wearing glasses during classes evidence from a … glasses during classes evidence from a regression...

Wearing Glasses during Classes

Evidence from a Regression Discontinuity Design

Draft of October 2011

By CHRISTOFFER SONNE-SCHMIDT*

This paper evaluates whether eyeglasses causes better test scores

for nearsighted youths by exploiting a natural experiment

discovered in the National Health Examination Survey of Youth

(NHES III, 1966-70). I argue, based on visual science, that

uncorrected vision is partly random. Then, I show that average test

scores form a V-shaped pattern over uncorrected vision, and that

this V shape kink at the selfsame point where the average youth

begins to wear glasses. I estimate the cumulative effect of visual

correction in middle and high school with a regression-

discontinuity approach near the knickpoint; it’s about 0.1 standard

deviations of the test-score distribution and as high as 0.3 standard

deviations for nearsighted youths. (JEL C36, I21)

Key words: Kinked-regression-discontinuity design; Instrumental-

variable regression; Eyesight; Academic skills.

* University of Copenhagen, Building 26, Øster Farimagsgade 5, DK-1353 Copenhagen K. (e-mail: christoffer.sonne-

[email protected]). I thank Thomas Barnebeck Andersen, Josh Angrist, Sascha Becker, Paul Bingley, Martin Browning,

Carl-Johan Dalgaard, Mette Ejrnæs, John N. Friedman, Meghan Jakobsen, Rafael Lalive, John Rand, Pablo Selaya, Finn

Tarp, Nina Torm, participants at the CAM December 2010 workshop, and participants at the third IWAEE for their keen

eye and helpful comments; it helped me shape and develop this paper and research topic.

Quite unknown to many students, they have a disadvantage in classroom

learning. They are nearsighted, and without eyeglasses, they have to squint to read

on the blackboard. While their classmates see things clearly, they have to struggle

all day to follow instructions, so they are more likely to tire in class, miss out on

classroom learning, and fail in school. But, wearing a simple pair of glasses, they

may refocus the blackboard—and so refocus classroom learning—yet many of

them aren’t wearing glasses during classes. In the United States today, 35 percent

of all middle- and high-school youths have some level of nearsightedness, which

98 percent of them can correct—and almost all can reduce—with a pair of

glasses, yet less than 76 percent have a pair, and merely 60 percent actually wear

them.1 So school-vision programs that, say, provide new glasses to students who

have none or have insufficient correction or that encourage students who have

glasses to start wearing them, could surely reduce visual blur in the classroom.

But will such remedies also improve proficiency in reading and math?

A creditable answer to this question cannot simply compare reading and math

skills of those who wear and do not wear glasses. Youths who for one develop

nearsightedness and afterwards choose to wear glasses may be quite different

from those who don’t, and that—that difference, not eyeglasses—may explain

success or failure in school. To separate these effects, I rely on a natural

experiment discovered in the National Health Examination Survey (NHES III,

1966-70) of youths from the late 1960’s.

Briefly, this experiment stems from partial randomization of nearsightedness;

nearsightedness is random close to its cutoff, and nearsightedness enables youths

to get and wear eyeglasses. Graphical evidence suggests that the consequence of

1 I’ve estimated these percentages upon a subsample of 9.961 adolescents (12-19 years old) from five rounds of the

National Health and Nutrition Examination Surveys (NHANES 1999 to 2008). Nearsightedness is defined as having an

uncorrected visual acuity of worse than 20/30 (inability to read the 20/40 line in an eye chart) in the better-seeing eye or

usually wearing glasses for distance visual correction; this definition is practically similar to the one use here, below. Similarly, best visual acuity defines as the ability to read the 20/40 line with an optimal correction.

this experiment is a V-shaped pattern between students’ performance and the

level of nearsightedness. Exactly at the cutoff of nearsightedness—when the

average youth begins to wear glasses—average performance kink, and increase.

I estimate the effect of this experiment with the instrumental-variable (IV)

version of regression-discontinuity (RD) design. Due the V shape and the fuzzy

treatment (enabling the use of glasses), the exact identification strategy combines

the kinked RD design in David Card, David S. Lee, and Zhuan Pei (2009) with

the fuzzy RD design in Jinyong Hahn, Petra Todd, and Wilbert van der Klaauw

(2001).

Results show that use of glasses strongly affects arithmetic and reading scores.

The cumulative use of glasses among middle- and high-school youths improves

average test scores between 0.1 to 0.2 standard deviations of the test-score

distribution. Among nearsighted youths, the average gain is even higher,

improving test scores between 0.3 to 0.6 standard deviations. So school-health

programs aimed at schoolchildren’s eyesight appear to be an effective educational

tool.

The ensuing section considers the identification problem and a natural

experiment that randomly assigns eyeglasses. Section II graph the identification

strategy and illustrate the effect of wearing glasses. It is complemented in Section

III, IV, and V with a formal description of the identification strategy, a description

of eyesight and test-score data in NHES III, and results from a regression-

discontinuity design. Section V also analyses robustness of the results. The final

Section—concludes.

I. Identification

Consider one of, if not the first study to look at the relationship between

eyesight and schooling: James Ware’s (1813) study on the prevalence of

nearsightedness and use of glasses among British foot guards and undergraduates

at Oxford and Cambridge.2 Ware discovered that one in four college students

wore glasses for nearsightedness whereas among foot guards this was “almost

utterly unknown,” J. Ware (1813, p 32). By simply comparing mean college

attendance of young men who wear and do not wear glasses from his study, one

might conclude that use of glasses causes academic success. But clearly, nineteen-

century foot guards and undergraduates were different in several ways other than

just their use of glasses; for example, upbringing and family background may

explain some—if not all—of the differences in academic success, not glasses. In

other—more recent—words, their parents may be “more educational ambitious

than the general run and so are not only more aware of the importance of

correcting vision in their children but also have stronger incentives to do so,” J.

W. B. Douglas et al. (1967, page 480). To my knowledge, the first study within

economics that looks at the relationship between eyesight and schooling.

