Consistent with existing literature (Heckman, 1974), let the desired hours in the crosssection of females be given by:
(1) h_{i}^{*} = δ_{0} + δ_{1}w_{i} + δ_{2}Z_{i} + ε_{i}
where Z includes nonlabor income and taste variables such as age and its square, education dummies, female is living with spouse, husband’s earnings, marital status, alimony dummy variable, and a dummy variable of the individual female living with a child less than six years old. We can think of ε_{i} as unobserved “tastes for work” (an unobserved, personspecific factor that makes female i work more or fewer hours than other observationallyidentical females). We will refer to (1) as the structural labor supply equation. It represents the behavioral response of the individual female’s labor supply decision to her economic environment and our goal here is to estimate δ_{1}.
Suppose the market wage that female i can command is given by:
(2) w_{i} = β_{0} + β_{1}X_{i} + µ_{i}
where X includes productivity and human capital variables such as age and its square, years of experience and it’s square, education and region dummies. In practice there may be considerable overlap between the variables in X and Z. It may be helpful to think of µ_{i} as “unobserved (wageearning) ability” here. We will refer to (2) as the structural wage equation. The wage equation includes dummy variables that distinguish between different regions of residence. Since there is no theoretical reason justifying the inclusion of region dummies, they are excluded from the labor supply equation.
In the above situation we already know that OLS estimates of either (1) or (2) on the sample of female workers only will be biased (in the case of (1) because the sample includes only those females with positive hours; in the case of (2) because the sample includes only those females with wages above their reservation wage). So we formalize the nature and size of these biases, and obtain unbiased estimates of the δ’s and β’s as shown below.
We begin by substituting (2) into (1), which yields:
(3) h_{i}^{*} = δ_{0} + δ_{1}[β_{0} + β_{1}X_{i} + µ_{i}] + δ_{2}Z_{i} + ε_{i}
(4) h_{i}^{*} = [δ_{0} + δ_{1}β_{0}] + δ_{1}β_{1}X_{i} + δ_{2}Z_{i} + [ε_{i} + δ_{1}µ_{i}]
(5) h_{i}^{*} = α_{0} + α_{1}X_{i} + α_{2}Z_{i} + η_{i}
where α_{0} = δ_{0} + δ_{1}β_{0}; α_{1} = δ_{1}β_{1}; α_{2} = δ_{2}; η_{i} = ε_{i} + δ_{1}µ_{i}. We will refer to equation (5) as the reduced form hours equation.
As a final step in setting up the problem, note that given our assumptions female i will work a positive number of hours if and only if (iff):
(6) h_{i}^{*} > 0; i.e. η_{i} >  α_{0}  α_{1}X_{i}  α_{2}Z_{i}
Note that conditional on observables (X and Z) either high unobserved tastes for work (ε_{i}) or (provided δ_{1} > 0) high unobserved wageearning ability (µ_{i}) tend to put all women into the sample of working women.
Next, to greatly simplify matters, we assume that the underlying error terms (ε_{i} and µ_{i}) follow a joint normal distribution. Note that (a) it therefore follows that the “composite” error term η_{i} is distributed as a joint normal with ε_{i} and µ_{i}; and (b) we have not assumed that ε_{i} and µ_{i} are independent. In fact, it seems plausible that work decisions and wages could have a common unobserved component. Indeed, one probably would not have much confidence in an estimation strategy that required them to be independent.
Recalling that an observation is in the sample iff equation (6) is satisfied for that observation we get:
(7) E(ε_{i}h_{i} > 0) = E(ε_{i} η_{i} >  α_{0}  α_{1}X_{i}  α_{2}Z_{i})
(8) ≡ θ_{1}λ_{i}
where in equation (8), the first term, θ_{1 }is a parameter that does not vary across observations. It is the coefficient from a regression of η_{i} on ε_{i}; therefore of ε_{i} + δ_{1}µ_{i} on ε_{i}. Unless δ_{1} (the true labor supply elasticity) is zero or negative, or there is a strong negative correlation between underlying tastes for work, ε_{i} and wageearning ability, µ_{i}, this will be positive. In words, conditioning on observables, women who are more likely to make it into the sample – i.e. have a high η_{i} – will on average have a higher residual in the labor supply equation, ε_{i}).
