589**
[0.278]
(0.034)
-0.904***
[0.300]
(0.003)
-0.630
[0.700]
(0.369)
0.659**
[0.262]
(0.012)
0.522*
[0.307]
(0.089)
-1.069***
[0.333]
(0.001)
-0.214
[0.745]
(0.774)
2.431
[2.815]
(0.388)
-14.765***
[5.676]
(0.009)
-3.886
[5.738]
(0.498)
0.668**
[0.262]
(0.011)
0.503
[0.309]
(0.104)
-1.057***
[0.334]
(0.002)
-0.192
[0.744]
(0.797)
1.420***
[0.402]
(0.000)
-0.709
[0.488]
(0.147)
-0.308
[0.471]
(0.513)
-4.059
[5.881]
(0.490)
1.117***
[0.285]
(0.000)
0.187
[0.299]
(0.531)
-15.028***
[3.402]
(0.000)
0.159
[0.330]
(0.631)
0.042
[0.329]
(0.898)
-1.327
[4.073]
(0.745)
0.153
[0.331]
(0.644)
0.039
[0.331]
(0.907)
-3.059
[4.776]
(0.522)
268
368
361
361
Black
Hispanic & multiracial
Free & reduced price meal
Special education
Inconsistent
Constant
Observations
1.401***
[0.356]
(0.000)
0.208
[0.376]
(0.579)
-18.710***
[4.323]
(0.000)
249
0.310
[0.423]
(0.463)
-0.009
[0.420]
(0.982)
-1.024
[5.397]
(0.849)
245
245
272
268
Standard errors in brackets, p-values in parentheses.
*** p<0.01, ** p<0.05, * p<0.10
34
Economically disadvantaged
38.5%
(7)
(8)
(9)
Table A3. Failure to complete high school in 4 years
Probit regressions on complete cases only
Blacks
37.8%
(2)
Rate of failure to graduate in 4 years
(1)
Discount rate (d.r.)
0.524**
[0.210]
(0.013)
0.537**
[0.231]
(0.020)
-12.723***
[2.649]
(0.000)
0.646***
[0.172]
(0.000)
0.420**
[0.209]
(0.045)
0.408**
[0.191]
(0.033)
0.448*
[0.232]
(0.054)
Math score (7th & 8th grade)
Math score (7th & 8th grade)×d.r.
Age
Male
Whites
28.5%
(5)
(3)
(4)
(6)
2.537
[2.319]
(0.274)
-8.825*
[5.023]
(0.079)
-4.140
[4.765]
(0.385)
0.429**
[0.192]
(0.026)
0.430*
[0.233]
(0.065)
0.149
[0.217]
(0.492)
-0.152
[0.245]
(0.535)
-9.592***
[1.915]
(0.000)
0.542***
[0.188]
(0.004)
0.380*
[0.205]
(0.063)
0.490**
[0.201]
(0.015)
0.327
[0.227]
(0.150)
-0.283
[2.050]
(0.890)
-9.804**
[3.814]
(0.010)
0.258
[4.015]
(0.949)
0.490**
[0.201]
(0.015)
0.327
[0.227]
(0.149)
0.192
[0.243]
(0.429)
-0.001
[0.241]
(0.998)
-3.170
[3.940]
(0.421)
0.983***
[0.202]
(0.000)
0.215
[0.240]
(0.371)
-0.108
[0.236]
(0.649)
-9.231***
[2.569]
(0.000)
0.804***
[0.224]
(0.000)
-0.403
[0.282]
(0.153)
-0.223
[0.263]
(0.395)
-2.894
[2.922]
(0.322)
0.399**
[0.162]
(0.014)
0.308*
[0.174]
(0.077)
-10.381***
[1.744]
(0.000)
0.554***
[0.141]
(0.000)
0.353**
[0.161]
(0.028)
-0.547***
[0.172]
(0.001)
-0.374
[0.396]
(0.345)
0.400***
[0.152]
(0.009)
0.322*
[0.176]
(0.067)
-0.637***
[0.188]
(0.001)
-0.123
[0.427]
(0.772)
1.394
[1.581]
(0.378)
-8.484***
[3.184]
(0.008)
-2.203
[3.183]
(0.489)
0.402***
[0.152]
(0.008)
0.311*
[0.177]
(0.078)
-0.626***
[0.189]
(0.001)
-0.102
[0.424]
(0.811)
0.804***
[0.224]
(0.000)
-0.403
[0.282]
(0.153)
-0.223
[0.262]
(0.395)
-2.793
[3.312]
(0.399)
0.662***
[0.167]
(0.000)
0.120
[0.172]
(0.484)
-8.809***
[1.938]
(0.000)
0.107
[0.192]
(0.578)
0.020
[0.188]
(0.917)
-1.067
[2.356]
(0.651)
0.107
[0.192]
(0.578)
0.016
[0.189]
(0.935)
-2.040
[2.731]
(0.455)
268
368
361
361
Black
Hispanic & multiracial
Free & reduced price meal
Special education
Inconsistent
Constant
Observations
0.809***
[0.205]
(0.000)
0.130
[0.214]
(0.544)
-10.882***
[2.395]
(0.000)
249
0.201
[0.243]
(0.406)
0.002
[0.239]
(0.992)
-1.014
[3.118]
(0.745)
245
245
272
268
Standard errors in brackets, p-values in parentheses.