Another more subtle identification problem comes from the reverse effect of

schooling on eyesight. Spurred by Ware’s finding, visual scientists have carried

out several studied that try to establish the causes of nearsightedness. See Seang-

Mei Saw et al. (1996) and Ian Morgan and Kathryn Rose (2005) for a review of

that literature, and Elena Tarutta et al. (2011) for an outline of resent advances in

the field.3 Similar to Ware’s study, they find strong relationships between the

visual environment and development of nearsightedness. Visual stress increases

the probability of developing nearsightedness in both humans and animals,

regardless of the subjects’ age, gender, race, and nationality. In particular, they

conclude that assiduous reading increase the prevalence of nearsightedness for all

2 Though Ware’s errand was different (he tried to establish that use of glasses worsened eyesight), the example still

serves to illustrate the main identification problem. 3

Among these visual-science papers, Angle and Wissmann (1980) use the same dataset that I’m using here, the

National Health Examination Survey of Youths (NHES III, 1966-70).

subgroups of humans, so youths who study hard may do well in school, but along

the way, they are also more likely to develop nearsightedness and therefore wear

glasses. A simple difference in mean achievements may therefore reflect nothing

more than a difference in study intensity.

In short, neither corrected nor uncorrected eyesight is random; the group of

youths who develop nearsightedness and afterwards choose to wear glasses might

be particularly studious. Creditable estimates of the effect of visual correction on

students’ achievements therefore require other identification strategies than

simple mean comparison.

The direction of the misestimation depends, as usual, on the relationship

between the unobserved factors that determine use of glasses and performance in

school. For example, glasses may reinforce grades of children who already do

good in school, so the mean difference is too large. Or parents may give glasses to

children who show poor achievements to compensate their grades, so the mean

difference is too small.

Paul Glewwe, Albert Park and Meng Zhao (2006) solve the identification

problem in a randomized controlled trial. Following an eye exam, they randomly

offered free glasses to primary-school students with poor vision who did not

already have glasses in two rural provinces of China. They conclude that wearing

eyeglasses for one year, on average, improves test scores between 0.1 and 0.2

standard deviation of the test score distribution.4+5

4 This source of randomization will estimates the potential effect of giving glasses to poor-vision students who

currently do not wear glasses. A well-known disadvantage, though, is that a randomized experiment may be specific to its

context and the effect of glasses for primary students in two rural provinces of China may not reflect the effect elsewhere. Potentially more problematic, however, is that the follow-up test scores, one year later, do not include students who drop

out of school (drop out of the sample) and lack of glasses may affect this probability, too. Even small dropout rates may

have a large impact on average scores because it’s the worst performing students who tend to drop out. At least, preliminary estimates here suggest that it is a potential source of bias.

5 In addition to Douglas et al. (1967) and Glewwe et al. (2010), mentioned here, three other studies—from the annals of

economics—consider the affect of eyesight and educational outcomes. Linda N. Edwards and Michael Grossman (1980)

use National Health Examination Survey from the early 1960’s (the version before the one used here); João B. Gomes-Netto, Eric A. Hanushek, Raimundo H. Leite and Roberto C. Frota-Bezzera (1997) use Brazilian panel data; and The-Wei

Here, I suggest an alternative source of randomization; rather than deliberate

randomization, I rely on natural randomization in exact eyesight, what David S.

Lee (2008) and D. S. Lee and T. Lemieux (2010) call local randomization. The

result of an eye exam depends partly on observed and unobserved characteristics

and choices of the examined youth and partly on random chance. Say, time spent

reading and family history of nearsightedness will influence your measure of

eyesight, but so will numerous random differences in test procedures, such as type

of eye chart, legibility of eye-chart letters, exact testing distance to the eye chart,

and light in the consulting room. The review by George Smith (2006) find that

with similar test procedures 95 percent of repeated eye tests still differ about one

eye-chart line above and below the patients average eyesight. Allowing test

procedures to differ, as well, increase this uncertainty even further. Y F Yang and

M Dcole (1996) find that one-half of all school-nurse examinations differ one

eye-chart line above or below the same examination in a controlled environment,

and another quarter of examinations differ even further. M. J. Hirsch (1956) and

K. Zadnik et al. (1992) find even larger uncertainties. Essentially, this means that

the exact outcome of an eye exam is random by one or two lines.

For youths bordering the cutoff of nearsightedness, that makes a huge

difference because the tag nearsightedness in general will enable them to get a

pair of glasses. When a school nurse gives such a student an eye exam, he or she

will randomly conclude that that the student is nearsighted and therefore

randomly refer the student to an optometrist. Within a tight boundary off cutoff,

the tag of nearsightedness is therefore as-good-as the purposefully randomized

offer to get a pair of glasses.

Hu (1977) use household data from a coal-mining county in Pennsylvania. But only Glewwe et al. (2010) estimate this relation from an explicit source of random variation.

II. Graphical Evidence

On the back wall of every school nurse’s consultation room hangs an eye chart

with which they measure the performance of students’ vision. As the familiar “big

E” chart in Figure 1, it depicts several lines of capital letters of decreasing size,

and in a routine eye exam, the nurse will first ask the student to name letters on

the smallest readable line without glasses and second do the same exercise with

glasses if the student brought any. The first measure, called uncorrected visual

acuity, UCVA, is a measure of intrinsic eyesight; the second, called presenting

visual acuity, PVA, is a measure of eyesight with whatever correction—none or

some—that the student brings along. Both measured on a 20/20 scale. I’ll follow

tradition of the visual science literature and refer to 20/20 acuity as normal vision.

Visual acuity of 20/30 then means the ability to see at 20 feet (at best) what

someone with normal-vision can see at 30 feet—two line nearsightedness.

FIGURE 1: THE SNELLEN EYE CHART

Notes: NHES III (1966-70) used a slightly modified version of this eye chart, the Sloan chart; while the Snellen chart contains letter of differing legibility, the Sloan chart does not. Please see Section IV. Properly scaled, target letters at the

20/20 line will subtend 5 arcmin at 20 feet, or 20*tan(5 arcmin) = 0.029 feet, or 0.349 inches.

