Untangling 

the Direct and Indirect Effects 

of Body Mass Dynamics 

on Earnings

Donna B. Gilleskie, Univ of North Carolina

Euna Han, Univ of Illinois at Chicago

Edward C. Norton, Univ of Michigan

October 20, 2010

Veteran’s Administration Cyber Seminar
Body Mass of Females as we Age

(same individuals followed over time)

Body Mass Index

Age
Flow of presentation

1. A closer look at body mass

2. Our model of behaviors over time that relate 

to body mass evolution and observed wages

3. A description of the data we use to 

understand the relationship between body 

mass and wages

4. Unobserved individual heterogeneity

5. Conditional Density Estimation

6. Preliminary results and discussion
Description of Average Body Mass by Age 

(using repeated cross sections from NHIS data)

Source: DiNardo, Garlick, Stange (2010 working paper)
Empirical Distribution of the Body Mass Index

•The distribution of BMI 

(among the US adult female population)

is changing over time.

•The mean and median 

have increased significantly.

•The right tail has thickened

(larger percent obese).

Source: DiNardo, Garlick, Stange (2010 working paper)
Trends in Body Mass over time 
1999

Obesity Trends* Among U.S. Adults

BRFSS, 1990, 1999, 2008

(*BMI ≥30, or about 30 lbs. overweight for 5’4” person)

2008

1990

No Data <10% 10%–14% 15%–19% 20%–24% 25%–29% ≥30%
Body Mass Index

Calories in < Calories out

Calories in > Calories out

Caloric Intake: 

food and drink

Caloric Expenditure: 

requirement to sustain life and exercise
• A function of weight & height; 

independent of age & gender (among adults)

• A commonly used diagnostic tool to identify weight 

problems

(original proponents stressed its use in population studies 

and considered it inappropriate for individual diagnosis) 

• A simple means for classifying (sedentary) individuals

• BMI < 18.5: underweight

• 18.5 to 25: ideal weight

• 25 to 30: overweight

• BMI 30+: obese

• Available in nationally representative data sets

Advantages of the Body Mass Index
• A function of self-reported weight and height 

- subjective measure, rounding issues, but can apply correction

• Does not fully capture, or capture correctly, adiposity

- may overestimate on those with more lean body mass (e.g. athletes)

- and underestimate on those with less lean body mass (e.g. the elderly)

• Other measures of “fatness”?

- percentage of body fat (skinfold, underwater weighing, fat-free mass index)

- measures that account for mass and volume location (body volume index)

• Does the functional relationship restrict measurement of the 

effect of body mass on wages? 

• Should weight and height be included separately/flexibly?

Concerns with using the Body Mass Index
Body Mass and Wages

• Evidence in the economic literature that 

wages of white women are negatively 

correlated with BMI.

• Evidence that wages of white men, white 

women, and black women are negatively 

correlated with body fat (and positively 

correlated with fat-free mass).

• Evidence of a height-related wage premium.
Potential Problems...

• Cross sectional data: we don’t want a snapshot of 

what’s going on, but rather we want to follow the same

individuals over time.

• Endogenous body mass: to the extent that 

individual permanent unobserved characteristics as well 

as time-varying ones influence both BMI and wages, we 

want to measure unbiased effects.

• Confounders: BMI might affect other variables that 

also impact wages; hence we want to model all avenues

through which BMI might explain differences in wages. 
An analogous situation...

• We may want to understand the effect of 

body mass on medical care expenditures.

• Body mass and medical care expenditures 

both tend to rise with age...and perhaps both 

decrease at oldest ages.

• Medical care expenditures are observed only 

if an individual consumes medical care.

