Voluntary Disclosure, Inference and the Strategic Behavior of

Colleges Michael Conlin Michigan State UniversityStacy Dickert-Conlin Michigan State UniversityGabrielle Chapman Syracuse University

Oregon State University– May 2009

Optional SAT Policies

“I SOMETIMES think I should write a handbook for college admission officials titled “How to Play the U.S. News & World Report Ranking Game, and Win!” I would devote the first chapter to a tactic called “SAT optional.”

The idea is simple: tell applicants that they can choose whether or not to submit their SAT or ACT scores. Predictably, those applicants with low scores or those who know that they score poorly on standardized aptitude tests will not submit. Those with high scores will submit. When the college computes the mean SAT or ACT score of its enrolled students, voilà! its average will have risen. And so too, it can fondly hope, will its status in the annual U.S. News & World Report’s college rankings.”

Colin Driver, President of Reed College, New York Times, 2006

Optional SAT Policies

The thesis, first stated last year by The New Republic, is that colleges are being less than honest about why they abolish requirements that applicants submit their SAT scores. Behind the rhetoric about "enhancing diversity" and creating a more "holistic approach" to admissions, the theory goes, many colleges "go optional" on the SAT to improve their rankings. The logic is rather simple: At an SAT-optional college, students with higher scores are far more likely to submit them, raising the institution's mean SAT score and hence the heavily test-influenced rankings.

Brownstein (2001) in The Chronicle of Higher Education

U.S. News & World Report (Criteria and weights for rankings colleges)

Prevalence of Optional Policy

As of Spring 2007, more than 700 colleges have SAT- or ACT- optional policies.

24 of the top 100 liberal arts colleges ranked by U.S. News & World are SAT- or ACT- optional.

Overview

Research Questions Data Voluntary Disclosure Literature Reduced Form Results

- Colleges’ Decisions to Accept

- Applicants’ Decisions to Submit SAT I

Structural Framework Future Work

Research Questions

Are Colleges’ Admission Decisions influenced by their incentive to increase their ranking in publications like U.S. News & World Report?

Do Applicants behave strategically when deciding whether to submit their SAT I scores and how does this inform the voluntary disclosure literature?

What is the college’s inference for applicants who choose not to submit their SAT I scores?

College Data Application data for 2 liberal arts schools in north

east Each with approximately 1800 students enrolled.

Both report a typical SAT I score in the upper 1200s/1600.

College X: 2 years ≈ 5 years after the optional policy

was instituted.

College Y: the year after the optional policy was instituted. Numerical Score from Admission Department Acceptance and Enrollment Decisions. Performance Measures for those who Enroll.

College Board Data

SAT scores for those who elected not to submit them to the college.

Student Descriptive Questionnaire (SDQ) SAT II ScoresSelf Reported incomeHigh school GPAHigh school activities

SATII are Subject Exams – 20 of them

Also have High School GPA from colleges but not standardized

Optional SAT I policies

College X Whether or not applicant submits SAT I scores,

require applicants to choose between submitting the ACT scores or three SAT II: Subject Tests.

College Y Along with their SAT I scores, applicants can submit

scores from their SAT II exams, ACT exam, and/or Advanced Placement exams. College Y applicants are required to submit at least one of these scores if they choose not to submit their SAT I scores.

Summary Statistics

15.3 percent of the 7,023 applicants to College X choose not to submit SAT I scores.

24.1 percent of the 3,054 applicants to College X choose not to submit SAT I scores.

Table 1 Summary Statistics College X

N=6,567 College Y N=3,504

Chose to Submit SAT I

(N=5550)

Chose Not to Submit

SAT I (N=1017)

SS

Chose to Submit SAT I

(N=2659)

Chose Not to Submit

SAT I (N=845)

SS

SAT I Combined (math+verbal) Score

1272 (124)

1139 (116)

***

1267 (144)

1229 (120)

***

SAT I Verbal Score

641 (74)

570 (67)

***

633 (84)

610 (68)

***

SAT I Math Score

632 (70)

569 (67)

***

634 (78)

619 (72)

***

SAT II Score(s) available (1=yes)

0.856 (0.351)

0.815 (0.388)

***

0.677 (0.468)

0.804 (0.398)

***

Average SAT II Score (when available)

633 (68)

590 (68)

***

632 (76)

632 (61)

ACT Score(s) available (1=yes)

0.015 (0.122)

0.013 (0.112)

0.200 (0.400)

0.141 (0.348)

***

Average ACT Score (when available)

24.6 (3.7)