Figure 2 illustrates the relationship between these two measures of eyesight for

youths in NHES III (1966-70). It plots average presenting visual acuity (PVA) for

each line of uncorrected visual acuity (UCVA), 95-percent confidence intervals,

and a kinked-linear prediction of PVA on UCVA. The kinked-linear prediction is

from an OLS regression that allows for different slopes, but not intercepts, on

either side of 20/20 acuity.6

FIGURE 2: PRESENTING VISUAL ACUITY OVER UNCORRECTED VISUAL

ACUITY (FIRST STAGE).

Notes: I’ve used a dummy-variable regression with robust standard errors to

estimate means and confidence intervals. The kinked regression allows for different slopes on either side of 20/20: PVA = π + πUUCVA + πUZUCVA·Z + ε,

where Z=1[UCVA<20/20], and where UCVA is centered to zero at the 20/20 line,

UCVA=20/20≡0.

Clearly, this relationship isn’t deterministic—it’s fuzzy. The level of UCVA

does not automatically determine the level of correction, nor does the tag of

nearsightedness force anyone to wear glasses, but UCVA undeniably influences

the probability of wearing glasses as well as the strength of these glasses, and

relationship seems to change discontinuously at 20/20. For high values of UCVA,

averages of PVA nestle on the 45-degree line; that is, youths with good visual

acuity hardly ever wear glasses, and those who do obviously have very little

6 The appendix shows similar graphs but with other functional predictions.

20/30

20/25

20/20

20/17

20/15

20/12

PV

A

lt 20/400

20/400

20/200

20/10020/70

20/5020/40

20/3020/25

20/2020/17

20/1520/12

UCVA

Mean PVA 95% Confidence limit

Kinked-linear prediction

First stage

Presenting Visual Acuity over Uncorrected Visual Acuity

correction. On the other hand, for low values of UCVA, averages of PVA increase

above the 45-degree line of no correction, so the average nearsighted youth do

wear glasses, and much stronger glasses, too. Development of nearsightedness

therefore seems to change youths’ incentives to wear glasses, as the kinked

regression indicates. Even close to the cutoff, there is a noticeable change;

averages nearest the 20/20 cutoff tend to kink, too. That is, among youth who

essentially have quite similar levels of UCVA, the nearsighted youths are still

more likely to have and wear glasses.

The differences may result from school nurse practices. Today, as well as in the

1960s, school nurses perform vision screenings to detect and refer nearsighted

students to an optometrist to get a pair of glasses. The guideline is, and was, to

refer fourth graders and above if visual acuity in one eye is strictly below 20/30,

or the difference between best and worse eye is more than two eye-chart lines

apart, see M. E. Doster (1971) and American Optometric Association, (2000). In

the appendix, I show that this monocular recommendation corresponds closely to

the binocular cutoff of 20/20, in Figure 2. School nurses are therefore more likely

to have referred students who are below the cutoff to an optometrist whereas

students above the cutoff are less likely to have been referred. Also, as vision

deteriorates, it becomes more and more obvious to teachers, parents, friends, and

the youth, that there is a visual problem. He or she will have to squint in the

classroom and will be the last one to recognize street signs or friends when

driving in a car or walking on the street. This could also change incentives to wear

glasses.

In any circumstance, falling below the cutoff, changes the use of glasses and

PVA, so if eyesight has an effect on performance in school, such a change should

show up in test scores, too.

Figure 3 shows average arithmetic (top) and reading (bottom) scores for each

line of UCVA, 95-percent confidence intervals, and a kinked-linear prediction,

similar to Figure 2—each test score conventionally normalized by its standard

deviation.

FIGURE 3: ARITHMETIC (TOP) AND READING (BOTTOM) SCORE OVER

UNCORRECTED VISUAL ACUITY (REDUCED FORM).

Notes: I’ve divided the test score by its standard deviation (sd) before averaging,

and used a dummy-variable regression with robust standard errors to estimate

means and confidence intervals. The kinked regression allows for different slopes on either side of 20/20: Score = γ + γUUCVA + γUZUCVA·Z + ε, where

Z=1[UCVA<20/20], and where UCVA is centered to zero at the 20/20 line, UCVA=20/20≡0.

Average test scores form a V-shaped pattern with knickpoint at 20/20,

suggesting that the gain in presenting vision does cause better test scores. Youths

with good uncorrected vision (on the right-hand side of Figure 3) do fine in

achievement tests, but as their vision deteriorates, so does average test scores until

vision reaches 20/20 acuity. At 20/20, exactly when the average youth begins to

wear glasses, the average test score regains. This alignment—of kinks in Figure 2

and Figure 3—suggests that as uncorrected vision reduces it becomes more and

3

3.5

4

4.5

Arith

me

tic S

co

re/s

d

lt 20/400

20/400

20/200

20/10020/70

20/5020/40

20/3020/25

20/2020/17

20/1520/12

UCVA

Mean score 95% Confidence limit


Reduced form

Arithmetic Score over Uncorrected Visual Acuity

3

3.5

4

4.5

Re

ad

ing S

core

/sd

lt 20/400

20/400

20/200

20/10020/70

20/5020/40

20/3020/25

20/2020/17

20/1520/12

UCVA

Mean score 95% Confidence limit


Reduced form

Reading Score over Uncorrected Visual Acuity

more difficult to follow classroom instructions, but as vision crosses the 20/20

cutoff, the average youth begins to wear glasses, can again see the blackboard,

refocus classroom learning, and increase test scores. Similar to Figure 2, these

kinks seem persistent even for youths close to the cutoff; that is, a kinked-linear

regression on a tighter sample would also predict a kink at 20/20. Not least

because these averages are very precise, most confidence intervals are less than

one-third of a standard deviation wide.

