Class Attendance and Exam Performance: A Randomized Experiment
*Jennjou Chen Department of Economics National Chengchi University
Taipei City 116, Taiwan (JENNJOU@NCCU.EDU.TW)
And
Tsui-Fang Lin
Department of Public Finance National Taipei University
Taipei City 104, Taiwan (TFLIN@MAIL.NTPU.EDU.TW)
This Version: April 2006
* Prepared for the Society of Labor Economists (SOLE) Eleventh Annual Meetings, May 5-6, 2006, Cambridge, Massachusetts, U.S.A. We thank Kumiko Imai for helpful comments. All errors are our own.
Abstract
The study of determinants of a college student’s academic performance is an important issue in higher education. Among all factors, whether or not attending lectures affects a student’s exam performance has received considerable attention. In this paper, we conduct a randomized experiment to study the average attendance effect for students who have chosen to attend lectures, which is the so-called the average treatment effect on the treated in program evaluation literature. This effect has long been neglected by researchers when estimating the impact of lecture attendance on students’ academic performance. Under the randomized experiment approach, least squares, fixed effects, and random effects models all yield similar estimates for the average treatment effect on the treated. We find that, class attendance has produced a positive and significant impact on students’ exam performance. On average, attending lecture corresponds to a 7.66% improvement in exam performance.
Keywords: class attendance, exam performance, random experiment
I. Introduction
The study of determinants of a college student’s academic performance is an important
issue in higher education. Among all factors, whether or not attending lectures and classroom
discussions affects a student’s exam performance has received considerable attention. In light
of the importance of the attendance effect, many researchers have explored the impact of a
student’s class attendance on his or her exam performance. Most studies find that attending
lectures yields a positive and significant impact on exam performance (Anikeeff, 1954;
Schmidt,1983; Jones,1984; Buckalew, et al., 1986; Brocato, 1989; Park and Kerr, 1990; Van
Blerkom, 1992; Romer, 1993; Gunn,1993; Durden and Ellis, 1995; Devadoss and Foltz,1996;
Marburger, 2001; Bratti and Staffolani, 2002; Dolton, et al., 2003; Kirby and McElroy, 2003;
Rodgers, 2001; Rocca, 2003; Stanca, 2006; Lin and Chen, 2006).
The investigation of the attendance effect is indeed an estimation of the treatment effect in
policy evaluation. In the program evaluation literature, two kinds of treatment effects are
frequently mentioned. They are the average treatment effect and the average treatment effect
on the treated. In the application of lecture attendance, the average treatment effect refers to the
expected effect of attendance on academic performance for a randomly drawn student from the
population. The average treatment effect on the treated refers to the mean attendance effect for
those who actually participated (or not participated) in the classroom.
Literature is prolific in the area of the exploration of the average treatment effect when
estimating the impact of a student’s class attendance on his or her exam performance. In the
estimation of the average treatment effect, it is of particular note that whether or not students
choose to attend classes is an endogenous choice. Therefore, the estimated coefficient of
attendance will suffer from the endogeneity bias and become inconsistent if researchers fail to
econometric models including instrumental variable (IV), proxy variable and panel data methods
are used to obtain consistent estimates in prior research (see Stanca (2006) for more details).
In stark contrast, existing studies in the area of attendance effect has not examined the
average treatment effect on the treated. As discussed above, the average treatment effect is the
expected effect of treatment on a randomly drawn student from the population while the average
treatment effect on the treated measures the average benefits a student could receive given that
he or she attends lectures. Most researchers focus on the estimation of the average treatment
effect since the issue of whether or not to make attendance compulsory has received great
attention in higher education. However, in the investigation of the attendance effect, we also
want to evaluate the lecture participation benefits for students who actually choose to attend
classes. This is similar to the example of job training program; what researchers and
policymakers want to know is the impact of job training on labor outcomes for program
participants but not for people from the population who probably would never be eligible for
such programs. Hence, to complement current attendance effect literature, the focus of this
paper is to investigate the average treatment effect on the treated.
One difficulty in the estimation of the average treatment effect on the treated arises from
finding the desired counterfactuals. In this case, we will need to estimate what would have
been the grades had the students not attended the class for those who actually do participate in
the classroom. One way to circumvent the problem of finding desired counterfactual is to run a
randomized experiment.
