multiple imputation approaches for right-censored wages in the german iab employment register...
Post on 27-Dec-2015
215 Views
Preview:
TRANSCRIPT
Multiple Imputation Approaches for Right-Censored Wages in the German IAB Employment Register
European Conference on Quality in Official Statistics 2008, 10 July 2008
Thomas BüttnerInstitute for Employment Research (IAB)
Susanne RässlerUniversity of Bamberg
3
For a large number of research questions it is interesting to use wage data- Analyzing the gender wage gap- Measuring overeducation- …
To address this kind of questions two types of data often are used:- Survey data- Administrative data from the social security
Advantages of administrative data- Large number of observations- No response burden - No interviewer bias
Motivation
4
Administrative data Represents 80 percent of the employees in Germany 2 percent random sample of all employees covered by social security 1.3 million persons
Problem:Wages can only be recorded up to the contribution limit of the social security system
The wage information is censored at this limit
The German IAB Employment Sample (1)
Sample drawn from the IAB register data (employment history) supplemented by information on benefit recipients
5
The German IAB Employment Sample (2)0
1.0e
+0
42.
0e+
04
3.0e
+0
4F
requ
ency
3 3.5 4 4.5 5lntentgelt
Daily wages in logs in Western Germany (2000)
Source: IAB Employment Sample
6
Several possibilities to deal with censored wages
Advantage of multiple imputation:
The imputed data set can be used for a multiplicity of questions and analyses
e.g. average wages of certain groups, Analyzing regional wage dispersions, effects of a modification of the contribution limit…
The conventional approaches assume homoscedasticity of the residuals
Censored Wages
Since in general the dispersion of income is smaller in lower wage categories than in higher categories, the assumption of homoscedasticity is highly questionable with wage data
7
Our Project
Step 1: Developing approaches considering heteroscedasticity
Step 2: Simulation study to confirm the necessity and validity of the new approaches
Step 3: Using uncensored wage information from an income survey (German Structure of Earnings Survey, GSES) to validate the approaches
Step 4: Using external wage information for the imputation model
8
Imputation Models
Single Imputation based on a homoscedastic tobit model
Single Imputation using a heteroscedastic model
Multiple Imputation based on a homoscedastic tobit model
Multiple Imputation considering heteroscedasticity
9
Single imputation based on a homoscedastic tobit model
if if
where a is the contribution limit
Imputation by draws of random values according to the parameters estimated using a tobit model
As the true values are above the contribution limit, draws from a truncated normal distribution
Single Imputation
),0(~, 2* Nxyiid
iiíi ayi
*ii yy
ayi *
ayi *
)ˆ,ˆ(~ 2* itrunci xNya
10
Development of an imputation approach considering heteroscedasticity (single
imputation) based on a GLS model for truncated variables
Imputation by draws from a truncated normal distribution
using individual variances
• Single imputation may lead to biased variance estimations (Little/Rubin 1987)
Single Imputation Considering Heteroscedasticity
)ˆ,ˆ(~ 2*iítrunci xNy
a
2ˆ i
11
Multiple Imputation (1)
1 Impute the data set m times2 Analyze each data set3 Combine the results
12
Multiple Imputation (2) 1. To be able to start the imputation based on MCMC, we first need to adapt starting values