III. Regression-Discontinuity Framework

While these graphs illustrate that use of glasses may cause better arithmetic and

reading performance of nearsighted students, and that this effect is sufficient large

to increase the average score, too, they only show a crude picture of what is going

on—size and statistical uncertainty of this effect is less clear. To refine it, I rely

on the regression-discontinuity (RD) framework, specifically, combining the

kinked-regression-discontinuity design in David Card, David S. Lee and Zhuan

Pei (2009) with the IV version of RD design—called fuzzy RD—in Jinyong

Hahn, Petra Todd, and Wilbert van der Klaauw (2001). In the kinked design,

treatment intensity gradually increases beyond some cutoff, and the outcome of

interest is a kinked function of the treatment intensity, but treatment is certain. In

the IV design, treatment isn’t certain; instead, it follows a two-stage model that

assigns an intention to treat beyond some cutoff, but students can afterwards

choose of refuse to take the treatment. Combining these, I estimating the

following two-stage model:

(1) i U i i iA UCVA PVA

and

(2) i U i UZ i i iPVA UCVA UCVA Z ,

where student i’s achievements, Ai, is a linear function of the treatment, PVAi, and

treatment assignment is a kinked-linear function of uncorrected eyesight, UCVAi,

above and below the 20/20 cutoff, Zi = 1[UCVAi<20/20]. This—fuzzy kinked—

design first assigns a gradually increasing intention to correct vision beyond the

20/20 cutoff (UCVAi·Zi), but students can afterwards choose their level of

correction (by choosing ξi).

To see the reduced-form of equation (1), please rearrange (2) into (1) to get:

(3) i U U i UZ i i i iA UCVA UCVA Z .

These first-stage and reduced-form equations in (2) and (3), mirror the kinked-

linear predictions drawn in Figure 2 and 3 if UCVAi is centered at 20/20 (takes the

value zero at 20/20). πUC and ρπUC are parametric representations of the kinks, and

the parameter of interest, ρ, determines the change in achievements for a visual

correction of one eye-chart line. As usual, dividing the reduced-form kink in

Figure 3, ρπUC, with the first-stage kink in Figure 2, πUC, will therefore estimate ρ.

That is, ρπUC estimates the kink in achievements and πUC adjust this kink for the

fuzzy take-up of visual correction.

Statistically, this framework is just instrumental-variable (IV) regression where

UCVAi·Zi is the instrument used to identify the first-stage equation, (2). It

therefore shares the pros and cons of IV regression: ρ is causal if the instrument is

uncorrelated with the error term, Cov(UCVAi·Zi,εi) = 0, and strongly correlated

with PVAi, so πUC is significantly different from zero.

For the same reason, it’s therefore closely related to standard RD with an

instrument. In fact, one can convert the kinked RD design into the standard

stairstepped RD design by simply replacing UCVAi·Zi with Zi. That is, kinked RD

gets identification from a changing slope—a kink—in the assignment variable

while stairstepped RD gets identification from a changing level—a jump. Thus,

similar to standard RD, plucking observed values of Ai, UCVAi, UCVAi·Zi, and

PVAi into equation (2) and (3) might not estimate a causal effect of PVAi but for a

slightly different reason. While stairstepped RD prohibits unobserved jumps,

kinked RD prohibits unobserved kinks.7 If some unobserved determinant of

achievements (in εi) kink at the cutoff, then UCVAi·Zi might capture this kink, and

the estimate of ρπUC will be either too big or too small. If assiduous reading—one

of the determinants of nearsightedness and achievements—changes nonlinearly

over the cutoff, then UCVAi·Zi will capture not only its causal effect but also the

effect of studying hard—the nonlinear effect of reading. On the other hand, if the

unobserved change is approximately linearly, then UCVAi will proxy its effect

and the estimate of ρπUC and, therefore, ρ is causal.

Again, as in stairstepped RD, to make causal estimation more probable, one can

estimate the IV model on a tighter sample over the cutoff. As the sample gets

tighter, it first becomes more and more likely that the linear function of UCVAi

will sufficiently approximate any unobserved differences.8 Second, it becomes

more and more likely that the tag of nearsightedness is random because results of

eye exams are partly random as explained in Section I.

The empirical section follows an alternative strategy, as well. In addition to

estimation on tighter samples, I include many predetermined covariates in

equation (1) and (2). If the kink is truly random, then this should have no effect on

ρ. The appendix goes even further and includes both nonlinear functions of

uncorrected vision for tight samples and with covariates. This has no qualitative

effect on results relative to the results presented here, below.

7 In fact, the kinked model can also allow for unobserved jumps in ε by including Z as an additional control,--not as an

instrument—in the IV regression. I do this in the appendix to allow for unobserved differences between nearsighted and non-nearsighted youths.

8 The strategy of estimation on ever smaller samples closer to a cutoff to retrieve the causal effect of a discontinuous

change on achievements is similar to the regression-discontinuity design in, among others, Joshua D. Angrist and Victor

Lavy (1999), Caroline M. Hoxby (2000), P. Bayer et al.(2007) , and Hanley Chiang (2009), what Angrist and Lavy call discontinuity samples.

IV. Data

The dataset that I use is the National Health Examination Survey of Youth

(NHES III, 1966-70). It is cross-section data of 7,514 U.S. youths between the age

of 12 and 17 years; the response rate is 90 percent; it was collected between 1966

and 1970.9+10

The main part is a physical and intellectual examination, focusing

on health of youths but also including demographic- and socioeconomic-

background information; in particular, for this study, it subjected youths to

arithmetic, reading and eye tests. To keep examination conditions constant across

the country, the examining team conducted all examinations in specially

constructed mobile clinics—approximately 12 examinations each day, six in the

morning and six in the afternoon. Because heavy snow could make roads

impassable to the mobile clinic, the survey avoided northern regions in the winter

and southern regions in the summer. For a through description of sampling

design, exact clinical examinations, validation studies, and the extensive data-

quality control see No. 8 (1969).

The precise wearing pattern of eyeglasses is for obvious reasons unobserved—I

do not know when, where, and how often nearsighted students wear glasses. But

survey staff would typically transport examinees to and from the examination

centre, picking up morning examinees at home and dropping them off at school

and the other way round for afternoon examinees. So students who normally wear

their eyeglasses to school are also likely to bring them at the examination centre

9 In this sample, I have changed a visual loss due to glasses to a zero gain for 76 farsighted youths; otherwise, the

sample is unchanged. Farsighted people may theoretically have worse distance vision when using their glasses (have PVA smaller than UCVA), and in the sample they do have that. But in practice they probably will not; it is likely just an artifact

of definitions. In school, or in the classroom, when looking at distant objects like the blackboard, farsighted students could

either look over the rim of their glasses or not use them at all. So in practice, they will also have PVA larger than or equal to UCVA. This change doesn’t affect results, see the appendix.

