Modelling multiple sample selection in intergenerational occupational mobility
Cheti NicolettiISER, University of Essex
Marco FrancesconiDepartment of Economics, University of Essex
Main aims of the paper
1. Estimation of intergenerational occupational mobility in Britain.
2. Correcting for potential sample selection problems in short panels using different estimation methods.
iii iAxy '
Sample selection problems
• Labour market selection: Intergenerational occupational mobility can be estimated only for people who are employed.
• Coresidence selection: Children must be living together with their parents in at least one wave of the panel.
1991 2003
BHPS 1991-1993
Child age
3
8
13
18
23
28
33
Child age
15
20
25
30
35
44
45
1997
Child age
9
14
19
24
29
34
39
Cohort
1988
1983
1978
1973
1968
1963
1958
Taking account of coresidence selection
Francesconi and Nicoletti (2006) find that the intergenerational mobility in occupational prestige is underestimated when using the subsample of sons born between 1966 and 1985.
They try different estimation methods to correct for sample selection and find that only the inverse propensity score is able to attenuate the selection problem
This sample selection evaluation is possible because all BHPS respondents are asked to report occupational characteristics of their parents when they were 14
Taking account of selection into employment for children
• Blanden (2005) and Ermisch et al (2005) consider two-step estimation procedures and find lower and unchanged βs
• Couch and Lillard (1998) and Nicoletti and Francesconi (2006) consider imputation methods and find lower βs
• Minicozzi (2003) use partial identification approach to produce bound estimates instead than point estimates for the intergenerational mobility and find higher βs when including unemployed and part-time workers.
Contributions of the paper
Propose new estimation methods to take account of sample selection problem in the intergenerational mobility models which are very parsimonious
Taking account of both coresidence and employment selection bias
Selection models
If εi and ui are not independent then we have selection due to unobservables
If εi depends on Zi then we have selection due to observables
If εi depends on Zi and ui then we have selection due to both observables and unobservables
iii xy
iuZdii
'*)0( *
iidId where
Selection due to unobservables
y=α+xβ+ε d*=Z γ+u d=l(d*>0)
Let E(ε|x)=0, ε ind Z, (ε, u) be N with means zeros, variances σ2 and 1 and covariance ρ
Then E((y-α-xβ) |x,d=1) ≠ 0 and OLSE is biased E(y|x,d=1)=α+xβ+E(ε|x,d=1)=α+xβ+ ρλ v=ε- ρλ is such that E(v|d=1,X)=0 We can consider an additional correction term
(Heckman 1979, Vella 1998)
Selection due to observables
y=α+xβ+ε d*=Z γ+u d=l(d*>0)
Let E(ε|x)=0, ε ╨ u but ε not ind Z Then E(ε|x,d=1)≠0 and OLSE is biased
because of selection on observables Since ε ╨ d|x,Z we can adopt (1) propensity score
methods, (2) regression adjustment methods or (3) combining methods. (see Rosembaum and Rubin, 1983; Robins and Rotnitzky, 1995; Hirano et al., 2003)
Propensity score weighting method
Let Pr(d=1|x,Z)=Pr(d=1|Z)=p(Z)
Then E(ε d|x) ≠ 0 but E(ε d p(Z)-1|x)=0E(ε d p(Z)-1|x)= EZE(ε d p(Z)-1|x,Z)
= EZ[E(ε |x,Z,d=1) Pr(d=1|x,Z)p(Z)-1]
Since ε ╨ d|x,Z
= EZ[E(ε |x,Z) Pr(d=1|x,Z)p(Z)-1]
=EZ[E(ε |x,Z)]=E(ε |x)=0
This holds even if some of the variables in Z are erroneously omitted from the main equation.
Regression adjustmenty=α+xβ+ε d*=Z γ+u d=l(d*>0)
• To take account that ε is not ind of Z
y=αN+xβN+Zδ+ω• If the linearity assumption is satisfied then
E(ω|X,Z,d=1)=E(ω|X,Z)=E(ω|X)=0 and• βN is consistently estimated• β=Cov(x,y)/Var(x)=βN+Cov(x,Z)Var(Z)-1δ
Combining regression adjustment and propensity score method
Estimation of the extended model y=α+xβ+Zδ+ωby using inverse propensity score weighting
E[(y-α-xβ-Zδ) d p(Z)-1|x]= EZE[(y-α-xβ- Zδ) d p(Z)-1|x,Z]= EZ[E(y-α-xβ- Zδ |x,Z,d=1) Pr(d=1|x,Z)p(Z)-1]
Notice that this expression is 0if either E(y-α-xβ- Zδ |x,Z,d=1)=E(ω|X,Z,d=1)=0 or Pr(d=1|x,Z)=p(Z) holds and not necessarilyboth.
Selection due to both observables and unobservables
y=α+xβ+ε d*=Z γ+u d=l(d*>0)
where ε depends on both Z and u
(ε, u) is N with means zeros, variances σ2 and 1 and
covariance ρ
v=(ε- ρλ) ind d |x,Z
We can use: (1) Heckman correction and propensity
score weighting or (2) Heckman correction and regression adjustment.