• Other endogenous factors affect medical care 

expenditures that may also be influenced by 

body mass.
Year 1984 Year 1999

Mean BMI

75% percentile

Females

Males

Body Mass Index Distribution 

of same individuals in 1984 and 1999

Source: NLSY79

% obese: 6.7 (1984) ; 30.2 (1999)

% obese: 5.4 (1984) ; 25.6 (1999)
Year 1984 Year 1999

Mean BMI

75% percentile

White

n = 1341

Black

n = 873

Body Mass Index Distribution 

of same females in 1984 and 1999

% obese: 3.9 (1984) ; 21.2 (1999)

% obese: 11.2 (1984) ; 43.4 (1999)

Source: NLSY79
The Big 

Schooling Picture

Observed Wagest

Body 

Masst

Employment

Body 

Masst+1

Schooling

Employment

Marriage

Children

Historyt of:

Current 

decisionst

regarding:

t-1 t t+1

Marriage

Children

Intaket

Exerciset

Info known 

entering period t
Model of behavior of individuals as they age

w

*

t st

, et, mt, kt

Bt+1

beginning 

of age t

beginning 

of age t+1

draw from wage 

distribution

schooling, 

employment, 

marriage, and 

child accumulation 

decisions

evolution of 

body mass

Ωt

= (Bt, 

St

,Et, Mt, Kt,

Xt, Pt )

Ωt+1= (Bt+1,

St+1,Et+1, Mt+1, Kt+1,

Xt+1, Pt+1 )

ci

t

, co

t

caloric intake, 

caloric output

dt ct 

dt = d(Ωt , Pd

t , Pct ) + edt

ct = c(Ωt , dt , wt , Pc

t

) + ec

t

Bt+1 = b(Bt , ci

t

, co

t ) + ebt

wt | et = w(Ωt) + ew

t

more frequent 

shocks

unobs’d biol. changes

= b’(Ωt , dt , wt , Pc

t

) + eb

t
Data: National Longitudinal Survey of Youth (NLSY)
Information entering period t (endogenous state variables)
Exogenous variables Xt : race, AFQT score, non-earned

income, spouse income if married,

ubanicity, region, and time trend 

Information entering period t (endogenous state variables)
Exogenous price and supply side variables
Exogenous price and supply side variables
Exogenous price and supply side variables
Decision/Outcome Estimator Explanatory Variables

Endogenous Exogenous Unobs’d Het

Initially observed

state variables

2 logit

7 ols

X1, P1, Z1 ρi

μ

Enrolled logit Bt

, St, Et, Mt, Kt Xt, Ps

t

, Pe

t

, Pm

t

, Pk

t

,

Pb

ρsμ, ωsυt

Employed mlogit Bt, St, Et, Mt, Kt Xt, Ps

t

, Pe

t

, Pm

t

, Pk

t

,

Pb

ρeμ, ωeυt

Married logit Bt, St, Et, Mt, Kt Xt, Ps

t

, Pe

t

, Pm

t

, Pk

t

,

Pb

ρmμ, ωmυt

Change in kids mlogit Bt, St, Et, Mt, Kt Xt, Ps

t

, Pe

t

, Pm

t

, Pk

t

,

Pb

ρkμ, ωkυt

Wage not obs’d logit Bt, St, Et, Mt, Kt Xt ρnμ, ωnυt

Wage if emp’d CDE Bt, St, Et, Mt, Kt Xt, Pe

t ρwμ, ωwυt

Body Mass CDE Bt, St+1, Et+1,

Mt+1, Kt+1

Xt, Pb ρbμ, ωbυt

Attrition logit Bt+1, St+1, Et+1,

Mt+1, Kt+1

Xt ρaμ, ωaυt
What do hourly wages look like among the employed?
... and ln(wages)?
Empirical Model of Wages

ln(wt ) | et ≠ 0 = η0 + η1 St

+ ρw μ + ωw νt + εw

t

+ η2 Et + η3 1[et = 1 (pt) ]