10 Sampling was based on the same sampling design as the previous, second cycle, of the survey, with some youths

examined in the Cycle II also examined in Cycle III. Cycle III sampled from the same 40 sampling areas as Cycle II, but

when time permitted, were previously (Cycle II) examined youths re-scheduled for examination as well. The selected sample is, because of this non-randomness in sampling, not completely unbiased of the U.S. population in the late 1960’s.

because they attended school either before or after the examination; except,

perhaps, for youth surveyed during their summer holydays.11

The arithmetic- and reading-achievements tests are subtests of the Wide Range

Achievement Test, WRAT. Psychologists, even today, use this psychometric tool

to measure basic academic skills such as reading and spelling of words and

performance of basic mathematical calculations. The complete battery of tests

lasts about 70 minutes, with the reading and arithmetic subtests always

administered as the first tests in each session. To increase accuracy of the test

score, a psychologist with at least a master’s degree and experience in

administering the WRAT test to adolescents gave the test. The reading test

consists of recognizing and naming letters and pronouncing words arranged in

order of increasing difficulty. The arithmetic test requires counting, reading

number symbols, solving oral problems, and performing written computations

usually taught in school. Initial validation studies compared the WRAT scores to

standardized measures of 7 to 12 graders school achievements in West Virginia,

Wisconsin, California, and Colorado.12

Both tests have reasonably high

correlations with these standardized tests, ranging from 0.47 to 0.84 conditional

on grade levels and geographic regions and with a slightly higher accuracy for the

arithmetic test. See D. C. Hitchcock and G. D. Pinde (1974). The raw arithmetic

score range from 0 to 56 points with mean of 23 and standard deviation of 6.9,

and the reading score range from 0 to 89 with mean and standard deviation of 48

and 14. Here, I follow common practice and divide each test score with its

standard deviations.

11 Before the examination households were given a pamphlet describing the program, among others outlining that there

would be “tests of vision and visual acuity”, No. 8 (1969, page 17); hence, some students who normally do not wear glasses to school may have brought them particularly for the purpose of the visual examination. Since academic outcomes

of these students are unaffected by the treatment, estimated treatment effects will tend to understate the effect of wearing

glasses. If they had worn glasses as they “claim”, estimated treatment effects would have been larger. 12

These standardized tests were The Stanford Achievement Test for junior high school students (grades 7-9) and

Metropolitan Achievement Tests for senior high school students.

The eye examination includes tests of eye diseases, color vision, and binocular

and monocular (one eye at a time) vision. It tested for nearsightedness,

farsightedness, and astigmatism and determined current eyeglasses prescription

by a lensometer reading. Most important for this paper, however, it includes eye-

chart tests for nearsightedness both with and without current glasses; that is, tests

of PVA and UCVA. It measured eyesight on an eye chart with four lines at or

above normal vision and eight lines below, giving thirteen possible outcomes of

the visual acuity tests, from less than 20/400 (unable to read any line) to 20/12

(you can read all lines).13

Again, to keep test procedures constant and ensure

consistency of test results, the mobile clinic had a specially designed room for the

eye test with stabile light and test distance, and throughout the survey, equipment

was periodically checked. All examiners received thorough training and practice

in vision testing techniques, and a team of ophthalmologists conducted a pilot

study prior to the survey and a further validation study midway through the

survey. See J. Roberts and D. Slaby (1973), J. Roberts and D. Slaby (1974), and J.

Roberts (1975) for more detail.

As described earlier, the eye-chart measure of visual acuity is on the 20/20

scale, and this scale has an inherent nonlinearity that may approximate

nonlinearity in the assignment variable. Technically, this scale determine the

patients viewing angle at 20 feet; that is, viewing angle standing 20 feet from the

chart—a larger viewing angle (larger letters) meaning poorer vision. To see this

use the formula for right-angled triangles: tan(patients viewing angle) = height of

letter / 20 feet. Essentially, the denominators in the 20/20 scale indicate the

viewing angle because all letters subtend a viewing angle of 5 arc minutes

(arcmin) at altering viewing distances. For example, letters at the 20/200 line

13 The actual eye chart use in NHES III is the Sloan chart, a slight modification of the eye chart in Figure 1, see L. L.

Sloan (1959). Concerned about the equal legibility of lines and letters, Sloan introduced five letters on each line, selected from ten capital letters of equal legibility.

subtend 5 arcmin at 200 feet, the smaller letters at 20/50 subtend 5 arcmin at 50

feet, and the 20/20 line subtend 5 arcmin at 20 feet, and so on, so in non-linear

estimations I can simply use the denominator instead of viewing angle.14

On the

eye chart, each line jump is approximately associated with a 0.1 change in log of

the patients viewing angle (or equivalently log of the denominator), so it’s

approximately 26 percent more difficult to read the line below. For example,

log10(25)-log10(20) ≈ log10(1+0.26) = 0.1, N. B. Carlson and D. Kurtz (2003, page

23) and L. L. Sloan (1959). This inherited nonlinearity in measurement of

eyesight has two important implications for the empirical model. First, the

estimated treatment effect, ρ, estimates the effect on achievements of a

proportional increase in eyesight of about 26 percent. Second, I can use it to allow

for a more flexible specification of the treatment assigning variable, UCVAi, by

including both the viewing distance in feet (the denominator) and its approximate

log10 transformation (the line number on the eye chart). The appendix pursues this

strategy in detail. Results from these nonlinear regressions are not qualitatively

different from results presented here.