23.7 (2.4)

26.6 (3.8)

26.1 (3.3)

Table 1 Summary Stats (cont) College X

(N=6,567) College Y (N=3,504)

Chose to Submit SAT I

Chose Not to Submit SAT I

SS

Chose to Submit SAT I

Chose Not to Submit

SAT I

SS

Attended Private HS

0.477 (0.500)

0.503 (0.500)

0.353 (0.478)

0.431 (0.495)

***

Female Student 0.657 (0.475)

0.778 (0.416)

***

0.487 (0.500)

0.548 (0.498)

***

Legacy (1=yes) 0.024 (0.153)

0.022 (0.146)

0.062 (0.242)

0.053 (0.225)

Apply Early 0.059 (0.235)

0.120 (0.325)

0.108 (0.311)

0.097 (0.296)

Intend to Apply for Financial Aid

0.499 (0.500)

0.515 (0.500)

0.594 (0.491)

0.505 (0.500)

Table 1 Summary Stats (cont) College X (N=6,567) College Y (N=3,504) Chose to

Submit SAT I Chose Not to Submit SAT I

SS

Chose to Submit SAT I

Chose Not to Submit SAT I

SS

White 0.835 (0.371)

0.834 (0.372)

0.877 (0.328)

0.859 (0.348)

African American 0.029 (0.168)

0.031 (0.175)

0.032 (0.175)

0.049 (0.215)

**

Native American 0.003 (0.052)

0.007 (0.083)

**

0.002 (0.043)

0.002 (0.049)

Asian American 0.043 (0.202)

0.041 (0.199)

0.054 (0.226)

0.041 (0.199)

Hispanic 0.037 (0.190)

0.046 (0.210)

0.035 (0.185)

0.047 (0.212)

Unknown Race 0.053 (0.224)

0.040 (0.197)

*

Voluntary Disclosure: Theory

Grossman & Hart (1980) – when disclosure is costless, complete unraveling occurs.

Grossman (1981) and Milgrom (1981) -generalizes Grossman & Hart (1980)

Jovanovic (1982) – when disclosure is costly, unraveling is not complete and it may not be socially optimal to mandate disclosure

Voluntary Disclosure Example

Student i has the following probability distribution in term of SAT I scores.

When disclosure is costless, Bayesian Nash Equilibrium results in every type except the worst disclosing and the worst being indifferent between disclosing and not disclosing.

SAT I Score Probability

1300 0.2

1200 0.4

1100 0.3

1000 0.1

Expected SAT I Score

1300(.2)+1200(.4)+1100(.3)+1000(.1)=1170

Voluntary Disclosure Models Comments:

Distribution depends on student characteristics that are observable to the school such as high school GPA.

With positive disclosure costs, the “unraveling” is not complete and only the types with the lower SAT I scores do not disclose.

Assumptions:Common Knowledge.Colleges use Bayesian Updating to Infer SAT I Score

of those who do not Submit/DiscloseColleges’ incentives to admit an applicant is only a

function of his/her actual SAT I score (not whether the applicant submits the score)

Voluntary Disclosure: Theoryo Eyster and Rabin (Econometrica,

2005) propose a new equilibrium concept which they call a Cursed Equilibrium. College correctly predicts the distribution of the other players’ actions but underestimates the degree these actions are correlated with the other players’ private information.

SAT I Score Probability

1300 0.2

1200 0.4

1100 0.3

1000 0.1

“Fully” Cursed Equilibrium (χ=1)– College infers if applicant doesn’t disclose that his/her expected SAT I score is

1300(.2)+1200(.4)+1100(.3)+1000(.1)=1170

“Partially” Cursed Equilibrium (χ=.4 for example)– College infers if applicant doesn’t disclose that his/her expected SAT I score is

(1-.4) [(1100(.3)+1000(.1))/.4]+ (.4)1170 = 1113

Voluntary Disclosure: Empirical

Mathios (2000) – fat content in salad dressings.

Jin and Leslie (2003) – hygiene quality grade cards for restaurants in Los Angeles.

Jin (2004) – HMO accreditation and summary statistics.

Robinson and Monk (2005) – applicants submitting SAT scores to Mount Holyoke College.

