Selection Bias, Comparative Advantage and Heterogeneous

Returns to Education: Evidence from China

James J. Heckman(University of Chicago)

Xuesong Li(Institute of Quantitative & Technical Economics,

Chinese Academy of Social Sciences)

2

1. Introduction

2. Models with and without Heterogeneity

3. Selection Bias and The Marginal Treatment Effect

4. Data Set and Empirical Results

5. Concluding Remarks

3

1. Introduction

Heterogeneity and missing counterfactual states are central features of

micro data.

This paper uses China’s micro data, to estimate the return to education

for China considering both heterogeneity and selection bias.

4

Our work builds on previous research by Heckman and Vytlacil (1999,

2001), and Carneiro (2002), which develops a semi parametric

framework.

5

2. Models with and without Heterogeneity

A conventional model of the return to education without heterogeneity in returns:

(1)

I for individuals (i=1, 2, . . . ,n),

lnYi is log income,

Si is schooling level or years of schooling,

Xi is a vector of variables

βis the rate of return to education,

γis a vector of coefficients.

iiii UXSY ln

6

OLS problem: omitted ability Ai,

Three strategies:

(1) IV.

But It is also very hard to find satisfactory instruments. In fact, most commonly

used instruments in the schooling literature are invalid because they are

correlated with the omitted ability.

(2) Fixed effect method: find a paired comparison such as a genetic twin or

sibling with similar or identical ability.

It needs enough information

7

.

(3) Proxy variables for ability

Many empirical analyses reveal that better family background and better family

resources are usually associated with better environments which raise ability.

In our empirical work we use parental income as a proxy for ability.

8

A model with heterogeneous returns to education (in random coefficient form)

(2)

βi is the heterogeneous rate of return to education, which varies among

individuals.

Xi is a vector of variables including the proxy for ability.

We focus on two schooling choices:

(1) high school Si=0

(2) college Si=1

iiiii UXSY ln

9

The two potential selection outcomes

(3b) 1 if ln

(3a) 0 if ln

111

000

iiii

iiii

SUXY

SUXY

10

Observed log earnings are:

where

(5)

is the heterogeneous return to education for individual i.

βi varies in the population, and the return to schooling is a random variable with

a distribution.

(4d)

(4c) )]()[(

(4b) ])([])[(

(4a) ln)1(lnln

00

000101

010001

01

iiii

iiiiii

iiiiiii

iiiii

UXγSβ

UXSUUX

SUUUXSX

YSYSY

)()( 0101 iiii UUX

11

The mean of βi given X is:

(6)

Decision rule:

(7)

Si* is a latent variable denoting the net benefit of going to school

Zi is an observed vector of variables.

])[()( 01 iii XEXE

* ( )

1 if 0 0 otherwise,

i i i si

*i i

S P Z U

S S

12

Pi = Pi (Zi) is the propensity score or probability of receiving treatment (going to

college). P(Z) can be estimated by a logit or probit model.

Usi is the unobserved heterogeneity for individual i in the treatment selection

equation. Without loss of generality, we may assume that Usi ~ Unif [0,1].

The decision of whether to go to college (or not) for individual i is determined

completely by the comparison of the observed heterogeneity Pi(Zi) with the

unobserved heterogeneity Usi.

The smaller the Usi, the more likely it is that the person goes to college.

13

3. Selection Bias and The Marginal Treatment Effect

(8)

ATE is the average treatment effect (the effect of randomly assigning a person to

schooling)

(9)

1 1 0 0

1 0

ˆplim( ) (ln , 1) (ln , 0)

( , 1) ( , 0)

[ ( 1) ( 0)]

OLS i i i i i i

i i i i i i i i

i i i i

E Y X S E Y X S

E X U X S E X U X S

E U S E U S

(ATE) (Bias)

)()( iiii XEXEATE

14

(10)

Selection bias is the mean difference in the no-schooling (S = 0) unobservable

s between those who go to school and those who do not.

