Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol95.Issue1.2000:

Journal of Econometrics 95 (2000) 131}156

Estimating the density of unemployment
duration based on contaminated samples
or small samplesq
Hang K. Ryu!, Daniel J. Slottje",*
!Department of Economics, Chung Ang University, Seoul, South Korea
"Department of Economics, Southern Methodist University, Dallas, TX 75275, USA
Received 1 August 1997; received in revised form 1 November 1998; accepted 1 April 1999

Abstract
In estimating a density function for the duration of unemployment, we consider two
departures from what would be ideal conditions. If the so-called digit preference e!ect
produces local distortion in observed samples, we can apply a maximum entropy density
estimation method. To establish the functional form of the density, we maximize entropy
subject to moment restrictions. The global shape of the density is determined by the lower
ordered sample moments which are not a!ected much by the digit preference e!ect. As
a by-product of this method, we can establish the local transition structure of the digit
preference e!ect. As a second case of departure from an ideal condition, we consider
coarse sample observations where unemployment duration was observed only for 4, 10,
14, 26, and 52 weeks. Once the unemployment duration density is derived, quintile

behavior over time, the Lorenz curve, and the Gini coe$cient for the distribution of
unemployment duration can be obtained. Finally, we discuss the rami"cations of only
focusing on the headcount ratio of unemployment when other information is available. ( 2000 Elsevier Science S.A. All rights reserved.
Keywords: Unemployment duration; Digit preference e!ect; coarse sample observations;
Global approximation; Gini coe$cient

* Corresponding author. Tel.: #1-214-7683555; fax: #1-214-7681821.
E-mail address: dslottje@mail.smu.edu (D.J. Slottje)
q
We thank the Associate Editor and two anonymous referees for many useful suggestions.

0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 3 3 - 0

132

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

1. Introduction
Economists for some time have understood that the duration of unemployment is as important as the traditional unemployment rate (or head count ratio)

in analyzing the problem of unemployment. In modeling unemployment duration, little attention has been paid to the problematic nature of data that comes
from retrospective surveys. An exception is the very recent work of Torelli and
Trivellato (1993). They note in their study that individuals tend to forget an
event occurred (memory e!ect) or remember the timing of an event incorrectly
(telescoping e!ects). These two e!ects can also be considered reasons for another
problem, the so-called &digit preference e!ect'. Individuals tend not to think that
they have been unemployed for precise units of time. They tend to clump weeks
of unemployment together. This &digit preference' makes it very di$cult to use
traditional hypothetical statistical distributions as descriptions of the distribution of unemployment duration.
One purpose of this paper is to determine the underlying probability density
function (pdf) when the observed samples are contaminated by the digit preference e!ect. The justi"cation for using a maximum entropy estimation method to
remove the digit preference e!ect is the following. Zellner and High"eld (1988)
and Ryu (1993) derived an exponential polynomial series for an unknown
density function when they maximized entropy subject to a given set of
moments. We call this maximum entropy estimated density a &pseudo'-true
density function because it approaches the true density function under certain
conditions which will be stated later. Under the assumption of a local departure
due to digit preference, it will produce an insigni"cant e!ect on the estimated
sample moments of lower order.1 It is hoped that the digit preference distortion
component has negligible in#uence on the lower ordered sample moments, and

out pseudo-true density function is a good approximation of the underlying true
density function. Based on this pseudo-true density function, we can decompose
each observation into two parts, a pseudo-&true' value part and a component
due to the digit preference distortion.
As an alternative way to remove the digit preference e!ect, local correction
approaches can be applied. Pickering (1992) introduced a Markov transition
where the model is misclassi"ed because there is a possibility of misclassifying
odd observations to adjacent even observations, while even numbers are
assumed correctly reported. Ridout and Morgan (1991) apply the Beta-geometric density function as the true underlying density function and make a local
1 Suppose "ve sample points (x!2), (x!1), x,(x#1), (x#2) were misreported as "ve observations of x. The mean is the same in both samples, but the variance of the observed samples will be
decreased from (x!x6 )2#2 to (x!x6 )2 if we use a digit distorted sample. However, if x6 is a large
number or if there are many observations that are not a!ected by the digit preference e!ect, then the
di!erence in variances will be negligible.