The second term in (8), λ_{i}, has an i subscript and therefore varies across observations. Mathematically, it is the ratio of the normal density to one minus the normal cdf (both evaluated at the same point, which in turn depends on X and Z). This ratio is sometimes called the inverse Mills ratio. For the normal distribution, this ratio gives the mean property: If x is a standard normal variate, E(xx > a) = φ(a)/(1 Φ(a)).
Now that we have an expression for the expectation of the error term in the structural labor supply equation (1) we can write:
(9) ε_{i} = E(ε_{i}h_{i} > 0) + ε_{i}^{*} = θ_{1}λ_{i}, where E(ε_{i}^{*}) = 0.
In a sample of participants, we can therefore write (1) as:
(10) h_{i}^{*} = δ_{0} + δ_{1}w_{i} + δ_{2}Z_{i} + θ_{1}λ_{i} + ε_{i}^{*}
We call this the augmented labor supply equation. It demonstrates that we can decompose the error term in a selected sample into a part that potentially depends on the values of the regressors (X and Z) and a part that does not. It also tells us that, if we had data on λ_{i} and included it in the above regression, we could estimate (1) by OLS and not encounter any bias. Thus, one can think of sample selection bias as a specific type of omitted variable bias [Heckman (1979)].
Following the same reasoning for the market wage equation we get:
(11) E(µ_{i}h_{i} > 0) = E(µ_{i} η_{i} >  α_{0}  α_{1}X_{i}  α_{2}Z_{i})
(12) ≡ θ_{2}λ_{i}
Note that λ_{i} in (12) is exactly the same λ_{i} that appeared in (8). The parameter θ_{2} is the supply coefficient from a regression of η_{i} on ε_{i}; therefore of ε_{i} + δ_{1}µ_{i} on µ_{i}. As before, unless δ_{1} (the true labor supply elasticity) is zero or negative, or there is a strong negative correlation between ε_{i} and µ_{i}, this will be positive (on average, conditioning on observables, women who are more likely to make it into the sample – i.e. have a high η_{i} – will have a higher residual in the wage equation, µ_{i}).
Equation (12) allows us to write an augmented wage equation:
(13) w_{i} = β_{0} + β_{1}X_{i} + θ_{2}λ_{i} + µ_{i}^{*}, where E(µ_{i}^{*}) = 0.
Thus, data on λ_{i} would allow us to eliminate the bias in wage equations fitted to the sample of working women only.
When (as we have assumed) all our error terms follow a joint normal distribution, the reduced form hours equation (5) defines a probit equation where the dependent variable is the dichotomous decision of whether to work or not (i.e. whether to be in the sample for which we can estimate our wage and hours equations). Note that all the variables in this probit (the X’s, Z’s and whether a female works) are observed for both female workers and female nonworkers. Thus we can estimate the parameters of this equation consistently. In particular (recalling that the variance term in a probit model is not identified) we can get consistent estimates of α_{0}/σ_{η}, α_{1}/σ_{η }and_{ }α_{2}/σ_{η}. Combined with the data on the X’s and Z’s, these estimates allow us to calculate an estimated λ_{i} for each observation in our data.
Now that we have consistent estimates of λ_{i}, we can include them as regressors in a labor supply equation estimated on the sample of participants only. Once we do so, the expectation of the error term in that equation is identically zero, so it can be estimated consistently via OLS. We can do the same thing in the wage equation. This procedure is known as the Heckit method. When we implement this, we will as a matter of fact get estimates of the θ parameters (θ_{1} in the case where the second stage is an hours equation; θ_{2} in the case where the first stage is a wage equation). These in turn provide some information about the covariance between the underlying error terms ε_{i} and µ_{i}.
In general, this technique is used whenever we are running a regression on a sample where there is a possible (or likely) correlation between the realization of the dependent variable and the likelihood of being in the sample. In principle, one can correct for sample selection bias by (i) estimating a reducedform probit in a larger data set where the dependent variable is included in the subsample of interest; then (ii) estimating the regression in the selected sample with an extra regressor, λ_{i}. According to the reasoning above, including this extra regressor should eliminate any inconsistency due to nonrandom selection in our sample.Continued on Next Page »
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Endnotes
1.) The author would like to thank Professor Craig Brett for his invaluable suggestions and comments.