Finally, to minimize transportation costs when picking up or dropping off

examinees, the daily consultation would typical schedule youths that either lived

in the same neighborhood or attended the same school, or both. So a variable for

day of examination likely proxy for school and neighborhood characteristics; say,

they may share the same school nurse, teachers, health care services, and

14 The denominator always indicating the distance at which a normal-vision person can barely read the line, where

normal is defined as a viewing angle of 5 arcmin at 20 feet, a 20/20 vision person. Hermann Snellen, the farther of the eye-

chart, also recognized this normalization of his measure: “Die Bestimmung der Sehschärfe had keinen absoluten sondern nur einen relativen Werth. [w]enn man nur als Ausgangspunkt für den Vergleich einen bestimten Sehwinkel wählt. Als

solchen haben wir einen Winkel von 5’ angenommen, weil diser durchschnittlich der kleinste Sehwinkel ist, unter welchem

normale Augen unsere Buchstaben erkennen.” Snellen 1863, page 4. Where, use of the term “normal eyes” dates back as far as Hooke’s experiments, in 1679, on the resolving power of the eye: “tis hardly possible for any unarmed eye well to

distinguish any Angle much smaller then that of a minute: and where two objects are not farther distant then a minute

[t]hey coalesce and appear one.” Levene 1977, page 43. Note that if the letter E subtends five arcminutes then each vertical line, in the E, is separated by one arcminute.

environment.15

As a robustness check, the empirical model among others include

day-of-examination fixed effects. Again, the appendix explores this issue in

further detail.

V. Regression Evidence

In Section II, reduced-form graphs illustrate that something causes test scores to

increase as visual acuity crosses normal vision, and the first-stage graph suggest

that it’s a change in presenting eyesight. This section refines this evidence by

estimating IV regressions in the kinked-regression-discontinuity design, first

without covariates, then with. The appendix provides a comprehensive set of

estimations while this section extracts key results from the appendix.

Panel A of Table 1 presents IV estimates of PVA on test scores, and Panel B

and C present associated reduced-form and first-stage estimates. Arithmetic

scores on the left-hand side, and reading scores on the right. For this table, I’ve

selected two tight samples: 20/30 to 20/17 and 20/50 to 20/12, and the full

sample.16

Figure 4 reproduces the estimates in Panel A, together with similar

estimates for gradually wider samples; each dot represents an estimate of ρ, in

equation (1), but on a different sample size. The second, the sixth, and the final

dot in Figure 4 correspond to estimates in Panel A.

Starting from the bottom, all first-stage estimates on UCVA·Z show significant

changes in visual correction at the cutoff of nearsightedness. That is, development

of nearsightedness is highly associated with the decision to wear glasses and the

level of correction. For tighter samples, the size of the effect is obviously smaller

than in the larger samples because the probability of wearing glasses and the level

of correction is smaller when youth only have slight nearsightedness. The F-

15 Thomas Barnebeck Andersen et al. (2010) show that UV radiation may affect development of poor vision.

16 The Appendix presents a table with estimates of all discontinuity samples.

statistic of this effect is nevertheless close to 50 in the small sample and reaches

well above 2000 in the full sample.17

But then, it’s probably not too surprising

that development and the level of nearsightedness is a—if not the—key

determinant in the decision to wear glasses and the level of visual correction. That

is, the first-stage seems strong.

TABLE 1—KINKED-RD ESTIMATES WITHOUT COVARIATES

(1) (2) (3) (4) (5) (6)

20/30 to

20/17

20/50 to

20/12

Full

sample

20/30 to

20/17

20/50 to

20/12

Full

sample

Arithmetic score Reading score

Panel A. Instrumental-variable regressions.

PVA 0.293 0.402 0.242 0.108 0.409 0.234 (0.14) (0.04) (0.02) (0.14) (0.04) (0.02)

[2.15] [9.55] [14.56] [0.77] [9.26] [13.84]

UCVA -0.213 -0.234 -0.108 -0.101 -0.241 -0.107 (0.08) (0.03) (0.01) (0.08) (0.03) (0.01)

[-2.62] [-8.02] [-18.23] [-1.24] [-7.91] [-18.72]

Panel B. Reduced-form regressions.

UCVA 0.046 0.142 0.142 -0.006 0.142 0.135

(0.05) (0.01) (0.01) (0.05) (0.01) (0.01) [0.95] [10.14] [10.67] [-0.11] [9.71] [9.86]

UCVA∙Z -0.164 -0.268 -0.255 -0.061 -0.272 -0.247

(0.08) (0.03) (0.02) (0.08) (0.03) (0.02) [-2.12] [-10.25] [-14.32] [-0.75] [-10.03] [-13.65]

Panel C. First-stage regressions PVA

UCVA 0.884 0.936 1.032

(0.03) (0.01) (0.01) [28.49] [95.25] [82.50]

UCVA∙Z -0.559 -0.666 -1.055

(0.08) (0.04) (0.02) [-7.06] [-15.82] [-50.01]

Observations 2,055 5,636 6,757 2,055 5,636 6,757

F-test of instrument 49.91 250.4 2501

The average gain of eye-chart lines from wearing glasses, and the share who uses glasses (in parenthesis): —in the sample 0.38 0.32 1.38

(0.23) (0.15) (0.28)

—in the sample of nearsighted youths 1.06 1.73 4.54 (0.43) (0.50) (0.73)

Notes: PVA is presenting visual acuity and UCVA is uncorrected visual acuity both measured in eye-chart

lines. Test scores are divided by their standard deviation, UCVA is centered at 20/20, and Z=1[20/20<0]. Under each estimate, parentheses and brackets hold robust standard errors and t-statistics. The table does not

report the estimated constant.

Source: Author calculations from NHES III (1966-70).

17 These F’s are well above the threshold of weak instrument bias, reported in (Douglas Staiger and James H. Stock,

1997)

Reduced-form estimates reflect this pattern; when visual correction is small so

is the reduced-form gain in test scores, and as the first-stage effects increase they

push up reduced-form effects, too. In the tighter samples, estimates are less

precise (less significant) but as sample size increases so do the level of

precision—as expected.

FIGURE 4. IV ESTIMATES OF PVA ON ARITHMETIC (TOP) AND READING (BOTTOM)

SCORES WITHOUT COVARIATES

Combined—dividing reduced form by first stage—IV estimates suggest that

visual correction is very important for nearsighted students’ achievements. A one-

line visual correction improves test scores between 0.1 and 0.4 standard

deviations of the test score distribution. Close to the cutoff, effects are larger—

although less precise—while in larger samples effects become smaller but

stabilize just above 0.2 standard deviations per line of correction.