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

133

correction in such a way that the probability of misreporting is symmetric
around the local maximum and decreases as the distance from the local maximum increases. However, these local correction models are based on the "xed

form approach so that their global properties may or may not be satisfactory.
The second purpose of this paper is to determine the unemployment duration
density for coarse sample observations. In particular, the Handbook of Labor
Statistics (1989) reports the number of persons unemployed for 4, 10, 14, 26, 52
weeks for the period 1958}1987. Additional observations for 1988}1994 were
taken from Bureau of Labor Statistics monthly reports on the Employment
Situation. The data we used are censored, and they are reported as 52 weeks
when in fact they could be for 200 weeks, this is always a problem since we only
know that individuals have been unemployed at least 52 weeks. Given "ve
sample points for each year, the global shape of an estimated density cannot be
very close to the true density at all points, but if the duration density is a smooth
function particularly for a longer spell of unemployment, an approximation with
an exponential series can do well over this region. Once the density function is
derived, we can easily "nd quintile values and attendant Lorenz curves. Once
the Lorenz curve is uncovered, we can calculate inequality in unemployment
duration over time. Finally, we can then "nd the Gini coe$cient of duration
density and discuss its relationship with the total accumulated duration of
unemployment in weeks.
There are several previous works for inference from grouped data. Heitjan
(1989) de"nes grouped data be the result of observing continuous variables only

up to the nearest interval. Heitjan and Rubin (1990, 1991) proposed a groupeddata likelihood function and estimated its parameters for coarse data where all
the data fall in one of a countable number of subsets of the sample space. They
extended their model for the more general case of the data coarsening at
random. In comparison, our approach suggests a curve-"tting method with an
exponential polynomial for the distribution function where only several points
of the distribution are known due to data grouping.
The paper proceeds as follows. Section 2 lays out the methodology we will use
to derive the maximum entropy density function and compare the observed
sample moments with the estimated true sample moments for dealing with
inaccurate samples. Section 3 deals with the problem of a coarse sample,
Section 4 presents the empirical results and Section 5 concludes the study.

2. Density estimation for contaminated observations
Our objective is to determine the underlying probability density function
when the observations are measured with error. Suppose
u "x #e
i
i
i

for i"1, 2,2, n,

(2.1)

134

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

where x is the unknown true unemployment duration of the ith person while
i
u is the reported observed value of the ith person's unemployment duration. For
i
example, some people have a preference for even numbers so that he (or she) will
report the nearest even number instead of the true x . The di!erence is e .
i
i
In general, we may consider several global smoothing approaches. We could
apply a nonparametric kernel estimation method. There are many good references. Nadaraya (1965), Silverman (1986), Devroye (1987), and HaK rdle (1990)
all provide good explanation of this method. Collomb (1984), Prakasa Rao
(1983), Devroye and GyoK r" (1985), GyoK r" et al. (1989) discuss the statistical

properties of kernel estimation explicitly. However, the data we are using are
biased towards 4, 12, and 26 weeks and the location of each kernel will be
biased accordingly. The kernel estimation method preserves local information
and there is some tendency that each observed peak will remain as a peak even
after smoothing. Another problem for out particular case is that many observations are located near the boundary (1}4 weeks of unemployment).
As an alternative way to estimate the density function, we can apply a locally
weighted regression method (LOWESS) introduced by Cleveland (1979), or an
orthonormal basis (ONB) method (Ryu, 1990); Prakasa Rao, 1983. To smooth
out local digit preference e!ect, it is necessary to include a su$cient number of
samples to average out the local e!ect in the LOWESS and to keep the size of
series expansions relatively small for the ONB method. The justi"cation for
using such a semiparametric or nonparametric approach is that the digit
preference e!ect is a local e!ect so that the contribution of the digit preference
e!ect can be averaged out when the global density function is estimated. In
comparison, in this paper, we determine the global density using a maximum
entropy method subject to the lower-ordered moments and we evaluate the
degree of in#uence caused by the digit preference e!ect by neglecting the higher
ordered moments.
2.1. Review of maximum entropy density estimation
Ryu (1993) shows that we can maximize entropy (=) subject to restrictions

given in the following general form:

P

max ="! f (x)log f (x) dx satisfying
f
m"0, 1,2, J

P

/ (x) f (x) dx"m ,
m
m
(2.2)

with the m having known values. The Lagrangian method produces the followm
ing maximum entropy distribution as a solution:

C

D

N
f (x)"exp + c / (x)
n n
n/0

satisfying

P

/ (x) f (x) dx"m ,
m
m

m"0, 1,2, J.

