Transcript Sample Selection Example Bill Evans 1

Sample Selection Example
Bill Evans
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• Draw 10,000 obs at random
• educ uniform over [0,16]
• age uniform over [18,64]
• wearnl=4.49 + 0.08*educ + 0.012*age + ε
• Generate missing data for wearnl
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•
drawn from standard normal [0,1]
• d*=-1.5+0.15*educ+0.01*age+0.15*z+v
• wearnl missing if d*≤0
• wearn reported if d*>0
• wearnl_all=wearnl with non-missing obs.
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• εi and vi are assumed to be bivariate
normal
• E(εi) = E(vi) =0
• Var(εi) = σ2
• Var(vi) = 1
• Corr(εi,vi) = ρ
• Cov(εi,vi) = ρ σ
• In this case, ρ=0.25 and σ=0.46
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• Yi = β0 + β1educi + β2agei + εi
• E[Yi | SSR] = β0 + β1educi + β2agei +
E[εi | SSR]
• E[εi | SSR] = E[εi | vi>-wiγ]
= ρ σ φ(wiγ)/Φ(wiγ)
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• λi = φ(wiγ)/Φ(wiγ)
• wiγ = γ0+educ γ1+age γ2+z γ3
• γ2 and γ3 are both constructed to be
positive
• cov(educ, λi) < 0 and
• cov(age, λi) < 0
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• The omitted variable λi is negatively
correlated with what is observed in the
model
• Therefore, the coefficients on educ and
age in the selected sample will be too low
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Numbe rof non-missing
observations
. * get frequency of missing data;
. tab missing;
weekly |
wages are |
missing |
Freq.
Percent
Cum.
------------+----------------------------------0 |
5,337
53.37
53.37
1 |
4,663
46.63
100.00
------------+----------------------------------Total |
10,000
100.00
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OLS on all data (no missing obs)
Generated by the equation
wearnl=4.49 + 0.08*educ + 0.012*age + ε
. * run ols model with real data;
. reg wearnl_all educ age;
Source |
SS
df
MS
-------------+-----------------------------Model |
1635.2229
2
817.61145
Residual | 2104.81874 9997 .210545037
-------------+-----------------------------Total | 3740.04164 9999 .374041568
Number of obs
F( 2, 9997)
Prob > F
R-squared
Adj R-squared
Root MSE
=
10000
= 3883.31
= 0.0000
= 0.4372
= 0.4371
= .45885
-----------------------------------------------------------------------------wearnl_all |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------educ |
.0803226
.0009926
80.92
0.000
.0783769
.0822684
age |
.0121512
.0003465
35.06
0.000
.0114719
.0128305
_cons |
4.483993
.016937
264.74
0.000
4.450793
4.517193
------------------------------------------------------------------------------
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OLS on reported data
Smaller MSE
. reg wearnl educ age;
Source |
SS
df
MS
-------------+-----------------------------Model | 559.207267
2 279.603633
Residual | 1079.42294 5334 .202366505
-------------+-----------------------------Total |
1638.6302 5336 .307089618
Number of obs
F( 2, 5334)
Prob > F
R-squared
Adj R-squared
Root MSE
=
5337
= 1381.67
= 0.00 00
= 0.3413
= 0.3410
= .44985
-----------------------------------------------------------------------------wearnl |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------educ |
.0703361
.0015006
46.87
0.000
.0673943
.073278
age |
.0118533
.0004631
25.60
0.000
.0109455
.0127611
_cons |
4.6703
.0258484
180.68
0.000
4.619626
4.720973
------------------------------------------------------------------------------
Notice that the estimates for educ and age are now smaller
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Probit, why is data non-missing
Generated by the equation
d*=-1.5+0.15*educ+0.01*age+0.15*z+v
. gen nonmissing=1-missing;
. label var nonmissing "=1 if data for wearnl is reported";
. * run probit, why data is reported;
. probit nonmissing educ age z;
Probit regression
Log likelihood = -5586.4551
Number of obs
LR chi2(3)
Prob > chi2
Pseudo R2
=
=
=
=
10000
2644.57
0.0000
0.1914
-----------------------------------------------------------------------------nonmissing |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------educ |
.1497386
.003211
46.63
0.000
.1434451
.1560321
age |
.0100424
.0010406
9.65
0.000
.0080029
.0120819
z |
.1403535
.0138183
10.16
0.000
.1132702
.1674369
_cons | -1.504516
.0526162
-28.59
0.000
-1.607642
-1.401391
