Multiple Imputation of Right-Censored Wages in the

German IAB Employment Register Considering

Heteroscedasticity

Thomas B ̈uttner and Susanne R ̈assler

Institute for Employment Research of the German Federal Employment Agency,

Research Unit for Statistical Methods

Regensburger Straße 104, 90478 Nuremberg

email: thomas.buettner@iab.de, susanne.raessler@iab.de

SUMMARY: In many large data sets of economic interest, some variables, as wages, are top-coded or rightcensored. In order to analyze wages with the German IAB-employment register we first have to solve the

problem of censored wages at the upper limit of the social security system. We treat this problem as a

missing data problem and use multiple imputation approaches to impute the censored wages by draws of a

random variable from a truncated distribution, based on Markov chain Monte Carlo techniques. In general,

the dispersion of income is smaller in lower wage categories than in higher categories and the assumption

of homoscedasticity in an imputation model is highly questionable. Therefore, we suggest a new multiple

imputation method which does not presume homoscedasticity of the residuals. Finally, in a simulation

study, different imputation approaches are compared under different situations and the necessity as well as

the validity of the new approach is confirmed.

Key words: multiple imputation, missing data, censored wage data, simulation study
Contents

1 Introduction 3

2 Imputation approaches for censored wages 4

2.1 Homoscedastic imputation approaches . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Homoscedastic single imputation . . . . . . . . . . . . . . . . . . . 5

2.1.2 Multiple imputation . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Heteroscedastic imputation approaches . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Heteroscedastic single imputation . . . . . . . . . . . . . . . . . . . 8

2.2.2 Multiple imputation considering heteroscedasticity . . . . . . . . . . 8

3 Simulation study 10

3.1 The IAB employment register . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Creating a complete population . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Simulation design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Results 15

4.1 Homoscedastic data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Heteroscedastic data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Conclusion 19

6 References 20

2
1 Introduction

For a large number of research questions, like analyzing the gender wage gap or measuring

overeducation with earnings frontiers, it is interesting to analyze income data. To address

this kind of questions two types of data are used: surveys and process generated data.

Process generated data have several advantages, like a large number of observations, no

nonresponse burden and no problems with interviewer effects or survey bias. Unfortunately, in many large process generated data sets of economic interest some variables,

such as wages, are top-coded or right-censored. This problem is very common with administrative data from social security systems like the IAB employment register (IABS),

which is edited from the data of the German unemployment insurance. The contribution

rate of this insurance is charged percentaged from the gross wage. Is the gross wage

higher than the current contribution limit, however only the amount of the limit is liable

for the contribution. In 2007 the contribution limit in the unemployment insurance is

fixed at a monthly income of 5,250 euros. As therefore wages are only recorded up to this

contribution limit, the income information in this register is censored at this limit.

In order to analyze wages with this register, we first have to solve the problem of the

censored wages. We treat this problem as a missing data problem and use imputation

approaches to impute the censored wages. Gartner (2005) proposes a single imputation

approach to solve the problem of the censored wages. A further approach - a multiple

imputation method based on draws of a random variable from a truncated distribution

and Markov Chain Monte Carlo techniques - is suggested by Gartner and R ̈assler (2005).

These two approaches presume homoscedasticity of the residuals, but since in general,

the dispersion of income is smaller in lower wage categories than in higher categories, the

assumption of homoscedasticity in an imputation model is highly questionable. Therefore

we use a third approach, a second single imputation approach based on GLS estimation

to consider heteroscedasticity.

A first simulation study using these three methods shows the necessity to develop a new

method that imputes the missing wage information multiple and does not presume homoscedasticity. Therefore we suggest a new multiple imputation method and finally compare in a simulation study the four different imputation approaches again under different

3
0 .5 1 1.5 2 2.5

Density

3 4 5 6

ln(wage)

Distribution of wages in the IABS

Figure 1: Distribution of wages in logs in the IAB employment register

situations in order to confirm the necessity as well as the validity of the new approach.