0 0

ˆplim( ) (ln , 1) (ln , 0)

, 1) [ ( 1) ( 0)]

OLS i i i i i i

i i i i i i i

E Y X S E Y X S

E(β X S E U S E U S

(TT) (Selection Bias)

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TT (treatment on the treated), the effect of treatment on those who receive it

(e.g. go to college) compared with what they would experience without treatm

ent (i.e. do not go to college), defined as

(11)

Sorting effect is the mean gain of the unobservables for people who choose ‘1’.

1 0 1 0

( , 1) ( , 1)

( 1) ( 1)

( )

i i i i i i

i i i i i i

TT E X S E X S

E U U S ATE E U U S

Sorting Effect

16

IV is not a consistent estimator In the presence of heterogeneity and selection

bias.

(12)

0 1 0

1 01 0

( ln ) ( ) [ ( ) ]ˆplim( ) ( ) ( )

[ ( ) 1][ ( ) ] ( ) ( )

i i i i i i i iIV

i i i i i i

i i i i ii i i i

i i i i

Cov I , Y Cov I , U Cov I , U U SCov I , S Cov I , S Cov I , S

Cov I , U U S PCov I , U U SCov I , S Cov I , S

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Neither OLS nor IV is a consistent estimator of the mean return to education in

the presence of heterogeneity and selection.

Under certain assumptions, it is possible to identify the heterogeneous return

to education with marginal treatment effect (MTE) via the method of local

instrument variables (LIV), where MTE is:

1 0 1 0

( ) ) )

( ) ( ) .i si s i i si s i i si s

i i si s

MTE X x, U u E( X x, U u E(β X x, U u

x E U U U u

18

The MTE is the average willingness to pay (WTP) for lnY1i (compared to lnY0i

) given characteristics Xi and unobserved heterogeneity Usi.

MTE can be estimated from the following relationship, where LIV can be

estimated by semi parametric methods for derivatives (Heckman, 2001):

ppx, PXYE

px, PXLIVpPUxXMTE iiiiiisii

)(ln)() ,(

19

All the other treatment variables can be unified using MTE:

10 )( ss duuMTEATE

10 )()( ssTTs duuhuMTETT

10 )()( ssTUTs duuhuMTETUT

20

Where the weights are:

)(

)(

)()(1

)(1

i

u

i

sPsTT PE

dppf

PEuF

uh s

)1()(

)1()(

)( 0

i

u

i

sPsTUT PE

dppfPE

uFuh

s

21

Treatment on the untreated (TUT) is the effect of treatment on those

who do not receive it (i.e. do not go to college) compared with what

they would experience with the treatment (i.e. go to college)

1 0

( , 0) ( , 0)

( 0).i i i i i i

i i i

TUT E X S E X S

E U U S

22

4. Data Set and Empirical Results

Data Source: China Urban Household Income and Expenditure Survey

(CUHIES) 2000

Conducted by the Urban Socio-Economic Survey Organization of the National

Bureau of Statistics.

Six provinces:

Guangdong Liaoning

Sichuan Shaanxi

Zhejiang Beijing.

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Sample size: 4250 households.

For each household, there is rich information on all household

members, including head, spouse, children and parents.

Age, sex, education level, employment status and enterprise

ownership, occupation, years of work experience and total annual

income are available for each household member.

There are seven education levels in the sample: university, college,

special technical school, senior high school, junior high school, primary

school, and other.

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The used sample consists of 587 individuals, including 273 people with

four-year college (or university) certificates and 314 people with only

senior high school certificates.

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Table 2. Summary Statistics

VariableAll (n=587) Treated (n=273) Untreated (n=314)