Burtless (1995) and Heckman and Smith (1995) both discuss advantages and disadvantages
of social experiments. The foremost advantage of controlled experimentation is that the
random assignment provides us with the clear causal link between treatment and outcome. In
outcome. Random assignment also eliminates systematic correlation between treatment status
and participants’ observed or unobserved characteristics. In addition, experimentation is simple
to understand to social scientists and policymakers. Some disadvantages of experimentation
include expensive costs, ethical issues of experimentation with human beings, limited duration,
attrition and interview non-response, partial equilibrium results and program entry effects.
In this study, we propose and implement a randomized experiment to estimate the average
treatment effect on the treated. This paper will provide useful insights on the issue of the
average treatment effect on the treated in the application of class attendance. Our empirical
results demonstrate that least squares model will provide a consistent estimator under the
randomized experiment. Class attendance has produced a positive and significant impact on
students’ exam performance. On average, attending lectures corresponds to a 7.66%
improvement in exam performance.
The details of the random experiment will be discussed in the next section. In section III,
the statistical models will be presented. In section IV, the data used for this study will be
examined. The estimation results are reported in Section V, and the conclusion is summarized
in Section VI.
II. The Random Experiment
The main goal here is to construct a random experiment to estimate the average treatment
effect on the treated. The following notation is similar to the ones in Heckman and Smith
(1995).
Y1: grade outcomes associated with attending the lecture
Y0: grade outcomes associated with not attending the lecture
d = 1, attending the lecture; d = 0, not attending the lecture
students who choose to attend classes. The average treatment effect on the treated can be
shown below:
E(Y1|d=1) – E(Y0|d =1) (1)
In order to estimate the average treatment effect on the treated, we need to know what
would have been the grades had the students not attended the class. This implies that we will
need an estimate for E(Y0|d=1) because it is unobserved by researchers. In general, E(Y0|d=0)
can not be used as a proxy for E(Y0|d=1) since students who choose not to attend lectures might
be different from those who choose to attend classes in many ways such as the unobserved
individual intelligence and motivation. As a result, the selection process to attend or not to
attend classes might become an issue and it will bias our results if we use E(Y0|d =0) to replace
E(Y0|d=1).
Our main focus then is to construct a randomized experiment to solve the problem of
selection bias. The goal is to generate an experimental group of students who would have
participated but were randomly denied access to the treatment. By doing so, we could use this
randomly selected group to be our control group and obtain their responses as the desired
counterfactuals, E(Y0|d=1).
Here is how the randomized experiment works. The instructor taught the same course in
two sections, section A and section B, in the spring semester of 2005. The course, Public
Finance, is a required course for all junior students who major in Industrial Economics at
Tamkang University in Taiwan. Students are allowed to register in any of these two sections1. At each class meeting, the same PowerPoint presentation is used in both sections, and the lecture
slides are posted on the course web site after each class meeting. During the sample semester,
the instructor randomly selects the dates, sections, and some materials/topics to cover or not to
cover in only one section. However, students are told to be responsible for those
materials/topics shown in the slides including the ones skipped by the instructor. This implies
that materials/topics not covered by the instructor might appear in the two exams, and students
will need to prepare and study those materials by themselves to be able to answer corresponding
exam questions. In this study, about 10% of the exam questions were not covered by the
instructor but yet appeared in the exams.
Let d* = 1 denote the students who would participate in a lecture in the presence of random
assignment, and d* = 0 for everyone else. Also, let r =1 denote the group of students who are
randomly assigned to the treatment group for particular corresponding exam questions (i.e. the
materials/topics corresponding to exam questions have been covered by the instructor), and r = 0
denote the group of students who are denied access to the treatment for particular corresponding
exam questions (i.e. the materials/topics corresponding to exam questions have been randomly
skipped by the instructor).