2 Imputation approaches for censored wages

Before we develop and evaluate a new approach, the first section of this paper describes

the three different approaches, which were already proposed to impute the missing wage

information in the IAB employment register. All of them assume that the wage in logs y

for every person i is given by

yi

∗ = xiβ + εi where ε ∼ N(0,σ2

). (1)

As the wages in the IAB employment register are censored at the contribution limit a we

observe the wage yobs = yi

∗ only if the wage is lower than the treshold a. If the wage is

censored, we observe the limit a instead of the true wage yi

∗

:

yi =

⎧

⎨

⎩

yobs if yi ≤ a

(2)

a if yi > a

To be able to analyze wages with our data set, we first have to impute the wages above

a. Thus we define y = (yobs, a) and yz = (yobs,z). So z is a truncated variable in the

4
range (a, ∞). We regard the missingness mechanism as not missing at random (NMAR,

according to Little and Rubin, 1987, 2002) as well as missing by design. The first because

the missingness depends on the value itself. If the limit is exceeded the true value will

not be reported but the value of the limit a. The latter because the data are missing due

to the fact that they were not asked.

2.1 Homoscedastic imputation approaches

2.1.1 Homoscedastic single imputation

One possibility to impute the missing wage information is using a single imputation approach. A homoscedastic single imputation (without consideration of heteroscedasticity)

can be performed based on a tobit model. The tobit model is used to estimate the parameter βb of the imputation model. According to the estimated parameters the censored

wage z can be imputed by draws of a random value. As we know that the true value is

above the contribution limit we have to draw a random variable from a truncated normal

distribution

∗ 0 ˆ zi ∼ Ntrunca

(x β,σ2). (3)

This means we add to the expected wage an error term ε with the standard deviation σ

(estimated from the tobit estimation):1

z

∗

i = x

0βb + ε (4)

Using a single imputation approach, we have to consider that this method may lead

to biased variance estimations. Thus Little and Rubin (1987) suggest that imputation

should rather be done in a multiple and bayesian way. Therefore we better use multiple

imputation approaches to impute the missing wage information.

1For more information on this approach see Gartner (2005).

5
2.1.2 Multiple imputation

To start with, let Y = (Yobs,Ymis) denote the random variables concerning the data with

observed and missing parts. In our specific situation this means that for all units with

wages below the limit a each data record is complete, i.e., Y = (Yobs) = (X, wages). For

every unit with a value of the limit a for its wage information we treat the data record

as partly missing, i.e., Y = (Yobs,Ymis) = (X, ?). Thus, we have to multiply impute the

missing data Ymis = wage. The theory and principle of multiple imputation originates

from Rubin (1978) and is based on independent random draws from the posterior predictive distribution fYmis|Yobs of the missing data given the observed data. As is it may be

difficult to draw from fYmis|Yobs directly, a two-step procedure for each of the m draws is

useful:

1. First, we perform random draws of the parameter Ξ according to their observed-data

posterior distribution fΞ|Yobs ,

2. Then, we make random draws of Ymis according to their conditional predictive

distribution fYmis|Yobs,Ξ.

Because

Z

fYmis|Yobs (ymis|yobs) = fYmis|Yobs,Ξ(ymis|yobs,ξ)fΞ|Yobs (ξ|yobs)dξ (5)

holds, with (1) and (2) we achieve imputations of Ymis from their posterior predictive

distribution fYmis|Yobs . Due to the data generating model used, for many models the

conditional predictive distribution fYmis|Yobs,Ξ is rather straightforward. That means it

can be easily formulated for each unit with missing data. In contrast, the corresponding

observed-data posteriors fΞ|Yobs are usually difficult to derive for those units with missing

data, especially when the data have a multivariate structure. The observed data

posteriors are often no standard distributions from which random numbers can easily be

generated. However, simpler methods have been developed to enable multiple imputation

based on Markov chain Monte Carlo (MCMC) techniques (Schafer 1997). In MCMC

the desired distributions fYmis|Yobs and fΞ|Yobs are achieved as stationary distributions of