Mean Std. Err Mean Std. Err Mean Std. Err

Log Wage 8.86 0.86 9.12 0.77 8.64 0.88

Age 26.25 4.72 26.48 4.14 26.06 5.16

Years of work experience 6.41 4.92 5.83 4.47 6.91 5.23

4-Year college attendance 0.47 0.50 1 0 0 0

Male 0.56 0.50 0.54 0.50 0.59 0.49

Lived in Guangdong Province (GD) 0.18 0.39 0.19 0.39 0.18 0.38

Lived in Liaoning Province (LN) 0.28 0.45 0.30 0.46 0.27 0.44

Lived in Shaanxi Province (SX) 0.10 0.30 0.08 0.27 0.12 0.33

Lived in Sichuan Province (SC) 0.16 0.37 0.15 0.36 0.17 0.38

Lived in Beijing (BJ) 0.15 0.36 0.15 0.36 0.14 0.35

Lived in Zhejiang Province (ZJ) 0.12 0.33 0.12 0.33 0.12 0.33

Worked in state owned enterprises (SOEs) 0.62 0.49 0.72 0.45 0.54 0.50

Worked in collective-owned firms 0.08 0.27 0.04 0.20 0.11 0.32

Worked in joint-venture or foreign owned firms 0.18 0.39 0.19 0.40 0.17 0.38

Worked in private owned firms 0.12 0.32 0.05 0.21 0.18 0.38

Worked in IND_CON sector* 0.26 0.44 0.21 0.40 0.32 0.47

Worked in TRA_COM sector* 0.03 0.17 0.03 0.17 0.03 0.18

Worked in HOU_RES sector* 0.08 0.27 0.07 0.26 0.09 0.29

Worked in SPO_SOC sector* 0.22 0.41 0.16 0.36 0.27 0.45

Worked in CUL_SCI sector* 0.10 0.29 0.14 0.34 0.06 0.24

Worked in FIN_INS sector* 0.11 0.32 0.09 0.28 0.13 0.34

Worked in GOVERN sector* 0.03 0.16 0.04 0.20 0.02 0.13

Worked in OTHER sector* 0.17 0.38 0.27 0.45 0.08 0.28

Years of father’s education 11.36 3.38 12.26 3.26 10.57 3.28

Years of mother’s education 9.90 2.99 10.41 3.31 9.46 2.60

Parental income (in 1000 yuan) 21.39 16.59 24.36 15.89 18.81 16.78

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VariableOLS IV

Coefficient Standard Error Coefficient Standard Error

Intercept 8.3189 0.1493 8.3040 0.1552

4-Year’s college attendance 0.2929 0.0630 0.5609 0.1695

Years of work experience 0.0380 0.0194 0.0196 0.0202

Experience squared -0.0016 0.0010 -0.0007 0.0010

Parental income in 1000 yuan 0.0117 0.0020 0.0098 0.0023

Male 0.1537 0.0602 0.1439 0.0607

Lived in Guangdong Province 0.7543 0.1255 0.7908 0.1267

Lived in Liaoning Province 0.2693 0.1085 0.3142 0.1092

Lived in Sichuan Province 0.2278 0.1181 0.2759 0.1192

Lived in Beijing 0.7246 0.1241 0.7775 0.1256

Lived in Zhejiang Province 0.6241 0.1297 0.6739 0.1314

Worked in state owned enterprises -0.3679 0.0855 -0.3873 0.0868

Worked in collective-owned firms -0.4786 0.1288 -0.5890 0.1298

Worked in private owned firms -0.4649 0.1179 -0.5304 0.1179

Worked in IND_CON sector* -0.2793 0.0788 -0.3048 0.0792

Worked in TRA_COM sector* -0.4512 0.1762 -0.4645 0.1779

Worked in SPO_SOC sector* -0.2880 0.0900 -0.3106 0.0905

Worked in FIN_INS sector* -0.3220 0.1050 -0.3327 0.1061

Table 3. Estimated Mincer Model

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Variable Coefficient Standard Error MeanMarginal Effect

Intercept -4.7370 0.7305 -

Years of father’s education 0.1017 0.0297 0.0211

Years of mother’s education 0.0605 0.0342 0.0126

Parental income in 1000 yuan 0.0190 0.0069 0.0040

Born before 1964 2.0008 0.7969 0.4159

Born in 1964 1.7285 0.9189 0.3593

Born in 1965 3.3423 0.8257 0.6947

Born in 1966 3.1813 0.8552 0.6613

Born in 1967 1.8455 1.1126 0.3836

Born in 1968 2.9030 0.8161 0.6034

Born in 1969 2.2569 0.7941 0.4691

Born in 1970 1.5076 0.7534 0.3134

Born in 1971 3.0771 0.7138 0.6396

Born in 1972 2.6424 0.7183 0.5492

Born in 1973 2.5395 0.6809 0.5279

Born in 1974 2.7740 0.6753 0.5766

Born in 1975 2.7931 0.6763 0.5806

Born in 1976 2.8634 0.6669 0.5952

Born in 1977 2.5890 0.6672 0.5381

Born in 1978 2.5572 0.6656 0.5315

Born in 1979 1.3631 0.7636 0.2833

Table 4. Estimated Logit Model For Schooling

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VariableHigh School College

Std. Err. Std. Err.