.497

.293.408 .38

.454.402

.282 .261 .234 .242 .242

0

.5

1

Std

. ga

inin

test sco

re

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA

95% confidence limit

Average gain in sample (2nd axis)

Av. gain for nearsighted (2nd axis)

Kinked-linear function

The Fuzzy-RD Effect of PVA on Arithmetic Scores

.313

.108

.322 .371.451 .409

.31 .277 .243 .235 .234

0

.2

.4

.6

.8

1

Std

. ga

inin

test sco

re

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA





The Fuzzy-RD Effect of PVA on Reading Scores

These estimate the average effect of correcting nearsightedness one eye-chart

line, but nearsighted youths may correct their eyesight more or less than one line,

on average. The bottom part of Table 1 lists the share of nearsighted youths who

wear glasses as well as the average correction. For example, 43 percent of

nearsighted youths in column 4 wear glasses, and the average level of correction

is slightly above one eye-chart line, 1.06 lines.18

Averaging up, use of eyeglasses

therefore improves nearsighted youths’ arithmetic scores by 0.29·1.06 = 0.31

standard deviations. The upper (orange) line in Figure 4 draws similar averages

for all samples and both test scores.

Alternatively, to compare this effect to other school interventions—say, a class

size reduction, which potentially affects every student—requires averaging over

everybody, both nearsighted and non-nearsighted youths, alike. The bottom part

of Table 1 also lists the share of all youths (in each sub-sample) who wear glasses

as well as the average correction. For example, 23 percent of youths in column 4

wear glasses, and the average level of correction is 0.38 lines. Averaging down,

use of eyeglasses therefore improves average arithmetic scores by 0.29·0.38 =

0.11 standard deviations. The lower (green) line in Figure 4 draws similarly

averaged effects for all samples and both test scores.

Overall, these effects are large, and for the full sample, maybe too large. As

explained above, if reduced-form estimates capture the effect of, say, being

studious, that might lead to overestimation. For example, if assiduous reading is

not approximately linear in equation (3) then UCVAi will not sufficient proxy

study intensity, and UCVAi·Zi might capture the non-linearity so the estimate of

ρπUZ is too large. In particular, the wider samples might overestimate the effect

while the tighter samples should more accurately estimate ρπUZ and therefore also

ρ.

18 Nearsightedness is defined as UCVA<20/20, so nearsighted youths in column 4 have either 20/25 or 20/30 vision.

A. Regression with covariates

Following, I explore validity of the identification strategy as sample size

increases over the cutoff. I present three sets of figures with covariate-adjusted

estimates and compare these to the non-adjusted estimates in Figure 4.

Conditioning on predetermined characteristics should not affect estimates of ρ in

samples where the kink (the instrument) captures random variation. On the other

hand, if the kink is related to observed determinants of student performance, then

conditioning on them will affect the estimate of ρ—and if observed characteristics

can affect the estimate, then maybe so can unobserved characteristics. Covariate-

adjusted estimates can therefore illustrate how far from the cutoff that the estimate

of ρ is causal. The appendix presents several other robustness analyses, among

others, regressing characteristics on the kink; these reach the same conclusion.

FIGURE 5. IV ESTIMATES FOR ARITHMETIC (TOP) AND READING (BOTTOM) SCORES

WITH YOUTHS COVARIATES.


WITH YOUTHS COVARIATES.

.606

.347 .306 .324 .35 .3.18 .175 .157 .167 .167

0

.2

.4

.6

.8

Std

. ga

inin

test sco

re

0

.5

1

1.5

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA






.346

.191 .243 .279 .337 .328.228 .209 .182 .173 .172

0

.2

.4

.6

.8

Std

. ga

inin

test sco

re

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA






.606

.347 .306 .324 .35 .3.18 .175 .157 .167 .167

0

.5

1

1.5

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA


Unconditional estimate



.346

.191.243 .279

.337 .328.228 .209 .182 .173 .172

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA






WITH YOUTHS AND PARENTAL COVARIATES.


WITH YOUTH AND PARENTAL COVARIATES.

.611

.361 .314 .315 .304 .26.149 .14 .126 .136 .134

0

.2

.4

.6

Std

. ga

inin

test sco

re

0

.5

1

1.5

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA






.217 .156 .205 .246 .266 .269.188 .164 .142 .131 .128

0

.2

.4

.6

Std

. ga

inin

test sco

re

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA






.611

.361 .314 .315 .304 .26.149 .14 .126 .136 .134

0

.5

1

1.5

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA





.217.156 .205 .246 .266 .269

.188 .164 .142 .131 .128

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA






WITH YOUTH, PARENTAL, AND NEIGHBORHOOD COVARIATES.

FIGURE 10. IV ESTIMATES FOR ARITHMETIC (TOP) AND READING (BOTTOM) SCORES WITH YOUTH, PARENTAL, AND NEIGHBORHOOD COVARIATES.

.364 .327 .381 .317 .252 .202 .139 .126 .114 .127 .129

0

.2

.4

.6

Std

. ga

inin

test sco

re

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA






.323.186 .238 .227 .206 .215 .178 .154 .135 .127 .126

0

.2

.4

.6

Std

. ga

inin

test sco

re

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA






.364 .327 .381.317

.252 .202.139 .126 .114 .127 .129

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA





.323.186 .238 .227 .206 .215 .178 .154 .135 .127 .126

-.5

0

.5

1

Std

. ga

inin

test sco

re

lt 20/400 to

20/12

20/400 to 20/12

20/200 to 20/12

20/100 to 20/12

20/70 to 20/12

20/50 to 20/12

20/40 to 20/12

20/40 to 20/15

20/30 to 20/15

20/30 to 20/17

20/25 to 20/17

UCVA samples

Coef. on PVA





Figure 5, Figure 7, and Figure 9 plot point estimates of PVA on test scores

similar to Figure 4, except; these figures condition on covariates. And Figure 6,

Figure 8, and Figure 10 compare these adjusted estimates to the non-adjusted

estimates in Figure 4. Estimates in Figure 5 and Figure 6 condition on youths’

characteristics: the age measured in months, gender, race measured as white or

non-white, birth weight measured in grams, gestation period measured in weeks,

and birth order of youth. In addition to these covariates, Figure 7 and Figure 8

also include household and parental characteristics given by the number of

children in the household, single parent home, household income, years of

parental education, and father and mothers age at birth. And finally, Figure 9 and

Figure 10 add a set of indicator variables for the day of examination. As explained

in Section IV, youths examined on the same day come from the same

neighborhood and attend the same school, so day-of-examination fixed effects

will proxy for differences in school and neighborhood characteristics. For

example, differences in school-nurse praxises. The appendix gives a more detailed

description of all these covariates as well as the complete set of estimation tables.