6
Markov chains based on the complete-data distributions, which is easier to compute. 2

Imputation model

To be able to start the imputation based on MCMC, we first need to adapt starting values

for β

(0) and the variance σ2(0) ( for programming reasons below defined as τ−2(0)) from a

ML tobit estimation. Second, in the imputation step, we randomly draw values for the

missing wages from the truncated distribution analog to the single imputation procedure

zi

∗(t) ∼ Ntrunca

(x

0β(t)

,τ −2(t)). (6)

Then a OLS regression is computed based on the imputed data sets according to

βˆ

z

(t) = (X

0X)−1X0

yz

(t)

(7)

After this, we can produce new random draws for the parameters according to their

complete data posterior distribution. Since drawings from a gamma distribution are

complicated to compute with STATA, we draw the variance in the posterior step as

follows:

g ∼ χ

2

(n − k) (8)

g

τ

2(t+1) = (9) RSS

n

where RSS is the residual sum of squares RSS =

P(y

z

(t) − xi

0βˆz

(t)

)

2 and k is the number

i

t=1

of columns of X.

Now we perform new random draws for the parameters β

β

t+1 ∼ N(βˆ(t)

, τ −2(t+1)(X

0X)−1

) z(10)

2For more details see Gartner and R ̈assler (2005)

7
We repeat the imputation and the posterior-step 5,000 times and use

2000 3000 11000 (z ,z ,. . ., z ) to obtain 5 complete data sets. 3

i i i

2.2 Heteroscedastic imputation approaches

2.2.1 Heteroscedastic single imputation

As we assume the dispersion of income to be smaller in lower wage categories than in

higher categories, we suppose the necessity of an approach considering heteroscedasticity.

Therefore we develop another single imputation procedure based on the first single imputation approach, a method that does not presume homoscedasticity of the residuals. We

develop this method in order to check wether we need to do further research towards a

multiple imputation approach considering heteroscedasticity.

Here we use a GLS model for truncated variables to estimate the parameters of the

ˆ imputation model, the coefficients β, like in the first approach, and furthermore the

parameters γˆ, describing the heteroscedasticity. Then the imputation can be done by

draws from a truncated normal distribution, similar to the first approach,

∗ 0 ˆ σ

2 m0γˆ

zi ∼ Ntrunca

(x β,σi

2

) where i = e . (11)

To consider the heteroscedastic structure of the residuals, we use here individual variances

for every person. This procedure is a solution that takes into consideration the existence

of heteroscedasticity, yet it does not solve the problem of biased variance estimations.

2.2.2 Multiple imputation considering heteroscedasticity

Since we assume the necessity of an approach that does not presume homoscedasticity

and since Little and Rubin show that single imputation approaches may lead to biased

variance estimations, consequently we suggest an approach that multiply imputes the

missing wage information and considers heteroscedasticity of the residuals. A first simulation study using the first three approaches shows as well the need for this approach. This

simulation study points out that, in case of a homoscedastic structure of the residuals,

3For further details on this approach see Gartner and R ̈assler (2005).

8
the multiple imputation leads to better results than a single imputation approach.

But in case of heteroscedasticity the single imputation considering heteroscedasticity is

superior to the multiple imputation approach suggested by Gartner and R ̈assler. This

indicates the necessity to develop another approach that combines these two properties:

an approach performing multiple imputation and considering heteroscedasticity.