for the parameters from a ML tobit estimation
2. In the imputation step, we randomly draw values for the missing wages from a truncated distribution
3. Based on the imputed data set, we compute an OLS regression
4. After this, we produce random draws for the parameters according to their complete data posterior distribution
5. We repeat the imputation and the posterior-step 5,000 times and use to obtain 5 complete data sets
),...,,()5000(*)2000(*)1000(*
iii yyy
),(~ )(2)()*( ttitrunc
ti xNy
a
RSSknXt )(~ 2)1(2
))(,ˆ(~ 1)1(2)()1( XXN ttt
13
Imputation Model Considering Heteroscedasticity (1)
Based on the multiple imputation approach with additional draws fordescribing the functional form of the heteroscedasticity
1. We now start the imputation by adapting starting values from a GLS estimation
2. Then we are able to draw values for the missing wages from a truncated distribution using individual variances
3. Then a GLS regression is computed based on the imputed data set
),(~ )(2)(i
)*( ti
ttrunc
ti xNy
a
14
Imputation Model Considering Heteroscedasticity (2)
4. Afterwards we perform random draws for and
5. Now the parameter can be drawn randomly according to their complete data posterior distribution
6. The steps 2 to 5 are repeated again 5,000 times and we use
to obtain 5 complete data sets
),...,,()5000(*)2000(*)1000(*
iii yyy
RSSknXt )(~ 2)1(2
)ˆ(ˆ,ˆ(~ )()()1( ttt VN
),ˆ(~1
)1(2)()1()1(
t
iz
ttt
e
XXN
)(ˆ
2)(2 )ˆ()ˆˆexp(ln t
iz
iitii e
xyzRSSmit
2
15
IAB Employment Sample 2000 (30 June 2000)
Only male persons from Western Germany
Only full time workers covered by social security
Simulation Study
About 210,000 Persons,about 23,000 or 11 percent with an income above the contribution limit
16
Creating Complete Data Sets
As the IAB Employment Sample is censored, we first have to create complete data sets
We create two different data sets:
one data set using an approach presuming homoscedasticity
another data set using an approach considering heteroscedasticity of the residuals
)ˆ,ˆ(~ 2xNynew
)ˆ,ˆ(~ 2inew xNy
17
.
1. IABS with censored wages
2. Creating complete data sets (with and without heteroscedasticity), calculating β
3. Defining a new limit
4. Drawing a random sample of 10 percent
5. Imputing the wage using the different approaches, computing a regression
Simulation Study
6. Calculating the fraction of confidence intervals of containing the true parameter β for the different approaches
18
Results of the Homoscedastic Data Set
HOM
complete data SI SI-Het MI MI-Het
coverage coverage coverage coverage coverage
educ1 0.1068 0.1069 0.959 0.1074 0.951 0.1073 0.95 0.1074 0.958 0.1073 0.958
educ2 0.1791 0.1790 0.965 0.1792 0.953 0.1790 0.952 0.1792 0.965 0.1790 0.961
educ3 0.1305 0.1310 0.954 0.1317 0.939 0.1330 0.935 0.1318 0.955 0.1330 0.957
educ4 0.2621 0.2623 0.963 0.2624 0.928 0.2654 0.888 0.2624 0.957 0.2653 0.949
educ5 0.4445 0.4446 0.948 0.4409 0.868 0.4466 0.759 0.4410 0.944 0.4469 0.922
educ6 0.5098 0.5096 0.962 0.5064 0.852 0.5121 0.719 0.5065 0.953 0.5118 0.929
level1 0.5449 0.5441 0.949 0.5440 0.952 0.5447 0.95 0.5440 0.949 0.5446 0.95
level2 0.6517 0.6512 0.95 0.6515 0.954 0.6524 0.951 0.6515 0.952 0.6523 0.951
level3 0.8958 0.8950 0.948 0.8973 0.95 0.8958 0.936 0.8976 0.948 0.8959 0.954
level4 0.8962 0.8956 0.953 0.8961 0.95 0.8962 0.949 0.8962 0.951 0.8963 0.951
age 0.0498 0.0498 0.955 0.0500 0.943 0.0500 0.93 0.0500 0.964 0.0500 0.957
sqage -0.0005 -0.0005 0.958 -0.0005 0.936 -0.0005 0.922 -0.0005 0.962 -0.0005 0.96