Years of work experience 0.0360 0.0225 0.0141 0.0278

Experience squared -0.0013 0.0011 -0.0009 0.0013

Parental income in 1000 yuan 0.0188 0.0038 0.0077 0.0038

Male 0.1365 0.0723 0.1913 0.0777

Lived in Guangdong Province 0.5712 0.1961 0.8853 0.1590

Lived in Liaoning Province 0.1901 0.1263 0.3929 0.1049

Lived in Sichuan Province 0.2612 0.1364 0.2296 0.1081

Lived in Beijing 0.7122 0.1695 0.7971 0.1301

Lived in Zhejiang Province 0.6930 0.1551 0.5461 0.1744

Worked in state owned enterprises -0.3368 0.1188 -0.4471 0.1093

Worked in collective-owned firms -0.6060 0.2065 -0.5868 0.1771

Worked in private owned firms -0.4205 0.1511 -0.6256 0.1677

Worked in IND_CON sector* -0.2297 0.0821 -0.3978 0.0990

Worked in TRA_COM sector* -0.3527 0.1318 -0.5040 0.1557

Worked in SPO_SOC sector* -0.3702 0.1282 -0.3040 0.1202

Worked in FIN_INS sector* -0.3345 0.1560 -0.3543 0.1331

Table 5. Estimated Coefficients from Local Linear RegressionGuassian Kernel, bandwidth = 0.4

29

Parameters Estimation

OLS 0.2929

IV* 0.5609

ATE 0.4336

TT 0.5149

TUT 0.3630

Bias -0.1407

Selection Bias -0.2220

Sorting Gain 0.0813

*Using propensity score as instrument ATEOLSBias

TTOLSBiasSelection

ATETTGainSorting

Table 6. Comparison of Different Parameters

30

Figure 1. Density of P(S=1)Urban areas of six provinces of China

From CUHIES 2000

0%

3%

6%

9%

12%

15%

0.0 0.2 0.4 0.6 0.8 1.0P

f(P)

31

Figure 2. Marginal Treatment EffectIncluding parental income as proxy for ability

in wage equation, all ownership and sectoral dummiesalso included, Bandwidth = 0.4

0.1

0.3

0.5

0.7

0.9

0.0 0.2 0.4 0.6 0.8 1.0

Us

MTE

32

Figure 3. Weights of Treatment ParametersFor Main Specifications

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0

Us

h(Us)

TT

TUT

ATE

33

Figure 4. Marginal Treatment Effect Excluding parental income in wage equation

But all ownership and sectoral dummies included Bandwidth = 0.3

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

Us

MTE

34

Figure 5. Marginal Treatment EffectAll specifications include parental income

in earnings equation, Bandwidth = 0.4

0.1

0.3

0.5

0.7

0.9

0.0 0.2 0.4 0.6 0.8 1.0

Us

MTE

A+B+C A+C

B+C C

A: with firms’ ownership dummies but not sectoral dummies

B: with sectoral dummies but not ownership dummies

C: no sectoral and ownership dummies

35

Figure 6. Marginal Treatment Effect All specifications exclude parental income

in earnings equation, Bandwidth = 0.3

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

Us

MTE

A+B A

B NO A & B

36

5. Concluding Remarks

Neglecting heterogeneity and selection bias leads to biased and inconsistent estimates, such as those obtained using conventional OLS and IV parameters.

We demonstrate the importance of proxying for ability in the wage equation

to identify returns to education. Excluding the proxy leads to implausibly high

estimates of the return to schooling.

37

In 2000 the average return to four-year college attendance is 43% (on

average, 11% annually) for young people in the urban areas of the six

provinces.

The results imply that, after more than twenty years of economic reform with

market orientation, the average return to education in China has increased

markedly compared with that of the 1980s and early 1990s.