*** p<0.01, ** p<0.05, * p<0.10
35
Economically disadvantaged
38.5%
(7)
(8)
(9)
Table A4. Failure to complete high school in 4 years
Maximum likelihood estimations accounting for missing outcome data1
Assumed rate of failure to complete high school in 4 years
0.25
0.275
0.30
0.325
0.35
0.375
(1)
(2)
(3)
(4)
(5)
(6)
Discount rate (d.r.)
5.202∗∗∗
5.066∗∗∗
4.877∗∗∗
4.681∗∗∗
5.134∗∗∗
4.950∗∗
[1.907]
[1.898]
[1.781]
[1.724]
[1.844]
[1.982]
(0.006)
(0.008)
(0.006)
(0.007)
(0.005)
(0.013)
Math score (7th & 8th grade)
-7.978∗∗∗
-8.264∗∗∗
-7.803∗∗∗
-7.467∗∗∗
-7.546∗∗∗
-9.451∗∗∗
[2.547]
[2.619]
[2.487]
[2.307]
[2.422]
[3.042]
(0.002)
(0.002)
(0.002)
(0.001)
(0.002)
(0.002)
Math score (7th & 8th grade)×d.r.
-10.511∗∗∗
-10.228∗∗∗
-9.767∗∗∗
-9.424∗∗∗
-10.314∗∗∗
-9.954∗∗
[3.949]
[3.932]
[3.675]
[3.559]
[3.809]
[4.108]
(0.008)
(0.009)
(0.008)
(0.008)
(0.007)
(0.015)
Age
0.616∗∗∗
0.619∗∗∗
0.599∗∗∗
0.595∗∗∗
0.591∗∗∗
0.654∗∗∗
[0.132]
[0.132]
[0.129]
[0.126]
[0.129]
[0.139]
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
Male
0.116
0.130
0.142
0.130
0.131
0.116
[0.163]
[0.164]
[0.160]
[0.156]
[0.160]
[0.170]
(0.476)
(0.427)
(0.375)
(0.403)
(0.414)
(0.495)
Black
-0.760∗∗∗
-0.765∗∗∗
-0.730∗∗∗
-0.764∗∗∗
-0.785∗∗∗
-0.785∗∗∗
[0.191]
[0.192]
[0.187]
[0.183]
[0.188]
[0.201]
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
Hispanic & multiracial
0.193
0.207
0.318
0.163
0.137
0.223
[0.381]
[0.383]
[0.369]
[0.366]
[0.377]
[0.393]
(0.614)
(0.589)
(0.389)
(0.655)
(0.716)
(0.570)
Free & reduced price meal
0.632∗∗∗
0.645∗∗∗
0.659∗∗∗
0.638∗∗∗
0.660∗∗∗
0.661∗∗∗
[0.224]
[0.227]
[0.224]
[0.213]
[0.223]
[0.240]
(0.005)
(0.004)
(0.003)
(0.003)
(0.003)
(0.006)
Special education
-0.221
-0.230
-0.197
-0.177
-0.208
-0.256
[0.194]
[0.195]
[0.187]
[0.182]
[0.190]
[0.205]
(0.254)
(0.240)
(0.294)
(0.330)
(0.274)
(0.211)
Inconsistent
-0.058
-0.058
-0.066
-0.032
-0.057
-0.083
[0.173]
[0.173]
[0.170]
[0.163]
[0.169]
[0.182]
(0.736)
(0.740)
(0.697)
(0.844)
(0.737)
(0.648)
Constant
-4.936∗∗
-4.850∗∗
-4.872∗∗
-4.912∗∗
-4.838∗∗
-4.761∗∗
[2.228]
[2.244]
[2.195]
[2.119]
[2.181]
[2.381]
(0.027)
(0.031)
(0.027)
(0.020)
(0.027)
(0.046)
Probability of observing a dropout
0.098∗∗∗
0.108∗∗∗
0.118∗∗∗
0.126∗∗∗
0.137∗∗∗
0.150∗∗∗
[0.010]
[0.011]
[0.012]
[0.013]
[0.014]
[0.015]
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
Probability of observing a graduation
0.639∗∗∗
0.617∗∗∗