Colleges’ Incentive to Institute Optional SAT Policy : Table 2

College X College Y Chose to

Submit SAT I

Chose Not to Submit

SAT I

SS Chose to Submit SAT I

Chose Not to Submit

SAT I

SS

SAT I Score (1600) – all applicants

1272 (124) [5550]

1139 (116) [1017]

*** 1267 (144) [2659]

1229 (120) [845]

***

Probability of Acceptance

.418 (0.493) [5550]

.395 (0.489) [1017]

.445 (0.497) [2659]

.488 (0.500) [845]

**

SAT I Score conditional on

Acceptance

1323 (107) [2320]

1172 (99) [402]

*** 1344 (115) [1182]

1260 (103) [412]

***

SAT I Score conditional on

Enrollment

1281 (107) [647]

1155 (100) [185]

*** 1299 (113) [351]

1227 (97) [135]

***

Predicted SAT I Score* (based on those that want

SAT I considered)

1272 (89)

[5547]

1219#

(82) [1017]

*** 1263 (78)

[2659]

1251#

(90) [845]

***

SS, statistical significance ; *** statistically different at 1% level, ** statistically different at 5% level, * # Statistically different than the actual SATI score at the 1% level. *Regression Results are in Table A1 of the Appendix.

College’s Acceptance Decision

Table 3 Columns I and III: Probit Regression(Dependent Variable =1 if accept)

College X College Y SAT1 Score/100 (16 max) 0.2939* 0.6164* (0.0214)

(0.0282)

Submitted SAT1 Score -0.3989* -0.1833** (0.0552)

(0.0813)

-2.6954* -2.4031* Submitted SAT2 Score

(0.2688)

(0.4883)

Submitted SAT2 Score* SAT2 Score/100 0.4554* 0.3688* (0.0422)

(0.0759)

-0.2171** 2.3137** Submitted ACT Score

(0.1079)

(1.1182)

Submitted ACT Score*ACT Score 0.0129* -0.0684 (0.0042)

(0.0422)

ME =.11

ME =-.07

ME =-.03

ME =.14ME =.17

ME =.24

ME =-.16

ME =.01

Possible Explanations for Negative Coefficient Estimate Associated with Submit SATI

1. For those who don’t submit, school might be “overestimating” their score

2. Not submitting may be correlated with error term – applicants who do not submit are “more mature” or are athletes.

3. School is behaving strategically when deciding who to accept.

College’s Acceptance Decision Table 3 Columns I and III (cont.)

College X College Y Attended Private High School 0.1250* 0.2016* (0.0400)

(0.0574)

Female -0.5383* 0.3179* (0.0393)

(0.0520)

No High School GPA reported 0.4608* 0.5356* (0.0896)

(0.1270)

High School GPA A+ 0.9152* 0.9240* (0.1179)

(0.1684)

High School GPA A 0.8472* 0.8642* (0.0858)

(0.1314)

High School GPA A- 0.6509* 0.7023* (0.0762)

(0.1207)

High School GPA B+ 0.4121* 0.5081* (0.0755)

(0.1182)

High School GPA B- -0.3198** -0.2292 (0.1578)

(0.2402)

High School GPA C -0.5827*** 0.1748 (0.3422)

(0.5214)

Note: High School GPA B is omitted category

ME =.08

ME =.12ME =-.21

ME =.05

College’s Acceptance Decision Table 3 Columns I and III (cont.)

College X College Y Missing Income 0.1916* 0.0802 (0.0526)

(0.0777)

Income <50K 0.3311* -0.1335 (0.0728)

(0.1145)

50K <Income <100K 0.1132*** -0.2186** (0.0580)

(0.0910)

Legacy (1=yes) 0.7501* 0.3033* (0.1220)

(0.1051)

Applied Early Decision 1.8209* 1.2020* (0.0859)

(0.0846)

Intend to Apply for Financial Aid -0.0814** -0.2519* (0.0409)

(0.0576)

African American 1.3637* 1.8549* (0.1079)

(0.1546)

Native American 0.1560 1.0906*** (0.3197)

(0.5791)

Asian 0.5073* 1.1541* (0.0855)

(0.1185)

Hispanic 0.5243* 1.5306* (0.0938)

(0.1387)

Note: White is omitted category

ME =.29 ME =.12

ME =.58 ME =.44

ME =-.03 ME =-.10

ME =.48 ME =.55

Is the college more likely to accept Applicant A or Applicant B if influenced by Ranking Organizations?

Applicant AWhite, Female, HS GPA is A-, Class Rank in top 10%, Private High School, Legacy, Submitted SATII of 600, Submitted SAT I of 1400.

Applicant B

White, Female, HS GPA is A-, Class Rank in top 10%, Private High School, Legacy, Submitted SATII of 600, Did not Submitted SAT I but college infers an SAT I score of 1400 (based on observables to college).