Imputation model

We develop this new method based on the multiple imputation approach proposed by

Gartner and R ̈assler (2005). The basic element of the new approach is that we need

additional draws for the parameters γ describing the heteroscedasticity. We start now

the imputation by adapting starting values for β

(0)

, γ

(0)and τ−2(0) from a GLS estimation

for truncated variables. Then we are able to draw values for the missing wages from

x

0γˆ a truncated distribution using individual variances σi = e like in the heteroscedastic

single imputation model:

∗(t) 2(t) 2(t) m0γˆ

z ∼ Ntrunca(x

0β(t)

,σ ) where σ = e . (12) i i i

Then a GLS regression is computed based on the imputed data set (comparable to the

OLS regression in the heteroscedastic multiple imputation approach). Afterwards we produce new random draws for the parameters according to their complete data posterior

distribution. As we consider now the existence of heteroscedasticity, some slight modifications to the algorithm are necessary. In the first step, we draw the variance τ

2(t+1)

according to

g ∼ χ

2

(n − k) (13)

n n (t) 0 ˆ

(t)

)

2

τ

2(t+1) g 0

(yi − xiβi = where RSS =

Xexp(ln εˆi − miγˆ) =X

em0ˆ

(14) RSS γ

t=1 t=1

In a additional step, we have to perform random draws for γ

9
γ

t+1 ∼ N(ˆγz

(t)

,τγ

−2(t)

(M0M)

−1

) (15)

Consequently the parameters β can be drawn like in the Gartner and R ̈assler approach,

again with a slight modification compared to the homoscedastic multiple imputation:

n X0X

,τ −2(t+1)( )−1 β )

t+1|γt+1,τ 2(t+1) ∼ N(βˆ(t) X (16) z 0γˆ em

t=1

We 2000 3000 11000 repeat these steps again 5,000 times and use (zi,zi,. . ., zi) to obtain the 5

complete data sets.

3 Simulation study

To evaluate the results delivered by these different approaches under different situations in

order to show the relevance of the suggested multiple imputation approach, we perform

a simulation study using a sample of the IAB employment register. We first create a

complete data set without censored wages, define a new limit and delete the wages above

this limit. Afterwards, the missing wages are imputed using the different approaches and

the results are compared to the complete data set.

3.1 The IAB employment register

The German IAB employment sample (IABS) is a 2 percent random sample of all employees covered by social security. Consequently self-employed, family workers and civil

servants are not included. The data set represents 80 percent of the employees in Germany. The IABS includes, among others, information about age, sex, education, wage and

the occupational group. For the register two sources of data are combined: Information

about employment coming from employer reports to the social security and information

about unemployment compensation coming from the German federal employment agency.

As already mentioned, the wage information in the IABS is censored at the contribution

limit of the unemployment insurance. 4

4For further insight on the data set, see Bender et al.(2000)

10
To simplify the simulation design, we restrict the data for the simulation to male WestGerman residents. We use all workers holding a full-time job covered by social security

effective on June 30th 2000. The data set contains 214,533 persons: 23,685 or 11 percent

of them with censored wages. The following table shows descriptive information about the

fraction of censored incomes of 6 educational and 5 age groups to demonstrate the need

to impute the missing wage information. Especially for analyzing highly-skilled workers

the table indicates the necessity to impute the missing wages.

<25 25-34 35-44 45-54 55+

educ1 0 .003 .008 .012 .17

educ2 .001 .021 .068 .116 .150

educ3 .010 .110 .232 .331 .371

educ4 .003 .110 .283 .393 .470

educ5 .024 .190 .450 .558 .604

educ6 .056 .256 .549 .686 .769

Table 1: Censored wages in the original data set

For the simulation study we assume a model containing the wage in logs as dependent

variable and age, squared age, nationality as well as six dummies for education levels and

four categories of job level as independent variables.