nation -0.0329 -0.0327 0.962 -0.0334 0.948 -0.0334 0.942 -0.0335 0.953 -0.0334 0.955
cons 2.4424 2.4433 0.953 2.4406 0.945 2.4405 0.932 2.4411 0.951 2.4406 0.949
19
Results of the Heteroscedastic Data Set
HET
complete data SI SI-Het MI MI-Het
coverage coverage coverage coverage coverage
educ1 0.1141 0.1145 0.952 0.1271 0.794 0.1136 0.945 0.1272 0.804 0.1136 0.955
educ2 0.1912 0.1915 0.955 0.2075 0.616 0.1903 0.948 0.2076 0.632 0.1903 0.955
educ3 0.1442 0.1444 0.961 0.0947 0.745 0.1406 0.942 0.0952 0.769 0.1420 0.963
educ4 0.2685 0.2686 0.961 0.2753 0.913 0.2688 0.922 0.2754 0.937 0.2689 0.96
educ5 0.4433 0.4435 0.963 0.4790 0.366 0.4372 0.761 0.4796 0.478 0.4377 0.917
educ6 0.5241 0.5248 0.954 0.5117 0.785 0.5164 0.718 0.5121 0.869 0.5161 0.896
level1 0.5422 0.5426 0.955 0.5415 0.946 0.5422 0.947 0.5416 0.946 0.5417 0.953
level2 0.6405 0.6411 0.95 0.6430 0.944 0.6412 0.944 0.6430 0.947 0.6407 0.95
level3 0.8856 0.8864 0.945 0.8780 0.941 0.8845 0.945 0.8782 0.948 0.8838 0.952
level4 0.8903 0.8908 0.952 0.8737 0.941 0.8919 0.943 0.8737 0.941 0.8913 0.951
age 0.0432 0.0431 0.955 0.0457 0.645 0.0431 0.948 0.0457 0.679 0.0431 0.97
sqage -0.0004 -0.0004 0.96 -0.0005 0.59 -0.0004 0.941 -0.0005 0.623 -0.0004 0.968
nation -0.0223 -0.0218 0.961 -0.0297 0.872 -0.0222 0.945 -0.0296 0.882 -0.0222 0.954
cons 2.5858 2.5865 0.947 2.5318 0.909 2.5868 0.945 2.5315 0.914 2.5875 0.952
20
Simulation study using external wage information (1)
Scientific-Use-File of the German Structure of Earnings Survey (GSES) 2001
Linked Employer-Employee data set
Information on about 22.000 establishments and about 846.000 employees
Information on
- individuals (e.g. sex, age, education)
- jobs (e.g. occupation, job level, working times)
- income (e.g. gross wage, net wage, income taxes)
- and establishments
21
Simulation study using external wage information (2)
Selection of a sample comparable to the first simulation study
Complete data set containing 382.710 persons
Censoring at the 85 percent quantile
22
coverage coverage coverage
educ2 0.0471 0.0472 0.946 0.0476 0.951 0.0474 0.952educ3 0.0933 0.0929 0.929 0.0709 0.841 0.0783 0.897educ4 0.1067 0.1069 0.907 0.0863 0.559 0.0894 0.674educ5 0.2095 0.2100 0.934 0.2086 0.963 0.2164 0.930educ6 0.2822 0.2826 0.906 0.2501 0.181 0.2685 0.790level3 0.0183 0.0181 0.949 0.0165 0.946 0.0164 0.944level4 0.0862 0.0833 0.951 0.0828 0.967 0.0848 0.964level5 0.0686 0.0652 0.951 0.0390 0.880 0.0359 0.856group2 -0.1378 -0.1377 0.956 -0.1338 0.912 -0.1336 0.909group3 -0.2691 -0.2691 0.955 -0.2634 0.874 -0.2631 0.870group4 -0.4151 -0.4150 0.951 -0.4108 0.927 -0.4104 0.920group5 0.6083 0.6117 0.942 0.5689 0.788 0.5797 0.869group6 0.1925 0.1956 0.951 0.2091 0.939 0.2118 0.928group7 0.0449 0.0482 0.950 0.0741 0.879 0.0763 0.862group8 -0.2738 -0.2701 0.952 -0.2421 0.863 -0.2393 0.843group9 -0.4865 -0.4840 0.947 -0.4578 0.901 -0.4554 0.895age 0.0332 0.0332 0.937 0.0336 0.953 0.0334 0.958sqage -0.0003 -0.0003 0.934 -0.0003 0.857 -0.0003 0.918region2 0.0491 0.0492 0.933 0.0651 0.123 0.0605 0.414region3 0.0060 0.0060 0.929 0.0147 0.623 0.0108 0.859region4 0.0732 0.0731 0.937 0.0711 0.931 0.0672 0.753contract -0.1659 -0.1660 0.959 -0.1624 0.947 -0.1619 0.943cons 3.8467 3.8461 0.939 3.8525 0.953 3.8572 0.952
complete data M I M I-Het
Simulation study using external wage information (3)
24
ReferencesBender, S., Haas, A. and Klose, C. (2000). IAB Employment Subsample 1975-1995. Opportunities for Analysis Provided by Anonymised Subsample. IZA Discussion Paper117, IZA Bonn.