0.597∗∗∗
0.580∗∗∗
0.555∗∗∗
0.529∗∗∗
[0.022]
[0.021]
[0.021]
[0.021]
[0.020]
[0.018]
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
N
828
828
828
828
828
828
1
Estimations are based on Ramalho and Smith (2013)
Standard errors in brackets, p-values in parentheses.
*** p<0.01, ** p<0.05, * p<0.10, + p<0.20
36
8
Appendix 2. Outer bounds of ML estimates with missing
data
Let (y, d, z) a dataset where y is an outcome, d is a treatment (preferences) and z is an
indicator that equals 1 if y is observed. We assume that d is always observed. The following
approach is proposed by Horowitz and Manski (2001). Let the likelihood of an observation
exp(dβ) y
1
be l(y, d, β). For instance, l(y, d, β) = ( 1+exp(dβ)
) ×( 1+exp(dβ)
)1−y . The expected likelihood
given (y, d, z) and β is E[l(y, d, β)|z = 1]P r(z = 1) + E[l(y, d, beta)|z = 0]P r(z = 0). We
do not observe E[l(y, d, β)|z = 0].
However, we can find a lower bound to the likelihood distribution. Suppose that b ∈
argmaxc E[l(y, d, c)|z = 1], then
E[l(y, d, b)] ≥ E[l(y, d, c)|z = 1]P r(z = 1) + minybl(b
y , d, c)P r(z = 0)
The expression says that the likelihood associated to parameter b cannot be smaller
than the likelihood that would have been obtained when the unobserved data is replaced
as to minimize the overall likelihood.
If a set of parameters a maximize the overall likelihood function, it must be the case
that
E[l(y, d, a)] ≤ E[l(y, d, a)|z = 1]P r(z = 1) + maxybl(b
y , d, a)P r(z = 0)
The expression says that the likelihood associated to parameter d has to be smaller
than the likelihood would have been obtained when the unobserved data is replaced as to
maximize the overall likelihood.
It follows that the overall solution “a” must satisfy the following inequality:
E[l(y, d, a)|z = 1]P r(z = 1) + maxybl(b
y , d, a)P r(z = 0) ≥
E[l(y, d, c)|z = 1]P r(z = 1) + minybl(b
y , d, c)P r(z = 0)
This can be rewritten as:
E[l(y, d, a)|z = 1] − E[l(y, d, c)|z = 1] ≥ [minybl(b
y , d, c) − maxybl(b
y , d, a)] PP r(z=0)
r(z=1)
Note that if the outcome is binary, i.e. l(y, d, c) = P r(y = 1|d, c)y (1−P r(y = 1|d, c))1−y ,
then minybl(b
y , d, c) = min{P r(y = 1|d, c), 1−P r(y = 1|d, c)} and maxybl(b
y , d, a) = max{P r(y =
1|d, a), 1 − P r(y = 1|d, a)}. This means that the computation of the outer bound of parameter a is simple. Note also that minybl(b
y , d, c) is calculated only once.