Is the college more likely to accept Applicant C or Applicant D if influenced by Ranking Organizations?

Applicant CWhite, Female, HS GPA is B, Class Rank in top quintile, Private High School, Legacy, Submitted SATII of 550, Submitted SAT I of 1100.

Applicant D

White, Female, HS GPA is B, Class Rank is top quintile, Private High School, Legacy, Submitted SATII of 550, Did not Submitted SAT I but college infers an SAT I score of 1100 (based on observables to college).

College’s Acceptance Decision Table 3 Columns II and IV: Probit Regression

College X College Y SAT1 Score/100 (16 max) 0.2207* 0.3978* (0.0456)

(0.0844)

Submitted SAT1 Score -1.3908** -3.0446* (0.5516)

(1.0482)

0.0849*** 0.2306* Submitted SAT1 Score* SAT1 Score/100 (0.0470)

(0.0843)

Submitted SAT2 Score -2.6729* -2.4895* (0.2690)

(0.4895)

0.4517* 0.3826* Submitted SAT2 Score* SAT2 Score/100 (0.0422)

(0.0761)

Submitted ACT Score -0.2254** 1.0074 (0.1080)

(1.2088)

Submitted ACT Score*ACT Score 0.0134* -0.0220 (0.0042)

(0.0452)

ME =.03 ME =.09

College’s Acceptance Decision Table 4 : Predicted rather than Actual SAT I Score

College X College Y I II III IV Predicted SAT1 Score/100 (16 max) 0.3095***

(0.0234)

0.2065*** (0.0652)

0.6260*** (0.0291)

0.3066*** (0.0897)

Requested school use SAT1 Score -0.1631***

(0.0518)

-1.4463* (0.7602)

-0.0340 (0.0806)

-4.1312*** (1.0941)

Requested school use SAT1 Score* SAT1 Score/100

0.1039* (0.0614)

0.3248*** (0.0864)

Requested school use SAT2 Score -2.4979***

(0.2795)

-2.5905*** (0.2852)

-2.2793*** (0.4887)

-2.5132*** (0.4947)

Requested school use SAT2 Score* SAT2 Score/100

0.4256*** (0.0439)

0.4401*** (0.0448)

0.3468*** (0.0760)

0.3852*** (0.0770)

Requested school use ACT Score -0.1884*

(0.1082)

-0.2157*** (0.1095)

0.4580 (1.0611)

-0.5234 (1.0994)

Requested school use ACT Score*ACT Score

0.0130*** (0.0042)

0.0139*** (0.0042)

0.0051 (0.0401)

0.0360 (0.0412)

In the spirit of Eyster & Rabin’s “fully” cursed equilibrium.

Interpretation of Point Estimates

College XAn applicant who scores a 1,000 on the SAT I score decreases her probability of being accepted by 9.7 percentage points if she submits her score while an applicant who scores a 1,500 increases her probability of being accepted by 3.8 percentage points if she submits.

College YAn applicant who scores a 1,000 on the SAT I score decreases her probability of being accepted by 16.8 percentage points if she submits her score while an applicant who scores a 1,500 increases her probability of being accepted by 12.6 percentage points if she submits.

Interpretation of Point Estimates

College X Applicants who submit their SAT I score are less likely to be accepted by College X if their SAT I score is less than 1,392 and are more likely to be accepted if their score is greater than 1,392.

College YApplicants who submit their SAT I score are less likely to be accepted if their SAT I score is less than 1,272 and are more likely to be accepted if their score is greater than 1,272.

Possible Explanations for Negative Coefficient Estimate Associated with Submit SATI

1. For those who don’t submit, school might be “overestimating” their score

2. Not submitting may be correlated with error term – applicants who do not submit are “more mature” or are athletes.

3. School is behaving strategically when deciding who to accept.

Submission on College X PerformanceTable 5B

Applicants’ Disclosure Decisions: Table 2

College X College Y Chose to

Submit SAT I

Chose Not to Submit

SAT I

SS Chose to Submit SAT I

Chose Not to Submit

SAT I

SS

SAT I Score (1600) – all applicants

1272 (124) [5550]

1139 (116) [1017]

*** 1267 (144) [2659]

1229 (120) [845]

***

Probability of Acceptance

.418 (0.493) [5550]

.395 (0.489) [1017]

.445 (0.497) [2659]

.488 (0.500) [845]

**

SAT I Score conditional on

Acceptance

1323 (107) [2320]

1172 (99) [402]

*** 1344 (115) [1182]

1260 (103) [412]

***

SAT I Score conditional on

Enrollment

1281 (107) [647]

1155 (100) [185]

*** 1299 (113) [351]

1227 (97) [135]

***

Predicted SAT I Score* (based on those that want

SAT I considered)

1272 (89)

[5547]

1219#

(82) [1017]

*** 1263 (78)

[2659]

1251#

(90) [845]

***

SS, statistical significance ; *** statistically different at 1% level, ** statistically different at 5% level, * # Statistically different than the actual SATI score at the 1% level. *Regression Results are in Table A1 of the Appendix.