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

135

The unknown constants c 's can be computed from known values of m . If we
n
m
choose a polynomial sequence, / (x)"xm, the density is called the exponential
m
series distribution.
Suppose a density function is given in an exponential power series. Zellner
and High"eld (1988) have shown how to determine the parameters numerically
from the J#1 moment restriction conditions:

C

f (x)"exp

J
+ c xj
j
j/0

D

satisfying

P

xmf (x) dx"k , m"0, 1,2, J. (2.3)
m

As an alternative way to determine the parameters, Ryu (1993) has shown an
explicit parameter determination rule assuming knowledge of 2J#1 moment
restriction conditions:

C

f (x)"exp

J
+ c xj
j
j/0

D

satisfying

P

xmf (x) dx"k , m"0, 1,2, 2J. (2.4)
m

See Theorem A.1 stated in the appendix for details. The major di!erence is that
only J#1 conditions are necessary to solve (2.3) numerically, but 2J#1
conditions are needed to solve (2.4) explicitly.
Until now, we have assumed that the moments have known values. If we
consider the e!ect of sampling error on parameter estimation that occurs
because the known moment k is replaced by its sample mean, k( "
m
m
(1/n)+n xm, two types of departures are expected. The true density function
i/1 i
may not be the same with the above exponential polynomial series and the
sample moments will be di!erent from the true moments (even if we assume
knowledge of the true density function).
Barron and Sheu (1991), Zellner and High"eld (1988), Aroian (1948) and
others have shown that maximum likelihood estimation of parameters are
mathematically the same as those given above in our maximum entropy procedure with sample moments replacing the given moments in (2.3). With
these estimated parameters, the model estimated moments will converge to the
sample moments. Similarly, Ryu (1993) shows how to estimate parameters when
sample moments replace the given moments in (2.4). See Theorem 2 of Ryu
(1993) stated in the appendix. In practice, the functional form of true pdf can be
well approximated with an exponential series if a su$ciently large number of
terms are included in the series and the true moments can be well approximated
with sample moments if the sample size becomes large. In Section 4, we will
apply both parameter estimation methods and compare their performance.
Barron and Sheu (1991) also approximated the density functions by sequences
of exponential polynomials but they considered the relative entropy (Kullback}Leibler distance) between the true density and the estimated density. The
di!erence in the conceptual foundations of the Maximum Entropy principle and

136

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

Kullback's Relative Entropy should be noted. See for example Barron and Sheu
(1991) and Kapur and Kesavan (1992). The maximum entropy principle is
founded on the concept of the maximization of Shannon's entropy which is
essentially a measure of uncertainty. Thus, we maximize the uncertainty about
the information not given to us subject to the use of all the information given to
us. On the other hand, with relative entropy (Kullback}Leibler distance), the
underlying concept is that of the probabilistic distance of one distribution from
another. Based on the KLIC (Kullback}Leibler information criterion)
: f (x)log [ f (x)/f (x)] dx, the object is to "nd a distribution f which, out of all
1
1
2
1
the distributions satisfying the given constraints, is closest to f . Here f is our
2
1
estimated density and f is a uniform density or measure. Thus by minimizing
2
this KLIC with respect to f subject to side conditions, we get as close as possible
1
to f , uniform measure. The counterpart : f (x)log [ f (x)/f (x)] dx measures the
2
2
2
1
closeness of f to f as the base measure. Note the "rst integral above is not equal
2
1
to the second. An invariant distance measure, as indicated by Je!reys (1961) is
the sum of the above two integrals. With the sum, distance from f to f is the
1
2
same as that from f to f .
2
1
2.2. Justixcation for using a ME method to remove the digit preference ewect
We provide two justi"cations. First, to understand the relationship between
the global shape and the local "ne details of a density function, we can expand
the logarithm of the density function in a polynomial series, log f (x)"+J c xj.
j/0 j
The lower-order terms determine the global shape with fewer roots and when we
increase the expansion size, we are superimposing the "ne details of the higherorder terms (which have many roots) to the previously derived global form of
the density function. Therefore, if we want only a globally smooth function, then
we can keep the size of the exponential series small such as J"4 (or J"6). To
estimate the parameters of a short series, we only require the lower order sample
moments.
As the second justi"cation for using observed sample moments which are
distorted by the digit preference e!ect, we separate the observed value u into
i
a true observation x and an error term e as stated in (2.1). Now let us
i
i
distinguish between the true sample moments (k( ), the observed sample
k
moments (m ), and the sample error moment (l( ),
k
l
1 n
True sample moments: k( " + xk.
k n
i
i/1
1 n
Observed sample moments: fK " + uk.
i
k n
i/1
1 n
Sample error moments: l( " + el.
l n
i
i/1