By introducing variables d* and r, we can re-write equation (1) as
E(Y1|d* = 1, r = 1) - E(Y0|d* = 1, r = 1), (1’)
where d = 1 is replaced by d* = 1, r = 1. As mentioned above, the instructor randomly selects
the dates, sections, and materials to cover or not cover. Also, when some materials are skipped
in the lectures, students are asked to study the materials by themselves. They are told again
about this information in their last class meeting before each exam. Thus, we could reasonably
expect that
E(Y0|d* = 1, r = 1) = E(Y0|d* = 1, r = 0). (2)
E(Y0|d* = 1, r = 0) is the expected grades for students who choose to attend lectures but do not
actually get the treatments since some materials/topics are randomly skipped for corresponding
E(Y0|d* = 1, r = 1) is the average grades that would have been obtained had students not
attended the lecture. This partial observation issue is a common problem in estimating the
average treatment effect on the treated. By running the randomized experiment, we can now
observe E(Y0|d* = 1, r = 0) and use it as a replacement for E(Y0|d* = 1, r = 1). Hence, by
using equation (2), the average treatment effect on the treated can be shown below:
E(Y1- Y0|d =1) = E(Y1- Y0|d* =1, r =1)
= E(Y1|d* = 1, r = 1) - E(Y0|d* = 1, r = 1)
= E(Y1|d* = 1, r = 1) - E(Y0|d* = 1, r = 0) (3)
Thus, randomization serves as an instrumental variable by creating variations among
students who choose to attend lectures, because some of them do receive the treatment randomly
while some of them do not receive the treatment. In so doing, we will be able to estimate the
average treatment effect on the treated accurately.
III. Statistical Models
In prior research, various econometric techniques including IV, proxy, and panel data
models are usually used to remedy the endogeneity bias problem when estimating the average
treatment effect. This paper sheds lights on the estimation of the average treatment effect on
the treated by employing a randomized experiment. Under our randomized experiment, the
mean outcomes of the experimental treatment group and the control groups provide estimates of
the average treatment effect on the treated without getting into the trouble of sample selection
bias.
This study uses a micro level data to explore the impact of a student’s attendance on his or
her exam performance. We use the following linear function to describe the relationship
between students’ exam performance and various input variables for learning.
I is the total number of students and J is the total number of exam questions. Yij corresponds to
student i’s exam performance on question j; xij is student i’s set of observed inputs in learning
question j. β represents the student input effect vector. rij is equal to one if student i attends
the lecture when question j is covered. η is the attendance effect. αi represents student i’s
time-invariant individual effect and εij is the random disturbance term.
To estimate the average treatment effect on the treated (i.e. η), Least Squares (LS), Fixed Effects (FE) and Random Effects (RE) models will be employed. In this paper, we employ a
linear probability model instead of a nonlinear probit model when estimating the individual
effects. The main reason for doing so is that we are concerned about the incidental parameters
problem in a typical nonlinear panel model (see Woldridge (2002) and Greene (2003) for
details).
The estimation results of LS, FE and RE models are all listed for the use of comparisons.
Under the experimental design, we would expect that all three models produce consistent
estimators. By the definition of a randomized experiment, the treatment is randomly assigned
within the estimation sample and will not be correlated with xij , αi and εij. This implies that LS
estimation of the attendance effect can produced consistent result. Also, LS, FE and RE will
produce similar estimates and all these three are consistent estimators.
IV. Data
We conduct a survey of 114 students who attended the Public Finance course at Tamkang
University in Taiwan in the Spring of 2005. All students who major in Industrial Economics
are required to take this course in their third-year of study. The students are in two separate
sections. There are 67 students in one section, and 47 students in the other section.
Attendance is recorded at each class meeting during the sample semester. There are 12 2-hour
Students’ demographic variables are collected form the survey distributed in the very first
class of the sample semester. They include students’ gender, average grades before taking this
course, living arrangement and family economic condition. Also, the commuting time between
students’ apartment and school, and the working hours for each student are
recorded. Moreover, two questions: hours students spent preparing for the exam and hours they
spent studying every week, are asked when students took their midterm and final exams. Table
1 reports the summary statistics of students’ characteristics.
In this paper, the dependent variable is a binary variable indicating students’ exam
performance. 50 multiple choice questions are asked in the midterm exam while 57 multiple
choice questions are asked in the final exam. There are 12,028 observations which come from
114 students and their responses to the 107 exam questions2. We assign 1 to the binary variable if students answer the exam question correctly; otherwise the binary variable is 0.
The binary variable, Actual Attendance, is coded as 1 if students have attended the lecture in
which the class material covered that day was relevant to the corresponding exam question, i.e.
dij = 1 as discussed in the random experiment section. Actual Attendance is coded as 0 if
students miss the class that day, i.e. dij = 0 as discussed in the random experiment section.