2.) Carliner et al. (1980) in their analysis of 1971 Canadian census data employ three measures of labor supply: labor force participation, hours per week, and weeks per year. Using education as a proxy for potential market wages they found that “greater education of the wife is associated with significantly increased labor supply for all three measures. This suggests that the … substitution effects of an increase in w_{f} [the wife’s wage] … outweigh the income effect.”
3.) The emphasis of the three papers is quite different. Nakamura, Nakamura, and Cullen (1979) report estimates for Canadian women using the 1971 Canadian census. Nakamura and Nakamura (1981) analyze both Canadian and U.S. census data emphasizing the role of taxes. Nakamura and Nakamura (1983) using these same data sets, distinguish further between fulltime and parttime workers.
4.) Robinson and Tomes (1985) used data from 1979 Quality of Life Survey, which is a survey conducted by the Institute for Behavioural Research, York University, to deal against the problems of using census data for their study. The survey contained a direct measure of the hourly wage rate and also presented hours of work directly rather than in intervals for a subset of Canadian women.
5.) Source: http://highered.mcgrawhill.com/sites/dl/free/0070891540/43156/benjamin5_sample_chap02.pdf.
6.) Source: See http://highered.mcgrawhill.com/sites/dl/free/0070891540/43156/benjamin5_sample_chap02.pdf for the original table.
7.) Standard hours are usually determined by collective agreements or company policies, and they are the hours beyond which overtime rates are paid. The data apply to nonoffice worker.
8.) Standard hours minus the average hours per week spent on holidays and vacations.
9.) This supply curve shows how the change in real wage rate affects the amount of hours worked by employees. Source: http://en.wikipedia.org/wiki/Backward_bending_supply_curve_of_labor. See the appendix section.
10.) Although the Heckman sample selection model is written in terms of hours of work H, the same equations
apply equally as well to the wage W.
11.) All the steps of the Heckit method is borrowed from lecture notes: CrossSection Regression Estimates of labor Supply Elasticities: Procedures and Problems.
12.) See http://www23.statcan.gc.ca/imdb/p2SV.pl?Function=getSurvey&SDDS=3889&lang=en&db=imdb&adm=8&dis=2 for more details on the Survey of labor and Income Dynamics (SLID).
13.) Census data is not used as the limitations of Census data in labor economics is well documented [Killingsworth (1983); Angrist and Krueger (1999)]. Income variables are based on respondents’ memory and willingness to disclose this information that is mostly underreported in the Census.
14.) To check for educational assortative mating, the husband’s education variable was added to the actual data that contains only females. After running a single regression of husband’s education on female’s education, a positive correlation for each level of education was found. Hence, husband’s education was added to the model to see how it affects the results. However, it must be noted that adding husband’s education to the model did not change the Heckit results that much. Most importantly, since adding husband's education to the model still results in a positive coefficient of nonfemale income in the Heckit, the sorting is not on education even though there is a positive correlation among husband's and wife's education. Therefore, the Heckit results with the inclusion of husband’s education to the model are not reported in this paper. Moreover, the existing literature of labor supply of women doesn't include this kind of variable.
15.) It has been mentioned by Adkins and Hill (2004) that “Donald (1995) has studied this problem and suggested a semiparametric estimator that is consistent in heteroscedastic selectivity models. Chen & Khan (2003) has also proposed a semiparametric estimator of this model. More recently, Lewbel (2003) has proposed an alternative that is both easy to implement and robust to heteroskedastic misspecification of unknown form.” The authors themselves proposed a “simple estimator that is easily computed using standard regression software,” and studied the performance of the estimator in a small set of Monte Carlo simulations.
Appendix
Table 1.2: Variable Descriptions
hours