+ η4 Bt + η5 Mt + η6 Kt

+ η7 Xt + η8 Pe

t + η9 t

Schooling (entering t)

work experience

and part time indicator

productivity

exogenous determinants

and changes in skill prices

unobserved permanent and 

time varying heterogeneity
Unobserved Heterogeneity Specification

• Permanent: rate of time preference, genetics

• Time-varying: unmodeled stressors 

ue

t = ρe μ + ωe νt + εet

where ue

t is the unobserved component for equation e decomposed into

• permanent heterogeneity factor μ with factor loading ρe

• time-varying heterogeneity factor νt with factor loading ωe 

• iid component εe

t 

distributed N(0,σ2

e) for continuous equations and 

Extreme Value for dichotomous/polychotomous outcomes
Individual’s optimal decisions

about schooling, employment, marriage, and child accumulation

Theory suggests that these decisions are jointly made, hence the 

observed behaviors are modeled jointly.

Because of the number of combinations, we specify equations 

for each behavior but estimate them jointly (correlated by the 

permanent and time varying unobserved heterogeneity). 
School



Body Mass Transition

Bt+1 = η0 + η1 Bt

+ ρb μ + ωb νt + εb

t

+ η2 St+1 + η4 Et+1 + η5 Mt+1 + η6 Kt+1

+ η7 Xt + η8 Pb

t

3.95 55.75 22.77 17.53 

0.74 46.25 37.91 15.10 

Under Normal Over Obese

Female

Male

body mass

production function
What does the distribution of body mass look like?

Females

Mean = 25.41

St.Dev. = 5.95
What does body mass look like as we age?

under

normal

over

obese
What does body mass look like as we age?

20 30 40
How should we estimate wages?

• OLS ? : quantifies how variation in rhs variables 

explain variation in the lhs variable, on average.

• It explains how the mean W varies with Z. 

• In estimation, we also recover the variance of W.

• The mean and variance of W define the 

distribution of wages (under the assumption of a 

normal density).
• So, using OLS, we can obtain the marginal 

effect of Z on W, on average.

• But what if Z has a different effect on W at 

different values of W?

• Might BMI have one effect on wages at low 

levels of the wage and a different effect on 

wages at higher levels of the wage?

• How can we capture that?

How should we estimate wages?
Might there be a more flexible way 

of modeling the density ?
Conditional Density Estimation

Determine cut points such that 1/K th of individuals are in 

each cell.

Let 

Replicate each observation K times 

and create an indicator of which cell 

an individual’s wage falls into.

Interact Z’s with α’s fully. 

Estimate a logit equation (or hazard), λ(k,Z).

Then, the probability of being in the kth cell , conditional on 

not being in a previous cell, is 

Define a cell indicator 
Variable Model 1

BMIt -0.008 (0.002)

***

BMIt ≤ 18.5 -0.047 (0.024)

**

25 ≤ BMIt< 30 -0.022 (0.013)

*

BMIt ≥ 30 -0.049 (0.025)

**

BMIt x Black 0.006 (0.002)

***

BMIt x Hisp 0.002 (0.003)

BMIt x Asian 0.001 (0.005)

Model 

Includes:

Xt, Bt

Marginal 

Effect of 5% 

0.122

0.057

Replication of Literature: ln(wages) of females

white females

black females

Results re-transformed to levels, so 

changes are in cents.
Variable Model 1 Model 2

BMIt -0.008 (0.002)

*** -0.008 (0.002)***

BMIt ≤ 18.5 -0.047 (0.024)

** -0.029 (0.019)

25 ≤ BMIt< 30 -0.022 (0.013)

* -0.008 (0.012)

BMIt ≥ 30 -0.049 (0.025)

** -0.010 (0.021)

BMIt x Black 0.006 (0.002)

***

0.004 (0.002)

**

BMIt x Hisp 0.002 (0.003) 0.001 (0.003)

BMIt x Asian 0.001 (0.005) 0.001 (0.004)

Model 

Includes:

Xt, Bt Xt, Bt, 

St, Et, Mt, Kt

Marginal 

Effect of 5% 

0.122

0.057

0.102

0.062

Replication of Literature: ln(wages) of females
Variable Model 1 Model 2 Model 3

BMIt -0.008 (0.002)