Among students who have attended lectures, we create a binary variable, Experimental
Attendance. Experimental Attendance is coded as 1 if students have attended the lecture and the instructor has taught the material in that lecture, i.e. d*ij = 1 and rij = 1 as discussed in the
random experiment section. Experimental Attendance is coded as 0 if students have attended
the lecture but the instructor has randomly choose not to cover some materials for corresponding
exam questions in that lecture i.e. d*ij = 1 and rij = 0 as discussed in the random experiment
2 There are two students missing the final exam (57*2), and some questions are not answered by some students (56). So 114 * 107 – 57*2 - 56 = 12,028.
section.
Table 2 reports the means and standard deviations of students’ actual attendance,
experimental attendance, and their exam performance by students’ demographic variables.
The average actual attendance rate is about 91% which is higher than that in some previous
studies (Romer (1993), Margurger (2001)). It is worth noting that the sample course, Public
Finance, is a required course for students in their junior year. In addition, students are more
likely to attend lectures when they are in their junior and senior years as pointed out by Rocca
(2003). Therefore, a 91% class attendance rate seems reasonable. If we further restrict our
sample to students who have chosen to attend lectures, we find that the experimental attendance
is about 92%.
V. Estimation Results
Table 4 presents the estimation results for the average treatment effect. In this regression
model, the independent variable is a binary variable indicating whether or not students answer
the exam question correctly. The dependent variables include Actual Attendance, exam
question dummies and individual time-invariant dummies. The number of observations is
12,028 in this case. The coefficient of Attendance in the panel linear probability model is about
4.32%, and it is about 7.03% in the least squares model. Thus, after controlling for individual
unobserved characteristics, the consistent fixed effects estimator obtains a much smaller estimate
on the attendance effect. Generally, our average treatment effect estimation results are
comparable to prior studies. For instance, Stanca (2006) also finds that the OLS estimates
overestimate the impact of attendance on exam performance.
Table 5 provides the estimation results for the average treatment effect on the treated. We
restrict our sample to those students who have chosen to attend lectures. The sample size is
whether or not students answer exam questions correctly. The dependent variables include
Experimental Attendance, exam question dummies and individual time-invariant dummies. The principle finding of Table 5 is that the coefficients of Experimental Attendance in LS,
FE ad RE models are almost the same. As mentioned earlier, randomization acts an instrument
variable, and simply running the least square models will give us a consistent estimator. Hence,
as we expected, LS, FE and RE models all produce similar results in the estimation of the
average treatment effect on the treated under our randomized experiment. We find that, among
students who choose to attend lectures, attending lectures yields a positive and significant impact
on students’ exam performance. On average, attending lectures corresponds to a 7.66%
improvement in exam performance.
From Tables 4 and 5, we also find that the average treatment effect is lower than the average
treatment effect on the treated. Our results suggest that the mean attendance effect for the
population of students who choose to attend classes is greater than the mean attendance effect
when students are randomly selected to attend lectures. This finding seems reasonable since we
would expect that students who decide to attend lectures should or might have a higher return
from attending classes than the ones who are randomly selected to attend lectures.
Some might be concerned about issues regarding the random assignment of treatment in this
study. For instance, our estimation results may still suffer from some biases since students
might expect that materials not covered by the instructor will be less likely to appear in the
exams even though they are told to be responsible for those skipped materials in the exams. In
order to examine whether students’ perception might become an issue and bias our results, we
further divide our sample into two sets: the midterm exam sample and the final exam sample.
If students think that materials not covered by the instructor are less likely to appear in the
example, in the midterm, if students assume that materials not covered by the instructor will not
appear in the exam and they realize that they were wrong after the midterm, then they will pay
the same attention to those skipped materials when preparing for the final exam. If this is the
case, we would expect to see different results in the midterm and final samples. Otherwise, we
would expect to find similar estimation results in the midterm and final exam samples.
Table 6 presents the estimation results for the average treatment effect on the treated in the
midterm and final samples. In these two models, the independent variable is a binary variable
indicating whether or not students answer exam questions correctly; the dependent variables
include Experimental Attendance, exam question dummies and individual time-invariant
dummies. It is of note that we find similar results in both cases. For the midterm exam, the
average treatment effect on the treated is 7.51%; for the final exam, the average treatment effect
on the treated is 7.91%. Taken together, we would not need to worry about the perception issue
here. Moreover, this also assures us of the robustness of our estimation results.
VI. Conclusion
In this paper, a randomized experiment is designed to control for students’ endogeneous
class attending choices when exploring the impact of class attendance on exam performance.
Under our randomized experiment, the mean outcomes of the experimental treatment and control
groups provide estimates of the average treatment effect on the treated without getting into the
trouble of sample selection bias.