Overall, covariate adjustments do not change the pattern in Figure 4: larger

effects initially that tend to reduce in wider samples. Close to the cutoff, the

kink—the natural experiment—is as good as a controlled experiment while

further from the cutoff unadjusted estimates may overestimate the causal effect of

wearing glasses. Point estimates from the tighter samples, the first two to three

estimates, still fluctuate about an effect size of a quarter of a standard deviation

(and are not statistically different from unadjusted estimates). The next three to

four estimates become smaller than their unadjusted kind but stabilize at the level

of the tighter samples, reducing some of the hump shape in Figure 4. The final

estimates are still very precise but smaller by about 0.1 standard deviations, and

stabilize at an effect of just above 0.1 standard deviations of the test score

distributions. That is, close to the cutoff, the kink is unrelated to covariates, so

estimation of equation (1) and (2), based on a tight sample, will retrieve and

estimate ρ. As the sample size increase, however, and include youths who are

more nearsighted, the unadjusted kinked-linear model begins to capture other

determinants of test scores. The figures suggest, however, that covariate-adjusted

estimates based on intermediate samples can also estimate ρ because these

estimates tend to stabilize at the same level as the tighter samples. Further from

the cutoff, covariate adjustments make a difference, so it is less reliable to draw

causal conclusions from these estimates.

Averaged effects are also more stabile in regressions with covariates. The

sample average is about 0.1 standard deviations, perhaps slightly lower for

reading scores, although not statistically. In the larger samples that include the

effect of glasses for very nearsighted youth, the effect is higher. Averaged over

nearsighted youth—the upper (orange) lines—the effect of wearing glasses seem

to stabilize around 0.3 close to the cutoff. That is, nearsighted youth gain about

0.3 standard deviations on their test scores if they are able to see clearly. Again,

this effect is slightly higher for arithmetic scores and slightly lower for reading

scores, but again not statistically different. Including youths with severe

nearsightedness, the effect of wearing glasses approaches 0.6 standard deviations

of the test score distribution for both reading and arithmetic scores, but still these

effects are less reliable.

VI. Conclusion

Nearsighted students who do not wear glasses will fall behind in school. The

evidence presented here suggests that nearsighted youths who wear glasses do

much better in school than nearsighted youths who do not wear glasses. For all

middle- and high-school aged youths, the cumulative effect of eyeglasses is about

0.1 standard deviation of the test score distribution, and for nearsighted youths

alone it is 0.3 standard deviations.

Essentially, the identification strategy goes as follows: nearsightedness is partly

random near its cutoff, but youths who develop nearsightedness are more likely to

wear glasses because school nurses refer them to an optometrist, and nearsighted

youths who wear glasses can follow classroom instructions and improve their

grades. So comparing performance of nearsighted youths to non-nearsighted

youths will estimate the causal effect of glasses.

Interpretation of these results requires some caution because I rely on

instrumental-variable estimation near the cutoff. First of all, I estimate the effect

of eyeglasses for youths who decide to wear glasses as they become nearsighted.

That is, the effect for current users of glasses. A vision-health program that

increases the use of glasses among current non-users may have a different effect.

Second, the estimate is only causal close to the cutoff of nearsightedness. That is,

the effect is not for youths who have severe levels of nearsightedness. On the

other hand, most people who develop severe nearsightedness will have crossed

the cutoff at some point (few children below the age of twelve are severely

nearsighted). Probably, severely nearsighted youths will have even larger benefits

of wearing glasses.

Overall, these findings suggest that use of glasses is an efficient way of

improving reading and math performance in school. Compared to the Tennessee

STAR experiment (Alan B. Krueger, 1999) or the class size reduction from

Maimonides’ rule in Israeli school (Joshua D. Angrist and Victor Lavy, 1999),

this effect of visual correction is about half the size of a seven student reduction.

But the average cost of the STAR experiment was about $1035 per student in the

1980’s (Alan B. Krueger, 1999)—clearly more than double the price of visual

correction even today.

***

“Quite unknown to myself, I was, while a boy, under a hopeless disadvantage in

studying nature. I was very nearsighted, so that the only things I could study were

those I ran against or stumbled over. It was that summer, [when I was about

thirteen,] that I got my first gun, and it puzzled me to find that my companions

seemed to see things to shoot at which I could not see at all. One day they read

aloud an advertisement in huge letters on a distant billboard, and I then realized

that something was the matter, for not only was I unable to read the sign but I

could not even see the letters. I spoke of this to my father, and soon afterwards got

my first pair of spectacles, which literally opened an entire new world to me. I

had no idea how beautiful the world was until I got those spectacles. I had been a

clumsy and awkward little boy, [a good deal,] due to the fact I could not see and

yet was wholly ignorant that I was not seeing.” Narrative by 26th

U.S. President,

Theodore Roosevelt (2010, page 16), from ca. 1871.

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8, No. 1969. "Plan and Operation of a Health Examination Survey of U.S. Youths

12-17 Years of Age." Vital Health Stat 1, (8), pp. 1-80.

Andersen, Thomas Barnebeck; Carl-Johan Dalgaard and Pablo Selaya. 2010.

"Eye Disease and Development," In Unpublished paper. University of

Copenhagen.

Angrist, Joshua D. and Victor Lavy. 1999. "Using Maimonides' Rule to

Estimate the Effect of Class Size on Scholastic Achievement*." Quarterly

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