*** -0.008 (0.002)*** -0.008 (0.00***2)

BMIt ≤ 18.5 -0.047 (0.024)

** -0.029 (0.019) -0.028 (0.019)

25 ≤ BMIt< 30 -0.022 (0.013)

* -0.008 (0.012) -0.010 (0.012)

BMIt ≥ 30 -0.049 (0.025)

** -0.010 (0.021) -0.013 (0.021)

BMIt x Black 0.006 (0.002)

***

0.004 (0.002)

**

0.004 (0.002)

*

BMIt x Hisp 0.002 (0.003) 0.001 (0.003) 0.000 (0.003)

BMIt x Asian 0.001 (0.005) 0.001 (0.004) 0.002 (0.004)

Model 

Includes:

Xt, Bt Xt, Bt, 

St, Et, Mt, Kt

Xt, Bt, 

St, Et, Mt, Kt, Pe

t

Marginal 

Effect of 5% 

0.122

0.057

0.102

0.062

0.099

0.063

Replication of Literature: ln(wages) of females
Variable Model 1 Model 2 Model 3 Model 4

BMIt -0.008 (0.002)

*** -0.008 (0.002)*** -0.008 (0.00***2) -0.003 (0.002)

BMIt ≤ 18.5 -0.047 (0.024)

** -0.029 (0.019) -0.028 (0.019) -0.046 (0.014)***

25 ≤ BMIt< 30 -0.022 (0.013)

* -0.008 (0.012) -0.010 (0.012) 0.008 (0.009)

BMIt ≥ 30 -0.049 (0.025)

** -0.010 (0.021) -0.013 (0.021) -0.006 (0.015)

BMIt x Black 0.006 (0.002)

***

0.004 (0.002)

**

0.004 (0.002)

*

0.004 (0.003)

BMIt x Hisp 0.002 (0.003) 0.001 (0.003) 0.000 (0.003) 0.004 (0.003)

BMIt x Asian 0.001 (0.005) 0.001 (0.004) 0.002 (0.004) -0.003 (0.006)

Model 

Includes:

Xt, Bt Xt, Bt, 

St, Et, Mt, Kt

Xt, Bt, 

St, Et, Mt, Kt, Pe

t

Xt, Bt, 

St, Et, Mt, Kt,Pe

t

, 

fixed effects

Marginal 

Effect of 5% 

0.122

0.057

0.102

0.062

0.099

0.063

0.034

-0.006

Replication of Literature: ln(wages) of females
White Females

Black Females

Predicted change in wage by BMI

Source: NLSY79 data

$

$

* Reference BMI = 22

Predicted change in wage by BMI

BMI Specification: Single index with fixed effects
Preliminary Results – quantile regression

Variable 25th

percentile

50th

percentile

75th

percentile

BMIt -0.053 (0.012)

*** -0.071 (0.010)*** -0.102 (0.015)***

BMIt ≤ 18.5 -0.241 (0.083)

*** -0.232 (0.082)*** -0.298 (0.124)***

25 ≤ BMIt< 30 -0.173 (0.085)

** -0.117 (0.083) -0.070 (0.109)

BMIt ≥ 30 -0.281 (0.126)

*** -0.225 (0.110)**

0.004 (0.170)

BMIt x Black 0.035 (0.008)

***

0.044 (0.009)

***

0.050 (0.013)

***

BMIt x Hisp 0.021 (0.011)

** -0.001 (0.010) -0.009 (0.018)

BMIt x Asian -0.007 (0.023) 0.006 (0.022) 0.061 (0.024)

Model 

Includes:

Xt, Bt, 

St, Et, Mt, Kt, Pe

t

Marginal 

Effect of 5% 

0.068

0.044

0.083

0.050

0.106

0.067

Quantile Reg

Averages:

0.086 whites

0.054 blacks

CDE Averages:

0.104 whites

0.064 blacks
Fit of preferred model

Outcome Observed Average Predicted Probability

Enrolled 15.88 15.89

Employment

- full time 58.48 57.59

- part time 23.09 23.14

- not employed 18.44 19.27

Married 52.80 52.83

Children

- no change 88.96 88.99

- acquire 8.06 8.61

- lose 2.44 2.40

BMI 25.68 25.64

Wage 10.71 10.32
Unobserved Heterogeneity Distribution

1.0



1

1

= 0



2



3

2

= 0.557

3

= 1

0.353

0.464

0.183



Probability

1.0



1

t1

= 0



2

t2

= 0.520

t

3

= 1

0.539

0.299

0.162

t

Probability



3

Permanent Time-varying

Preliminary Results: preferred model
Simulated reduction in weight

Age Predicted Wage

By Age

Predicted 

Difference

if BMI reduced 5%

Age Predicted Wage

By Age

Predicted 

Difference

if BMI reduced 5%

White Black White Black White Black White Black

19 6.38 5.84 0.000 -0.030 33 11.87 10.34 0.028 -0.006

20 6.78 5.97 -0.001 -0.032 34 12.09 10.59 0.028 -0.001

21 7.17 6.37 0.001 -0.035 35 12.15 10.83 0.036 -0.005

22 7.87 6.95 0.002 -0.034 36 12.64 11.02 0.036 0.011

23 8.59 7.47 0.005 -0.033 37 12.80 11.46 0.045 0.009

24 9.18 7.88 0.004 -0.035 38 13.14 11.78 0.043 0.018

25 9.65 8.25 0.008 -0.034 39 13.09 11.79 0.036 0.019

26 10.07 8.55 0.010 -0.033 40 13.69 12.39 0.045 0.024

27 10.45 8.91 0.012 -0.030 41 13.83 12.67 0.036 0.024

28 10.80 9.19 0.015 -0.025 42 14.17 12.95 0.051 0.020

29 11.14 9.45 0.018 -0.020 43 14.28 13.40 0.033 0.044

30 11.34 9.78 0.019 -0.019 44 14.17 14.03 0.050 0.016

31 11.58 10.09 0.021 -0.017 45 15.11 13.53 0.015 0.067

32 11.83 10.13 0.023 -0.005 Ave 10.96 9.57 0.020 -0.015
• No updating: simply compute the effect of a 

change in BMI given the values of one’s observed 

RHS variables entering the period.

• Hence, this calculation provides only the direct effect 

of BMI on wages.

• We find that a 5% decrease in BMI leads to a small 

increase in wages for white females and a slight 

decrease in wages of black females.

• This simulation does not update state variables over 

time; hence no change in behaviors associated with 

the reduction in BMI.

Preliminary Results: using preferred model
• Specification of BMI (more moments, interactions)

• Selection into employment

• Endogenous BMI

• Endogenous state variables (related to history of 

schooling, employment, marriage, and children)

• Random effects vs fixed effects

• Permanent and time-varying heterogeneity

• Modeling of effect of BMI on density of wages

Sources of differences from preliminary models
Preliminary Conclusions

• Permanent and time-varying unobservables that influence BMI 

also affect wages (and therefore must be accounted for)

• In addition to its direct effect on wages, BMI has a significant 

effect on wages through particular endogenous channels: 

schooling, work experience, marital status, and number of 

children.

• Conditional density estimation results suggest that it is 

important to model the effect of BMI across the distribution of 

wages, not just the mean. (And to model the different effects 

of past decisions across the distribution of current BMI.)

• BMI appears to have only a very small statistically significant 

direct effect on women’s wages (an effect that remains 

unexplained).
Additional Feedback ...

• The body mass index (BMI) specifies a 

particular relationship b/w weight and height

• In light of the positive height effect and 

conjectured “beauty” effect on wages, are 

the appropriate measures simply weight and 

height?

• How might we simulate a change in BMI or 

weight when updating?
Thank you!

• I look forward to further discussion.

• Please email me with any comments, 

questions, or suggestions:

donna_gilleskie@unc.edu