Our estimation results show that, under the randomized experiment, simply running the
least squares model will yield a consistent estimator. In addition, the LS, FE and RE models all
produce similar estimates of the attendance effects. On average, attending lectures corresponds
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Sample Size % Mean Standard Deviation All Students 114 1.000 Gender Female 65 57.02 Male 49 42.98
Average Grade Before Entering the Course 72.047 7.7553
60-70 40 35.09
70-80 47 41.23
80-90 27 23.68
Housing
Live with Relatives 43 37.72
Not Live with Relatives 71 62.28
Family Economic Condition (1 to 5) 2.8596 0.5781
Poor (1) 4 3.509 Below Average (2) 15 13.16 Average (3) 89 78.07 Above Average (4) 5 4.386 Wealthy (5) 1 0.877 Commute Time 29.355 32.425
less than 10 Minutes 37 32.46
30-10 Minutes 41 35.96
60-30 Minutes 7 6.140
more than 60 Minutes 29 25.44
Work Hours 6.9599 10.546
zero 65 57.02
10-zero Hours 15 13.16
20-10 Hours 15 13.16
30-20 Hours 13 11.40
more than 30 Hours 6 5.263
Hours Studied Before the exam 8.6327 6.4431
Below 5 Hours 25 21.93
5-10 Hours 58 50.88
10-15 Hours 20 17.54
Above 15 hours 11 9.649
Hours Studied Every Week 1.5035 1.9138
Below 1 Hours 41 35.96
1-2 Hours 47 41.23
Above 2 hours 26 22.81
Mean Standard Deviation Mean Standard Deviation Mean Standard Deviation Mean Standard Deviation All Students 0.9088 0.0026 0.6313 0.0044 0.9198 0.0026 0.6347 0.0046 Gender Female 0.9261 0.0032 0.6387 0.0058 0.9165 0.0035 0.6404 0.0060 Male 0.8860 0.0044 0.6215 0.0067 0.9243 0.0039 0.6267 0.0071
Average Grade Before Entering the Course
60-70 0.8425 0.0056 0.5914 0.0076 0.9209 0.0045 0.5950 0.0083
70-80 0.9296 0.0036 0.6259 0.0069 0.9177 0.0040 0.6256 0.0071
80-90 0.9654 0.0034 0.6978 0.0086 0.9225 0.0051 0.6996 0.0087
Housing
Live with Relatives 0.9143 0.0032 0.6306 0.0056 0.9216 0.0032 0.6353 0.0058
Not Live with Relatives 0.8996 0.0045 0.6324 0.0072 0.9167 0.0043 0.6336 0.0076
Family Economic Condition
Poor 0.8066 0.0191 0.6840 0.0226 0.9213 0.0144 0.7026 0.0247 Below Average 0.9296 0.0064 0.6258 0.0121 0.9210 0.0070 0.6232 0.0126 Average 0.9110 0.0029 0.6264 0.0050 0.9196 0.0029 0.6311 0.0052 Above Average 0.9132 0.0122 0.6811 0.0203 0.9196 0.0123 0.6722 0.0213 Wealthy 0.7925 0.0396 0.6887 0.0452 0.9048 0.0322 0.7024 0.0502 Commute Time
less than 10 Minutes 0.9070 0.0047 0.6203 0.0079 0.9160 0.0047 0.6263 0.0082
30-10 Minutes 0.8974 0.0046 0.6252 0.0074 0.9225 0.0043 0.6260 0.0078
60-30 Minutes 0.9636 0.0069 0.6806 0.0171 0.9147 0.0105 0.6783 0.0175
more than 60 Minutes 0.9125 0.0051 0.6425 0.0086 0.9213 0.0051 0.6473 0.0090
Work Hours
zero 0.9254 0.0032 0.6444 0.0059 0.9228 0.0034 0.6460 0.0061
10-zero Hours 0.9635 0.0047 0.6560 0.0119 0.9080 0.0074 0.6562 0.0121
20-10 Hours 0.8893 0.0078 0.6094 0.0122 0.9104 0.0075 0.6104 0.0130
30-20 Hours 0.8919 0.0084 0.5965 0.0132 0.9301 0.0073 0.6016 0.0140
more than 30 Hours 0.6745 0.0186 0.5849 0.0196 0.9209 0.0129 0.6047 0.0236
Hours Studied Before the exam
Below 5 Hours 0.8671 0.0067 0.6124 0.0097 0.9203 0.0057 0.6089 0.0104
5-10 Hours 0.9098 0.0037 0.6397 0.0062 0.9190 0.0037 0.6468 0.0064
10-15 Hours 0.9304 0.0058 0.6309 0.0110 0.9324 0.0059 0.6368 0.0113
Above 15 hours 0.9456 0.0060 0.6325 0.0127 0.9086 0.0078 0.6276 0.0131
Hours Studied Every Week