Figure 2: Predicted versus Actual SAT I Score for those who Chose not to Submit

Figure 2: Predicted versus Actual SAT I Score for those who Chose not to Submit

Conclusions from Reduced Form1. College admission departments are behaving

strategically by more (less) likely accepting applicants who do not submit their SAT I scores if submitting their scores would decrease (increase) the average SAT I score the colleges report to the ranking organizations.

2. Applicants are behaving strategically by choosing not to reveal their SAT I scores if they are below a value one might predict based on their other observable characteristics.

Note that the reduced form estimates do not address directly the college’s inference for those applicants who do not submit.

Model and Structural Estimation

Summary Statistics for College X

N=895N=122 N=5216N=324

Summary Statistics for College Y

N=294 N=83 N=2440 N=785

Notation

μ(Xi)+εap +εen ,expected utility from attending the college for applicant i

μ(Xi) is portion of the applicant specific preferences for attending College X that depends on the observables variable.

εap is unobservable applicant specific preferences for attending College X that is known to the applicant at the time she submits her application.

εen is unobservable applicant specific preferences for attending College X that is known to the applicant at the time she makes enrollment decision but not at time she submits her application

Notation (cont)

UR ,expected utility if applicant does not attend College X and does not apply early decision at College X.

UR-C , expected utility if applicant does not attend College X and does apply early decision at College X.

K , fixed cost of applying εen , unobserved cost of submitting SAT I

Applicant’s Decision to apply early decision and/or submit SAT I

Expected Utility if applicant applies early and submits

Pa(X,ed,s) Pe(X,ed,s) [μ(X)+εap+εen]+

[1-Pa(X,ed,s) Pe(X,ed,s)][UR-C]-K- εs

Expected Utility if applicant doesn’t apply early or submit

Pa(X,ned,ns) Pe(X,ned,ns) [μ(X)+εap+εen]+

[1-Pa(X,ned,ns) Pe(X,ned,ns)][UR]-K

Apply early, don’t submit and Don’t apply early, submit are analogous

Applicant applies early decision and submits [assuming E(εen)=0] if

εs +[Κ1]εap > [Κ1][-μ(X)+C] ,

[Κ2]εap > [Κ2][-μ(X)]+[1- Pa(X,ed,ns)Pe(X,ed,ns)][C]

and

εs+[Κ3]εap > [Κ3][-μ(X)]+[1Pa(X,ed,ns)Pe(X,ed,ns)][C]

Κ1=Pa(X,ed,ns)Pe(X,ed,ns)-Pa(X,ed,s)Pe(X,ed,s),

Κ2=Pa(X,ed,ns)Pe(X,ed,ns)-Pa(X,ned,ns)Pe(X, ned,ns)

Κ3=Pa(X,ed,ns)Pe(X,ed,ns)-Pa(X,ned,s)Pe(X,ned,s)

Whether to Apply Early Decision and/or Submit SATI Score

Literature on College Objective Function

Ehrenberg (1999) single well-defined objective function may explain “fairly well the behavior of small liberal arts colleges…” (page 101).

Epple, Romano, and Seig (2006) GE model assume a school maximizes quality (average quality of the

student body, school expenditure per student, and the mean income of the student body)

s.t. balanced budget constraint and a fixed student body size. Our Model

To account for the college’s concern for the quality of its current and future students and the understanding that future student quality depends on the college’s ranking, we allow the college’s objective function to depend on the perceived ability of the incoming students, the “reported” ability of these students, and the demographic characteristics of the student body.

College’s Decision to Accept Applicant

College accepts applicant i if:

Pe(Xi,k,l) [ΠP(X+iP)+εqi + ΠR(X+i

R)+ ΠD(X+iD)]

+(1- Pe(Xi,k,l)) [ΠP(X-iP)+ ΠR(X-i

R)+ ΠD(X-iD)]

> ΠP(XriP)+ ΠR(Xri

R)+ ΠD(XriD)

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