------------------------------------------------------------------------------
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Syntax for Heckman model in STATA
. heckman
wearnl educ age, select(educ age z);
Equation of interest
Variables
in selection
equation
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Notice β’s have increased over OLS w/ missing data
Heckman selection model
(regression model with sample selection)
Log likelihood = -8893.456
Number of obs
Censored obs
Uncensored obs
=
=
=
10000
4663
5337
Wald chi2(2)
Prob > chi2
=
=
428.68
0.0000
-----------------------------------------------------------------------------|
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------wearnl
|
educ |
.0816778
.0064328
12.70
0.000
.0690698
.0942858
age |
.0125495
.0006062
20.70
0.000
.0113614
.0137376
_cons |
4.445103
.1268036
35.06
0.000
4.196572
4.693633
-------------+---------------------------------------------------------------select
|
educ |
.1495845
.0032106
46.59
0.000
.1432918
.1558771
age |
.010029
.0010394
9.65
0.000
.0079919
.0120661
z |
.1390889
.0138789
10.02
0.000
.1118867
.1662911
_cons | -1.502784
.0525991
-28.57
0.000
-1.605877
-1.399692
-------------+---------------------------------------------------------------/athrho |
.3085357
.1723748
1.79
0.073
-.0293128
.6463841
/lnsigma | -.7759725
.0268514
-28.90
0.000
-.8286003
-.7233447
-------------+---------------------------------------------------------------rho |
.2991043
.1569536
-.0293044
.5692308
sigma |
.460256
.0123585
.4366601
.4851269
lambda |
.1376646
.0756984
-.0107016
.2860307
-----------------------------------------------------------------------------LR test of indep. eqns. (rho = 0):
chi2(1) =
1.96
Prob > chi2 = 0.1618
-----------------------------------------------------------------------------Cannot reject null 13
Sigma right on
Rho is a little off
Rho=0
Comparison of Estimates
Covariate
Educ
OLS w/
All data
0.0803
(0.0010)
OLS w/
Selected
sample
MLE of
Heckman
SS model
0.0703
(0.0015)
0.0817
(0.0064)
Age
0.0122
(0.0035)
0.0119
(0.0046)
0.0125
(0.0006)
Constant
4.484
(0.169)
4.670
(0.258)
4.445
(0.127)
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Comparison of Estimates
Covariate
Educ
OLS w/
All data
0.0803
Age
0.0122
OLS w/
Selected
sample
MLE of
Heckman
SS model
0.0703
[-12.5%]
0.0817
[1.7%]
0.0119
[-2.5%]
0.0125
[2.5%]
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[% difference from OLS w/ all data]
• * run heckman sample selection
correction;
• . * but use functional form to identify the
model;
• . heckman wearnl educ age, select(educ
age);
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Heckman selection model
(regression model with sample selection)
Log likelihood = -8946.485
Number of obs
Censored obs
Uncensored obs
=
=
=
10000
4663
5337
Wald chi2(2)
Prob > chi2
=
=
295.99
0.0000
-----------------------------------------------------------------------------|
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------wearnl
|
educ |
.0650405
.0113713
5.72
0.000
.0427531
.087328
age |
.0115308
.0008296
13.90
0.000
.0099048
.0131568
_cons |
4.775379
.2251815
21.21
0.000
4.334031
5.216727
-------------+---------------------------------------------------------------select
|
educ |
.1487287
.0031911
46.61
0.000
.1424743
.1549832
age |
.009841
.0010369
9.49
0.000
.0078088
.0118733
_cons | -1.491731
.0524251
-28.45
0.000
-1.594483
-1.38898
-------------+---------------------------------------------------------------/athrho | -.1419248
.3029263
-0.47
0.639
-.7356494
.4517998
/lnsigma | -.7939694
.0238974
-33.22
0.000
-.8408074
-.7471314
-------------+---------------------------------------------------------------rho | -.1409795
.2969056
-.6265094
.4233773
sigma |
.4520469
.0108027
.4313621
.4737235
lambda | -.0637293
.1356091
-.3295183
.2020596
-----------------------------------------------------------------------------LR test of indep. eqns. (rho = 0):
chi2(1) =
0.07
Prob > chi2 = 0.7894
-----------------------------------------------------------------------------17
No where close on rho
Comparison of Estimates
OLS w/
Selected
sample
Covariate
OLS w/
All data
Educ
0.0803
0.0703
[-12.5%]
MLE of
Heckman
SS model
Function
form Ident.
0.065
[-19.2%]
Age
0.0122
0.0119
[-2.5%]
0.0115
[-5.7%]
[% difference from OLS w/ all data]
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