Below 1 Hours 0.9522 0.0037 0.6333 0.0084 0.9223 0.0048 0.6352 0.0086
1-2 Hours 0.9008 0.0041 0.6430 0.0066 0.9208 0.0039 0.6474 0.0070
Above 2 hours 0.8775 0.0056 0.6129 0.0083 0.9171 0.0050 0.6159 0.0088
Table 2: Actual Attendance, Experimental Attendance, and Exam Performance
Actural Attendance
Experimental
Attendance Exam Performance
Exam Performance
All Samples (N = 12,028)
Samples with Actural Attendance = 1 (N = 10,919)
Mean Standard Deviation Mean Standard Deviation Number of Students Male 0.4776 0.5033 0.3617 0.4857
Average Grade Before Entering the Cou 72.097 8.1414 71.909 7.3211
Live with Relatives 0.6716 0.4732 0.5532 0.5025
Family Economic Condition 2.8358 0.5928 2.8936 0.5608
Commute Time 29.750 32.351 29.213 33.098
Work Hours 5.5758 9.3002 9.1444 11.997
Hours Studied Before the exam 8.6119 6.4894 8.5333 6.4583
Hours Studied Every Week 1.5209 2.2242 1.3222 0.8691
67 47
Table 3: Sample Statistics for Section "A" and "B"
Fixed Effects Model Random Effects Dependant Variable Independant Variable Attendance 0.0703** 0.0432** 0.0519** (0.0142) (0.0158) (0.0151) R-squares 0.2199 0.2473 . F-value or X2 value* 53.17** 50.04** 3435.06** Sample Size 12,028 12,028 12,028
Table 4: Estimation Results for the Average Treatment Effects
Note: "**" is at 5% significant level and "*" is at 10% significant level. White (1980) robust standard errors are in parentheses. Exam question dummies are used in all regressions. *: the F-values are reported in the Least Squares and Fixed Effects model
Correctly Answer the Question (yes = 1, no = 0) Least Squares Method
Fixed Effects Model Random Effects Dependant Variable Independant Variable Attendance 0.0779** 0.0763** 0.0766** (0.0210) (0.0202) (0.0202) R-squares 0.2228 0.2479 . F-value or X2 value* 48.69** 29.83** 3168.39** Hausman Test (X2107) Sample Size 10,919 10,919 10,919
Table 5: Estimation Results for the Average Treatment Effect on Treated
Note: "**" is at 5% significant level and "*" is at 10% significant level. White (1980) robust standard errors are in parentheses. Exam question dummies are used in all regressions. *: the F-values are reported in the Least Squares and Fixed Effects model
Correctly Answer the Question (yes = 1, no = 0) Least Squares Method
Panel Linear Probability Model
22
Fixed Effects Model Random Effects Fixed Effects Model Random Effects
Dependant Variable Independant Variable Attendance 0.0751** 0.0761** 0.0756** 0.0791** 0.0715** 0.0738** (0.0393) (0.0364) (0.0363) (0.0249) (0.0246) (0.0245) R-squares 0.2176 0.2534 . 0.2261 0.2611 . F-value or X2 value* 39.81** 29.37** 1475.63** 56.62** 30.20** 1694.86** Hausman Test (X250 or X257) Sample Size 5,247 5,247 5,247 5,672 5,672 5,672 6.53
Table 6: Estimation Results for the Average Treatment Effect on Treated by Midterm and Final Exams
Note: "**" is at 5% significant level and "*" is at 10% significant level. White (1980) robust standard errors are in parentheses. Exam question dummies are used in all regressions. *: the F-values are reported in the Least Squares and Fixed Effects model
Final Exam
Least Squares Method
Panel Linear Probability Model
Correctly Answer the Question (yes = 1, no = 0)
3.30 Correctly Answer the Question (yes = 1, no = 0) Least Squares Method
Panel Linear Probability Model Midterm Exam