Heckman correction & propensity score weighting
E[(y-α-xβ- ρλ) d p(Z)-1|x]
= EZE[(y-α-xβ- ρλ) d p(Z)-1|x,Z]
= EZ[E(y-α-xβ- ρλ|x,Z,d=1) Pr(d=1|x,Z)p(Z)-1]
Since (y-α-xβ- ρλ) ╨ d|x,Z
= EZ[E(y-α-xβ- ρλ |x,Z) Pr(d=1|x,Z)p(Z)-1]
=EZ[E(y-α-xβ- ρλ |x,Z)]= E(y-α-xβ- ρλ |x)= 0
Heckman correction & regression adjustment
Estimation of the extended model with
additional variables Z and correction term λ
y=α+xβ+Zδ+ ρλ +ω
d*=Z γ+u
• ρλ controls for the dependence of ε1 on u
• Zδ controls for the dependence of ε2 on Z
How can the BHPS help us?All BHPS respondents are asked to report occupational
characteristics of their parents when they were 14THEREFORE
• We know the occupational prestige even for daughters and fathers living apart during the panel.
• We can estimate the intergenerational mobility without any coresidence selection.
• We can consider the subsample of daughters coresident with the fathers at least once during the panel and assess the relevance of the coresidence selection.
• We can then compare different methods to correct for the coresidence selection.
BHPS Samples
• FULL SAMPLE: 2691 women (daughters) born between 1966 and 1985 with at least one valid interview over the first 13 waves of the BHPS (aged between 16-37, average 24)
• RESTRICTED SAMPLE: 745 individuals from the full sample who can be matched with their father (aged between 16-37, average age 21).
• We consider an average occupational prestige over all waves available for daughters. We consider instead the occupation prestige reported retrospectively by daughters for fathers (average age 46).
Estimation Methods Used
• Inverse propensity score weighting (Weights)
• Regression adjustment
• Regression adjustment & weights
• Heckman correction method (Heckman)
• Heckman & weights
• Regression adjustment & Heckman
Coresidence selection model
y=α+xβ+Aμ+ε d*=Z γ+u d=l(d*>0)where y is the daughter’s occupational prestige (log Hope-
Goldthorpe score) x is her father’s occupational prestige A age and age2
d=1 for daughters living together with their father in at least one wave and 0 otherwise
Z=dummies for education, age, regions, ethnicity, religiosity and two house price indexes
The intergenerational equation is too parsimonious
y=α+xβ+Aμ+ε d*=Z γ+u d=l(d*>0)
Education dummies are important to explaining both the daughters occupational prestige and their probability to be coresident
The assumption that ε ╨ d is not acceptable.
Regression adjustment when x is missing
y=αN+xβN+Zδ+ω
• If the linearity assumption is satisfied
• βN is consistently estimated
• β=Cov(x,y)/Var(x)=βN+Cov(x,Z)Var(Z)-1δ
• If x is missing it is not possible to estimate Cov(x,Z) consistently
Correcting for coresidence selection only
β SE
Full sample 0.250 0.028
Restricted sample 0.147 0.044
Weights 0.208 0.084
Heckman 0.145 0.043
Heckman and weights 0.206 0.083
Regression adjustment 0.135 0.043
Regression adjustment & Heckman 0.132 0.043
Regression adjustment & weights 0.206 0.063
Employment selection model
y=α+xβ+Aμ+ε d*=Z γ+u d=l(d*>0)where y is the daughter’s occupational prestige (log Hope-
Goldthorpe score) x is her father’s occupational prestige A age and age2
d=1 for daughters are employed at least in at least one wave and 0 otherwise
Z=occupation prestige father, dummies for education, age, regions, ethnicity, religiosity, a house price index, marital status and number of children aged between 0-2, 3-4, 5-11, 12-15, 16-18.
Correcting for employment selection only
β SE
Full sample 0.250 0.028
Weights 0.265 0.031
Heckman 0.209 0.041
Heckman and weights 0.227 0.032
Regression adjustment 0.249 0.028
Regression adjustment & Heckman 0.255 0.029
Regression adjustment & weights 0.253 0.030
Correcting for employment and sample selection simultaneously
β SE
Full sample 0.250 0.028
Restricted sample 0.147 0.044
Weights Bivariate selection 0.208 0.084
Regression adj & weights Bivariate selection 0.145 0.043
Weights 0.206 0.083
Regression adjustment & weights 0.135 0.043
Heckman 0.132 0.043
Heckman & Regression adjustment 0.132 0.043
Heckman & weights 0.206 0.063
Matching selection in quantile regressions
Quantile Full Restricted Weights
10 0.386 0.125 0.384
0.069 0.100 0.133
25 0.257 0.219 0.281
0.052 0.077 0.131
50 0.248 0.164 0.215
0.063 0.071 0.109
75 0.240 0.109 0.138
0.045 0.054 0.096
90 0.079 0.059 0.141
0.033 0.070 0.067
Employment selection in quantile regressions
Quantile Full Weights
10 0.386 0.311
0.069 0.070
25 0.257 0.266
0.052 0.046
50 0.248 0.292
0.063 0.049
75 0.240 0.279
0.045 0.041
90 0.079 0.195
0.033 0.040
Double selection in quantile regressions
Quantile Full Restricted Weights
10 0.386 0.125 0.389
0.069 0.100 0.132
25 0.257 0.219 0.389
0.052 0.077 0.136
50 0.248 0.164 0.244
0.063 0.071 0.146
75 0.240 0.109 0.109
0.045 0.054 0.131
90 0.079 0.059 0.008
0.033 0.070 0.112
Conclusions
• The intergenerational equation is too parsimonious and there are probably omitted variables such as education dummies.
• In this situation correcting for selection on observables is much more important than correcting for selection on unobservables.
• The coresidence selection seems to cause an underestimation of β.
• The selection into employment does not seem to cause a large bias in β.