total hours paid all jobs during 2009

wage

composite hourly wage all paid jobs in 2009

wagesqrd

the square of composite hourly wage all paid jobs

age

female's age, 2009, external crosssec file

agesqrd

the square of female's age

marst

marital status of female as of December 31 of 2009
1 – female is married
2 – female is in a commonlaw relationship
3 – female is separated
4 – female is divorced
5 – female is widowed
6 – female is single (never married)

fslsp

female is living with spouse in 2009
1 – Yes
2  No

province

Province of residence group, household, December 31, 2009
10  Newfoundland and Labrador
11 – Prince Edward Island
12 – Nova Scotia
13 – New Brunswick
24 – Quebec
35 – Ontario
46 – Manitoba
47 – Saskatchewan
48 – Alberta
59 – British Columbia

exper

number of years of work experience, fullyear fulltime

expersqrd

the square of number of years of work experience, fullyear fulltime

alimo

Support payments received

educ

Highest level of education of female, 1st grouping
1  Never attended school
2  14 years of elementary school
3  58 years of elementary school
4  910 years of elementary and
secondary school
5  1113 years of elementary and
secondary school (but did not
graduate)
6  Graduated high school
7  Some nonuniversity postsecondary (no certificate)
8  Some university (no certificate)
9  Nonuniversity postsecondary
certificate
10  University certificate below
Bachelor's
11  Bachelor's degree
12  University certificate above
Bachelor's, Master's, First
professional degree in law, Degree
in medicine, dentistry, veterinary
medicine or optometry, Doctorate
(PhD)

nonfemaleincome

income of nonfemale in the household

kidslt6

female with a child less than six years old

working

total hours paid all jobs greater than zero






Table 1.3: Summary Statistics of Canadian women
Variable

Observations

Mean

Standard Deviation

Minimum

Maximum

puchid25(id)

32065

4012858

7414.513

4000001

4025693

province

31819

33.74845

14.69714

10

59

agyfm

32065

38.72475

25.07988

0

80

agyfmg46

32065

5.924965

2.56457

1

9







alimo46

32065

263.0711

1860.065

0

45000

earng46

31745

51132.91

63660.3

0

1387250

age

17042

43.26998

10.50669

24

60

marst

28264

2.8629

2.118468

1

6

fslac

28325

1.907326

.2899806

1

2







fslsp

28325

1.406884

.4912616

1

2

hours

24009

1129.835

922.4242

0

5200

wage

16371

19.89017

11.81493

6

142

exper

24864

14.9928

13.18434

0

50







alimo

28325

249.0071

1825.297

0

45000

earng42

28108

20899.72

28372.56

0

539000

mtinc42

28179

25065.66

30446.84

0

680000

oas42

28325

1210.796

2430.963

0

7750

ogovtr42

28325

33.60018

181.1052

0

2400







ottxm42

28325

561.278

4202.446

0

120000

prpen42

28325

2120.96

7977.688

0

185000

sapis42

28325

406.2242

2022.65

0

25000

uccb42

28325

139.9682

495.2109

0

7800

uiben42

28325

757.8279

2789.844

0

31000







wgsal42

28325

19643.28

27591.65

0

525000

wkrcp42

28325

130.5137

1279.867

0

32000

educ

28204

7.580946

2.599754

1

12

totalfemincome

28179

50174.48

56331.64

0

1110900

nonfemincome

28067

29301.02

32393.06

0

680000







wagesqrd

16371

535.2028

918.9231

36

20164

agesqrd

28325

2642.723

1801.337

256

6400

expersqrd

24864

398.6038

528.935

0

2500







kidslt6

32065

.0902542

.28655

0

1

working

32065

.8051458

.3960946

0

1

Table 1.4: Marital Status of Canadian women
Marital Status

Frequency

Percent

Cumulative

1 – female is married

13,841

48.97

48.97

2 – female is in a commonlaw relationship

2,485

8.79

57.76

3 – female is separated

982

3.47

61.24

4 – female is divorced

1,900

6.72

67.96

5 – female is widowed

2,776

9.82

77.78

6 – female is single (never married)