Buchinsky, M. (1994). Changes in the U.S. wage structure 1963–1987: Applicationof quantile regression. Econometrica 62(2), 405–458.
Gartner, H. (2005). The imputation of wages above the contribution limit with the GermanIAB employment sample. FDZ Methodenreport 2/2005.
Gartner, H. and Rässler, S. (2005). Analyzing the changing gender wage gapbased on multiply imputed right censored wages. IAB Discussion Paper 05/2005.
Jensen, U., Gartner, H. and Rässler, S. (2006). Measuring overeducation with earnings frontiers and multiply imputed censored income data. IAB Discussion Paper Nr. 11/2006.
Khan, S. and Powell, J.L. (2001). Two-step estimation of semiparametric censored regression models. Journal ofEconometrics 103, 73–110.
Little, R.J.A and Rubin D.R. (1987). Statistical Analysis with Missing Data. John Wiley,New York, 1 edn.
Meng, X.L. (1994). Multiple Imputation Inferences with Uncongenial Sources of Input. Statistical Sciences Volume 9, 538-558.
Powell, J.L. (1986). Symmetrically Trimmed Least Squares Estimation for Tobit Models. Econometrica 54(6),1435-1460.
Rässler, S. (2006). Der Einsatz von Missing Data Techniken in der Arbeitsmarktforschung des IAB. Allgemeines Statistisches Archiv.
Rubin, D.B. (1987). Multiple Imputation for Nonresponse in Surveys. J.Wiley & Sons, New York.
Schafer, J.L. and Yucel, R.M (2002). Computational Strategies for Multivariate Linear Mixed-Effects Models With Missing Values. Journal of Computational and Graphical Statistics Volume 11 437-457.
Schafer, J.L. (1997). Analysis of Incomplete Multivariate Data. Chapman & Hall, New York.
25
Combining Rules
m
t
tMI m 1
)(ˆ1ˆ
m
t
traVm
W1
)()ˆ(ˆ1
Bm
mWT
1
m
tMI
t
mB
1
2)( )ˆˆ(1
1
• The associated variance estimate has two components. The within-imputation-variance is the average of the complete-data-variance estimates:
• The between-imputation-variance is the variance of the complete-data point estimates:
• The total variance is defined as:
• Multiple Imputation point estimate for is defined as:
26
First Results
The simulation study using these three approaches shows the necessity of a new method
that multiply imputes the missing wages and does not presume heteroscedasticity
Second step: Development of a new multiple imputation approach considering
heteroscedasticity
Finally we perform a new simulation study to compare the four approaches under different
situations in order to confirm the necessity as well as the validity of the new approach
29
Simulation Study (2)
The simulation procedure consisting of
drawing a random sample, deleting the wages above the limit imputing the data using the different approaches, computing a regression, and calculating the confidence intervals
is repeated 1000 times.
Coverage: The fraction of confidence intervals of containing the true parameter β for the different approaches
30
Summary of Results
In case of a homoscedastic structure of the residuals the same quality of imputation
results can be expected from the two multiple imputation approaches
In case of heteroscedasticity the simulation study confirms the necessity of our new
approach
Since the structure of the wages in the IAB employment register is heteroscedastic, the
results of the simulation study necessitate the use of the new approach to impute the
missing wage information in this register
top related