These bounds are very conservative. The true parameter cannot be outside these bounds.
They are useful to determine the sign of the parameter. The table below shows estimates
of the failure to complete high school assuming that the dependent variable follows a logit
model. The first 3 columns present estimates assuming that students with unknown graduation rate who had above the median test scores did not graduate and that those with below
the median test scores did graduate. The last 3 columns present estimates assuming that
students with discount rates below 0.8 (relatively patient students) did not graduate and
that those with a discount rate above 0.8 (relatively impatient students) did graduate. We
observe that under these assumption the relationship between test scores and graduation
37
becomes not significant and the relationship between impatience and failure to graduate
become negative. This confirms that bounds on parameter estimates even under parametric assumptions are too wide to derive conclusive estimates with the level of missing data
in our study.
Table B1. Failure to complete high school in 4 years - Logit regression
Assumption on missing outcome:
Students with high math scores
in 7th & 8th are less
Patient students are
likely to graduate
less likely to graduate
Variables
(1)
(2)
(3)
(4)
(5)
(6)
Discount rate (d.r.)
0.164
[0.170]
(0.334)
0.107
[0.174]
(0.539)
-1.437
[1.148]
(0.211)
0.592***
[0.152]
(0.000)
0.377**
[0.168]
(0.025)
-0.970***
[0.193]
(0.000)
-0.179
[0.386]
(0.642)
0.876***
[0.203]
(0.000)
0.071
[0.191]
(0.709)
-0.172
[0.185]
(0.351)
-9.683***
[2.078]
(0.000)
845
Math score (7th & 8th grade)
-1.298***
[0.182]
(0.000)
-1.541***
[0.195]
(0.000)
-9.360***
[1.441]
(0.000)
0.519***
[0.157]
(0.001)
0.402**
[0.172]
(0.020)
-1.050***
[0.198]
(0.000)
-0.281
[0.397]
(0.479)
0.887***
[0.209]
(0.000)
-0.038
[0.208]
(0.856)
-0.263
[0.191]
(0.170)
-7.839***
[2.373]
(0.001)
0.401
[1.049]
(0.702)
-0.991
[1.935]
(0.609)
-0.569
[1.998]
(0.776)
0.519***
[0.158]
(0.001)
0.401**
[0.172]
(0.020)
-1.051***
[0.198]
(0.000)
-0.279
[0.397]
(0.482)
0.888***
[0.209]
(0.000)
-0.041
[0.208]
(0.845)
-0.261
[0.192]
(0.173)
-8.070***
[2.506]
(0.001)
0.892***
[0.160]
(0.000)
0.322*
[0.176]
(0.068)
-0.813***
[0.200]
(0.000)
-0.288
[0.411]
(0.483)
0.929***
[0.212]
(0.000)
0.488**
[0.194]
(0.012)
0.045
[0.193]
(0.816)
-12.876***
[2.172]
(0.000)
0.682***
[0.166]
(0.000)
0.338*
[0.186]
(0.069)
-1.055***
[0.211]
(0.000)
-0.370
[0.439]
(0.399)
0.799***
[0.221]
(0.000)
-0.088
[0.215]
(0.684)
-0.163
[0.203]
(0.421)
-4.491*
[2.493]
(0.072)
-1.687
[1.352]
(0.212)
-9.562***
[2.359]
(0.000)
0.291
[2.674]
(0.913)
0.682***
[0.166]
(0.000)
0.339*
[0.186]
(0.069)
-1.055***
[0.211]
(0.000)
-0.371
[0.439]
(0.398)
0.799***
[0.221]
(0.000)
-0.086
[0.216]
(0.689)
-0.164
[0.203]
(0.419)
-4.393*
[2.653]
(0.098)
831
831
840
828
828
Math score (7th & 8th grade)×d.r.
Age
Male
Black
Hispanic & multiracial
Free & reduced price meal
Special education
Inconsistent
Constant
Observations
Standard errors in brackets, p-values in parentheses.
*** p<0.01, ** p<0.05, * p<0.10
38