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

137

Next decompose the true sample moments with the observed sample moments
and the sample error moments,

AB

1 n
k k
1 n
k( , + xk" + (u !e )k" +
(!1)lf
l(
i n
i
i
k n
(k~1) l
l
i/1
i/1
l/0
k(k!1)
f
l( !2 .
"f !kf
l( #
(k~2) 2
k
(k~1) 1
2

(2.5)

In (2.5), higher-order terms can be neglected if f Akf v( A(k(k!1)/2)
k
k~1 1
f
l( A(k )f
l( A2.
(k~2) 2 3 (k~3) 3
To evaluate the e!ect of digit preference, we can decompose the error term
into two parts:
e "u !x "(u !x8 )#(x8 !x ),
(2.6)
i
i
i
i
i
i
i
where x8 is the theoretical sample generated by the maximum entropy pdf using
i
the observed sample moments f ,
k
J
fK (x)"exp + c( xj ,
(2.7)
j
j/0
where parameter estimation methods are already discussed. We refer to the
above maximum entropy density as a pseudo true density function because it
converges to the true density under certain conditions.2 In (2.6), we have
decomposed the error term into two parts, the "rst part is the di!erence between
the observed value (u ) from the model estimated value (x8 ) and the second part is
i
i
the di!erence between the true value (x ) and the model estimated value (x8 ).
i
i
We assume that the true population pdf and the maximum entropy estimated
pdf's are very smooth so that all the digit preference e!ects will be con"ned to
the "rst part of the RHS term of (2.6).3
Now let us approximate the sample error moments with the "rst part of (2.6),

C

D

1 n
1 n
l( " + el+ + (u !x8 )l for l"1, 2,2
(2.8)
l n
i n
i
i
i/1
i/1
Once we have approximated the values of l( , we can evaluate the role of l( in
l
l
(2.5). The digit preference e!ect is negligible if f Akf l( A[k(k!1)/2]f
l( .
k
k~1 1
(k~2) 2
2 Suppose the true underlying density function is a smooth function. If the observed sample
moments approach the true moments, and the size of the exponential series goes to in"nity, then we
can show that the maximum entropy density function converges globally to the true density
function.
3 Suppose unemployment duration is measured in weeks. For a 52 week measurement period,
there will be roughly 26 peaks of even numbers if more people report even numbers compared to odd
numbers. Therefore any exponential polynomial function of order much less than 26 (for example 6)
cannot in principle show the digit preference e!ect. The exponential polynomial function of lower
order is a smooth function and thus cannot show rapid oscillation.