3.2 Creating a complete population

To perform the simulation study, a complete population is needed in to order to be

able to compare the results of the different approaches with a complete data base. As

the IABS employment register is right-censored, we first have to impute our sample to

obtain this control population. The fact that the data set has to be imputed before

starting the simulation study allows us to produce control populations with different

characteristics: We create one data set where homoscedasticity is existent and another

with heteroscedasticity of the residuals. To obtain the first data set (Data set A) we use

the homoscedastic single imputation procedure as described in section 2.1.1 to impute new

11
wages for every person regardless if the wage was originally censored or not, according to

y

0 b new ∼ Ntrunca

(x β, σ2). (17)

A test for heteroscedasticity shows constant variances for this data set. To receive the

second data set (Data set B), the heteroscedastic single imputation method described in

section 2.2.1 is used in order to get a control population with heteroscedasticity of the

residuals, according to

0

0ˆ

ynew ∼ Ntrunca(x β, em γ b ). (18)

A test for heteroscedasticity shows no constant variances, which refers to heteroscedasticity of the residuals in this data set. This two data sets will later be used as complete

populations for the analysis of the results we receive by using the different approaches.

3.3 Simulation design

The simulation study consists of four steps. Each of these steps is simultaneously done

for the homoscedastic data set A and the heteroscedastic population B: We draw random

samples from the complete population repeatedly, define a new threshold and impute the

wage above this threshold using the four approaches. Then we compare the imputed data

sets with the complete population. To clarify the proceeding we describe the procedure

only for one of the two options.

Step 1: Drawing of a random sample

In the first step a random sample of n=21,453 persons is drawn without replacement

from the population of N=214,533 persons (equivalent to 10 percent). This 10 percent

random sample is kept to illustrate the results of the different imputations later. For the

simulation study we define a new threshold. To point out the differences between the

four approaches we choose a limit lower than in the original IABS (censoring the highest

30 percent of incomes appears adequate) and delete the incomes above this limit.

12
Step 2: Imputation of the missing wage information

The deleted wage information above the threshold of this (now right-censored) sample is

imputed by using the four different approaches described above:

• Homoscedastic single imputation

• Heteroscedastic single imputation

• Multiple imputation

• Multiple imputation considering heteroscedasticity

That means by each of the single imputation methods, one complete data set is obtained

and by each of the multiple imputation methods, five complete data sets. This imputed

data sets now can be used to evaluate the quality of the different approaches by comparing

them with the original complete population.

Step 3: Analysis of the results

To analyze the results of the imputations we run OLS regressions on the imputed data sets

and the 10 percent complete random sample on the one side, on the complete population

on the other side. We use as estimation model - simulating an analysis which is typically

done with income data - the same model as the imputation model. Afterwards we can

evaluate which approach delivers the best imputation quality compared to the original

complete data. Therefore we compare the parameter βb - estimated based on the imputed

data sets - with the results β of the regression on the complete population.

For this purpose we need to use the parameter βb as well as the corresponding confidence

intervals. Since the multiple imputation approaches lead to five complete data sets, the

estimations have to be done five times as well and afterwards, the results have to be

combined following Rubin (1987) in order to get valid parameters and confidence intervals. Thus, the multiple imputation point estimate for βb is the average of the five point

estimates

1

m

β =

Xβ

(t)

. (19)

m

b

t=1

13
The variance estimate associated with βb has two components. The within-imputationvariance is the average of the complete-data variance estimates,

1

m

U =

XU

(t)

. (20)

m

t=1

The between-imputation variance is the variance of the complete-data point estimates

1

m

B =

X(βb(t) − β)

2

. (21)

m − 1

t=1

Subsequently the total variance is defined as

T = U + (1 + m−1

)B. (22)

For large sample sizes, tests and two-sided (1 − α) ∗ 100% interval estimates for multiply

imputed data sets can be calculated based on the Student‘s t-distribution

√ √ (β ̄ − β)/ T ∼ tv and β ̄ ± tv,1−α/2 T (23)

with the degrees of freedom

U

)

2

v = (m − 1)(1 + . (24) (1 + m−1

)B

We save for every approach in every iteration the parameter βb and the corresponding

standard error of βb, as well as the 95 percent confidence interval of βb. Besides, we keep the

information if the confidence interval of βb contains the parameter β of the original data set.

