Thomas Büttner Institute for Employment Research (IAB) Susanne Rässler University of Bamberg

1. Multiple Imputation Approaches for Right-Censored Wages in the German IAB Employment Register Thomas Büttner Institute for Employment Research (IAB) Susanne Rässler University of Bamberg European Conference on Quality in Official Statistics 2008, 10 July 2008

2. Motivation • 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

3. The German IAB Employment Sample (1) • 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 • Sample drawn from the IAB register data (employment history) supplemented by information on benefit recipients

4. The German IAB Employment Sample (2) Daily wages in logs in Western Germany (2000) Source: IAB Employment Sample

5. 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 Censored Wages • 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

6. 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

7. 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

8. Single Imputation • 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

9. Single Imputation Considering Heteroscedasticity • 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)

10. Multiple Imputation (1) 1 Impute the data set m times 2 Analyze each data set 3 Combine the results

11. Multiple Imputation (2) • 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 • In the imputation step, we randomly draw values for the missing wages from a truncated distribution • Based on the imputed data set, we compute an OLS regression • After this, we produce random draws for the parameters according to their complete data posterior distribution • We repeat the imputation and the posterior-step 5,000 times and use to obtain 5 complete data sets

12. Imputation Model Considering Heteroscedasticity (1) Based on the multiple imputation approach with additional draws for describing the functional form of the heteroscedasticity • We now start the imputation by adapting starting values from a GLS estimation • Then we are able to draw values for the missing wages from a truncated distribution using individual variances • Then a GLS regression is computed based on the imputed data set

13. 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 • The steps 2 to 5 are repeated again 5,000 times and we use to obtain 5 complete data sets

14. About 210,000 Persons, about 23,000 or 11 percent with an income above the contribution limit Simulation Study • IAB Employment Sample 2000 (30 June 2000) • Only male persons from Western Germany • Only full time workers covered by social security

15. 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

16. 6. Calculating the fraction of confidence intervals of containing the true parameter β for the different approaches Simulation Study 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

17. Results of the Homoscedastic Data Set

18. Results of the Heteroscedastic Data Set

19. 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

20. 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

21. Simulation study using external wage information (3)

22. References Bender, 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: Application of quantile regression. Econometrica 62(2), 405–458. Gartner, H. (2005). The imputation of wages above the contribution limit with the German IAB employment sample. FDZ Methodenreport 2/2005. Gartner, H. and Rässler, S. (2005). Analyzing the changing gender wage gap based 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 of Econometrics 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.

23. Combining Rules • Multiple Imputation point estimate for is defined as: • 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:

24. 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

25. 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

26. 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