6,280

22.22

100.00

Total

28,264

100.00


Table 1.5: Canadian women living with spouse or not
Living with spouse or not

Frequency

Percent

Cumulative

1  Yes

16,800

59.31

59.31

2  No

11,525

40.69

100.00

Total

28,325

100.00


Table 1.6: Residence of Canadian women
Province

Frequency

Percent

Cumulative

10  Newfoundland and Labrador

1,390

4.37

4.37

11 – Prince Edward Island

870

2.73

7.10

12 – Nova Scotia

1,877

5.90

13.00

13 – New Brunswick

1,849

5.81

18.81

24 – Quebec

6,136

19.28

38.10

35 – Ontario

8,976

28.21

66.31

46 – Manitoba

2,124

6.68

72.98

47 – Saskatchewan

2,304

7.24

80.22

48 – Alberta

3,172

9.97

90.19

59 – British Columbia

3,121

9.81

100.00

Total

31,819

100.00


Table 1.7: Highest level of education attained by Canadian women
Highest level of education

Frequency

Percent

Cumulative

1  Never attended school

111

0.39

0.39

2  14 years of elementary school

227

0.80

1.20

3  58 years of elementary school

2,025

7.18

8.38

4  910 years of elementary and
secondary school

2,037

7.22

15.60

5  1113 years of elementary and
secondary school (but did not
graduate)

1,869

6.63

22.23

6  Graduated high school

4,449

15.77

38.00

7 Some nonuniversity postsecondary (no certificate)

2,037

7.22

45.22

8  Some university (no certificate)

1,584

5.62

50.84

9  Nonuniversity postsecondary
certificate

8,548

30.31

81.15

10  University certificate below
Bachelor's

617

2.19

83.34

11  Bachelor's degree

3,447

12.22

95.56

12  University certificate above
Bachelor's, Master's, First
professional degree in law, Degree
in medicine, dentistry, veterinary
medicine or optometry, Doctorate
(PhD)

1,253

4.44

100.00

Total

28,204

100.00


Table 1.8: Canadian women with or without a child less than six years old
Child less than six years old or not

Frequency

Percent

Cumulative

1  Yes

29,171

90.97

90.97

2  No

2,894

9.03

100.00

Total

32,065

100.00


Table 2.9: OLS Estimates for Canadian Women
Dependent Variable: hours of work

Independent Variables

Coefficient

composite hourly wage of all paid jobs

1.42
[2.70]

the square of composite hourly wage of all paid jobs

.065**
[.0327]

female's age

39.23***
[5.01]

the square of female's age

.49***
[.06]

1  Never attended school (base group)



2  14 years of elementary school

104.7
[201.5]

3  58 years of elementary school

65.4
[115.8]

4  910 years of elementary and
secondary school

90.5
[110.1]

5  1113 years of elementary and
secondary school (but did not
graduate)

21.25
[111]

6  Graduated high school

117.3
[105.5]

7  Some nonuniversity postsecondary (no certificate)

4.36
[106.9]

8  Some university (no certificate)

14.6
[107.8]

9  Nonuniversity postsecondary
certificate

85.8
[105.2]

10  University certificate below
Bachelor's

61.8
[109.1]

11  Bachelor's degree

48
[106.2]

12  University certificate above
Bachelor's, Master's, First
professional degree in law, Degree
in medicine, dentistry, veterinary
medicine or optometry, Doctorate
(PhD)

51.84
[107.8]

female is living with spouse (base group)



female is not living with spouse

77.14 ***
[28]

income of nonfemale in the household

.008***
[.0007]

1 – female is married (base group)



2 – female is in a commonlaw relationship

29.3
[15.98]

3 – female is separated

18.8
[35.14]

4 – female is divorced

20.65
[33.11]

5 – female is widowed

78.7
[54.71]

6 – female is single (never married)

23.2
[29.9]

Support payments received

.013***
[.003]

female without a child less than six years old (base group)



female with a child less than six years old

193.2***
[17.73]

constant

606.7
[149.9]

Sample size

12469

Rsquared

0.143

* Statistical significance at the 90% level
** Statistical significance at the 95% level
*** Statistical significance at the 99% level
[ ] Heteroskedasticityrobust standard error
Table 2.0: Probit Estimates for Canadian women
Independent Variables