Step 4: 1000 iterations

The whole simulation procedure - consisting of drawing a random sample, imputing the

data using the different approaches, running a regression on the different imputed data

sets and calculating the confidence intervals - is repeated 1000 times. Finally the fraction

of confidence intervals of βb containing the true parameter β can be calculated for the

different approaches. The results of this iterations are described in the following chapter.

14
4 Results

This chapter contains tables showing the results of the simulation study comparing the

four different approaches. The first column contains the true parameter β, estimated using

the original complete population. The following columns show the parameter βb (here the

average of the 1000 iterations) of the regression using the 10 percent complete random

samples and the regressions using the data sets imputed by the different approaches. The

tables show as well the fraction of iterations where the 95 percent confidence interval of

βb contains the parameter β (coverage).

4.1 Homoscedastic data set

The first table shows the results of the simulation based on the homoscedastic data set A.

As expected, the simulation study shows the necessity of a multiple imputation approach,

since the coverage of the two multiple imputation approaches is higher than of the single

imputations throughout almost all variables. Using a homoscedastic data set, the results

do not show serious differences between the homoscedastic and the heteroscedastic multiple imputation. We receive a coverage for both of this approaches around 95 percent

(between 0.965 and 0.922) - similar to the coverage received by the estimations using the

complete random samples (between 0.965 and 0.948) - which refers to a good imputation

quality. The coverage of the single imputations, especially the single imputation considering heteroscedasticity, is for most of the variables lower than 0.95. Consequently, it can

be concluded, that in case of a homoscedastic structure of the residuals, it is advisable

to use a multiple imputation approach. However it does not matter if the algorithm considering heteroscedasticity is chosen in the homoscedastic case, since it just represents a

generalization of the homoscedastic approach and therefore works just as well in case of

homoscedasticity.

15
complete data single homosc. single heterosc. mult

β β

b

Coverage β

b

Coverage β

b

Coverage β

b

educ1 0.1068 0.1069 0.959 0.1074 0.951 0.1073 0.95 0.107educ2 0.1791 0.1790 0.965 0.1792 0.953 0.1790 0.952 0.179educ3 0.1305 0.1310 0.954 0.1317 0.939 0.1330 0.935 0.131educ4 0.2621 0.2623 0.963 0.2624 0.928 0.2654 0.888 0.262educ5 0.4445 0.4446 0.948 0.4409 0.868 0.4466 0.759 0.441educ6 0.5098 0.5096 0.962 0.5064 0.852 0.5121 0.719 0.506

level1 0.5449 0.5441 0.949 0.5440 0.952 0.5447 0.95 0.544level2 0.6517 0.6512 0.95 0.6515 0.954 0.6524 0.951 0.651level3 0.8958 0.8950 0.948 0.8973 0.95 0.8958 0.936 0.897level4 0.8962 0.8956 0.953 0.8961 0.95 0.8962 0.949 0.896

age 0.0498 0.0498 0.955 0.0500 0.943 0.0500 0.93 0.050

sqage -0.0005 -0.0005 0.958 -0.0005 0.936 -0.0005 0.922 -0.000

nation -0.0329 -0.0327 0.962 -0.0334 0.948 -0.0334 0.942 -0.033

cons 2.4424 2.4433 0.953 2.4406 0.945 2.4405 0.932 2.441

Table 2: Results of the homoscedastic data set

16
4.2 Heteroscedastic data set

The results based on the heteroscedastic data set B show a different situation. First the

results recommend as well the use of a multiple imputation approach, since the coverage

of the single imputation approaches is again lower than 0.95 for all variables. Concerning

the heteroscedastic structure of the residuals, it reveals the necessity of an approach

considering heteroscedasticity. The homoscedastic approaches deliver in several cases

a significant lower coverage than the procedures that consider heteroscedasticity. The

coverage of the heteroscedastic multiple imputation approach amounts again to around

95 percent and is similar to the coverage of the complete random samples (the coverage

ranges between 0.97 and 0.917, except the dummy for the highest education level where

the coverage is 0.896). In this case, the coverage of the conventional multiple imputation is

significant lower (between 0.948 and 0.478, for some variables even lower than the coverage

received by the heteroscedastic single imputation approach, where the coverage ranges

between 0.948 and 0.718). Therefore the results suggest the use of an approach considering

heteroscedasticity to impute the missing wage information in case of a heteroscedastic

structure of the residuals .

17
complete data single homosc. single heterosc. mult

β β

b

Coverage β

b

Coverage β

b

Coverage β

b

educ1 0.1141 0.1145 0.952 0.1271 0.794 0.1136 0.945 0.127educ2 0.1912 0.1915 0.955 0.2075 0.616 0.1903 0.948 0.207educ3 0.1442 0.1444 0.961 0.0947 0.745 0.1406 0.942 0.095educ4 0.2685 0.2686 0.961 0.2753 0.913 0.2688 0.922 0.275educ5 0.4433 0.4435 0.963 0.4790 0.366 0.4372 0.761 0.479educ6 0.5241 0.5248 0.954 0.5117 0.785 0.5164 0.718 0.512

level1 0.5422 0.5426 0.955 0.5415 0.946 0.5422 0.947 0.541level2 0.6405 0.6411 0.95 0.6430 0.944 0.6412 0.944 0.643level3 0.8856 0.8864 0.945 0.8780 0.941 0.8845 0.945 0.878level4 0.8903 0.8908 0.952 0.8737 0.941 0.8919 0.943 0.873

age 0.0432 0.0431 0.955 0.0457 0.645 0.0431 0.948 0.045

sqage -0.0004 -0.0004 0.96 -0.0005 0.59 -0.0004 0.941 -0.000

nation -0.0223 -0.0218 0.961 -0.0297 0.872 -0.0222 0.945 -0.029

cons 2.5858 2.5865 0.947 2.5318 0.909 2.5868 0.945 2.531

Table 3: Results of the heteroscedastic data set

18
5 Conclusion

There is a diversity of ways to deal with censored wage data. We propose to use imputation approaches to estimate the missing wage information and to so solve this problem.

Nevertheless, there are also different possibilities to impute the wages in the IABS, for example single and multiple imputation approaches. Another important question is wether

the wages should be imputed considering heteroscedasticity or not.

In this paper we propose a new approach to multiply impute the missing wage information above the limit of the social security in the IAB employment register. We have

assumed the dispersion of income to be smaller in lower wage categories than in higher

categories. Thus we have suggested and developed a multiple imputation approach considering heteroscedasticity to impute the missing wage information. The basic element

of this approach is to impute the missing wages by draws of a random variable from a

truncated distribution, based on Markov chain Monte Carlo techniques. The main innovation of the suggested approach is to perform additional draws for the parameter γ

describing the heteroscedasticity in order to be able to allow individual variances for every individuum. To confirm the necessity and validity of this new method we have used

a simulation study to compare the different approaches. The results of the simulation

study can be subsumed as follows: The missing wage information should be imputed

rather multiple. This is due to two reasons: First, because single imputations may lead to

biased variance estimations and second (as the simulation study shows), because multiple

imputation approaches lead to better imputation results. Furthermore, the imputation

should be done considering heteroscedasticity. In case of homoscedastic residuals the same

quality of imputation results can be expected compared to the Gartner and R ̈assler (2005)

approach. But if heteroscedasticity is existent the simulation confirms the necessity of our

new approach. As the assumption of homoscedasticity is highly questionable with wage

data, the simulation study shows it is preferable to use the new approach considering

heteroscedasticity, as this approach is more general.

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