Coefficient

∆P(working) per unit ∆independent variable

female's age

.041***
(.012)

.0164

the square of female's age

.001***
(.0001)

.0004

number of years of work experience, fullyear fulltime

.09***
(.004)

.036

the square of number of years of work experience, fullyear fulltime

.001***
(.0001)

.0004

1  Never attended school (base group)





2  14 years of elementary school

.61
(.441)

.244

3  58 years of elementary school

.87**
(.35)

.348

4  910 years of elementary and
secondary school

1.11***
(.351)

.444

5  1113 years of elementary and
secondary school (but did not
graduate)

1.16***
(.354)

.464

6  Graduated high school

1.36***
(.35)

.544

7  Some nonuniversity postsecondary (no certificate)

1.25***
(.35)

.5

8  Some university (no certificate)

1.34***
(.35)

.536

9  Nonuniversity postsecondary
certificate

1.56***
(.35)

.624

10  University certificate below
Bachelor's

1.64***
(.36)

.656

11  Bachelor's degree

1.8***
(.35)

.72

12  University certificate above
Bachelor's, Master's, First
professional degree in law, Degree
in medicine, dentistry, veterinary
medicine or optometry, Doctorate
(PhD)

1.96***
(.353)

.784

10  Newfoundland and Labrador (base group)





11 – Prince Edward Island

.301***
(.108)

.1204

12 – Nova Scotia

.1
(.082)

.04

13 – New Brunswick

.001
(.083)

.0004

24 – Quebec

.08
(.07)

.032

35 – Ontario

.146
(.07)

.0584

46 – Manitoba

.044
(.081)

.0176

47 – Saskatchewan

.0454
(.081)

.0182

48 – Alberta

.052
(.076)

.0208

59 – British Columbia

.106
(.076)

.0424

constant

1.22
(.427)



Pseudo Rsquared

0.17



Proportion of women who worked

0.42



Final value of log of likelihood function

5637.7



* Statistical significance at the 90% level
** Statistical significance at the 95% level
*** Statistical significance at the 99% level
( ) Usual standard error
Table 2.1: Heckit Estimates for Canadian Women
Dependent Variable: hours of work

Independent Variables

Coefficient

composite hourly wage of all paid jobs

9.1***
(1.41)

the square of composite hourly wage of all paid jobs

.006
(.015)

female's age

19.8***
(5.13)

the square of female's age

.235***
(.061)

1  Never attended school (base group)



2  14 years of elementary school

84.8
(334.3)

3  58 years of elementary school

18.02
(280)

4  910 years of elementary and
secondary school

50.34
(278.3)

5  1113 years of elementary and
secondary school (but did not
graduate)

148.6
(279.2)

6  Graduated high school

67.43
(277.2)

7  Some nonuniversity postsecondary (no certificate)

188.9
(277.7)

8  Some university (no certificate)

202.9
(278.2)

9  Nonuniversity postsecondary
certificate

125.9
(277.1)

10  University certificate below
Bachelor's

183.8
(279.1)

11  Bachelor's degree

171.1
(277.4)

12  University certificate above
Bachelor's, Master's, First
professional degree in law, Degree
in medicine, dentistry, veterinary
medicine or optometry, Doctorate
(PhD)

174.3
(278.1)

female is living with spouse (base group)



female is not living with spouse

60.9**
(25.6)

income of nonfemale in the household

.01***
(.0003)

1 – female is married (base group)



2 – female is in a commonlaw relationship

19.5
(17)

3 – female is separated

24.6
(34.52)

4 – female is divorced

20.3
(31.3)

5 – female is widowed

73.3
(50.9)

6 – female is single (never married)

7.3
(27.8)

Support payments received

.013***
(.003)

female without a child less than six years old (base group)



female with a child less than six years old

172.2***
(17.3)

constant

1292.3
(299.2)

(Selectivity bias)

314.8
(18.12)

Sample size

13515

* Statistical significance at the 90% level
** Statistical significance at the 95% level
*** Statistical significance at the 99% level
( ) Usual standard error