138

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

To summarize, we have assumed that the model estimated pdf (2.7) is smooth
so that this function cannot pick up the digit preference e!ect. The second term
of (2.6) is negligible relative to the "rst term of (2.6), so our observed sample
moments can approximate the unknown true sample moments well even though
each observation may be contaminated with a digit preference e!ect. Our
suggested model will be successful if the true density function is smooth so that it
can be well approximated with an exponential series of lower j while the digit
preference e!ect is a phenomenon which impacts the higher j terms. It should be
noted that other smoothing methods which removes local #uctuations can also
be e!ective in removing the second term of (2.6). There is an analogy between
our smoothing method and the time-series method of spectral analysis. One can
remove high-frequency #uctuations by expanding the time series in a Fourier
series of lower order. Similarly, local digit preference e!ects are removed in an
exponential power series of lower order.
3. Density estimation for coarse observations
A major "nding of the previous section is that the functional form of a density
can be written in an exponential series. Furthermore, Ryu and Slottje (1996,
1998) showed that an income distribution, which has a fat tail in the right-hand
side, can be approximated well by an exponential polynomial series.
We interpret an unemployment duration share as a probability density (pdf )
function because the duration share s(z ) is the probability associated with the
i
possibility that each week of total measured unemployment will end up in the ith
person.4 The population index z is de"ned such that the person at z"0 is
employed full time and the person at z"1 is unemployed for 52 weeks.
4 Suppose we have unemployment sample observations. Let this sample be rearranged in order from
lowest to greatest values and let the ordered values be (u , 2, u ) where I is the sample size. Dividing
0
I
each measured unemployed weeks by total measured unemployed weeks, the share function is de"ned as
u
i .
s,
i +I u
i/0 i
We interpret the share function as a probability density function because the share s is the
i
probability associated with the probability density that each weeks of total measured unemployed
weeks will end up with the ith person. Since each individual has di!erent attributes and a di!erent
locational position, each unemployment week will end up in the hand of the ith individual with
probability s so that this share function can be considered as a pdf.
i
Once we have considered the share function as a density function, we can apply the maximum
entropy method to determine the functional form of the share function. See Ryu (1993) for examples
of this method. Let us approximate the logarithm of s with a polynomial series:
i
N
log s(z)" + a zn#f,
n
n/0
where s is replaced with s(z) for continuous z.
i

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

139

We introduce a very simple model,
(3.1)
log s(z)"a@ #a z#a z2#a z3#a z4#e.
0
1
2
3
4
Since the share of the person is proportional to his measured unemployed
weeks, we write
log D(z)"a #a z#a z2#a z3#a z4#e,
(3.2)
0
1
2
3
4
where D(z) refers to duration of unemployed weeks. To estimate the parameters
of (3.2), we applied the least-squares method. To evaluate the usefulness of the
above simple model, we can compare the performance of our model based on
"ve sample points with those of a full data model (or a pseudo-&true' model).
Once the share function s(z) is derived, quintile values are

P

Q,
j

s(z) dz,
(3.3)
Rj
where R is the domain of Q . Similarly, a Lorenz curve can be derived,
j
j
z
¸(z)" s(z@) dz@.
(3.4)
0
Each year, di!erent numbers are observed for 4, 10, 14, 26, and 52 weeks of
unemployment. The duration transition mechanism over time can be followed
through the changes in the Gini coe$cient, in a Lorenz curve, and through the
estimated quintiles themselves.

P

4. An application
4.1. The duration density for observations contaminated with the digit
preference ewect
This section serves two purposes. The "rst one is to evaluate the usefulness of
the maximum entropy method as an approach to determining the unknown
unemployment duration density function. We study how much distortion is
produced by the digit preference e!ect relative to the sample moments. Once the
pseudo-&true' density is derived, the distribution produced by the digit preference e!ect is plotted using a Boxplot for each observation group.
The second purpose of this section is to determine the density function given
the coarseness of the sample observations. Finally, we also look at the level of
inequality in the density of unemployment duration. We uncover the Gini
coe$cient for the duration density and its relationship with the total accumulated unemployment duration in weeks. We conclude the section by comparing

140

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

Table 1
Maximum entropy estimated unemployment duration probability density
Weeks Sample
obs

Observ
%

Model
PDF

Zellner
PDF

Weeks Sample
obs

Observ
%

Model
PDF

Zellner
PDF

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

2.3
4.4
5.0
11.6
2.3
5.1
2.2
6.6
1.9
4.3
0.6
9.8
3.1
1.5
1.0
3.3
2.2
1.1
0.6
3.8
0.5
3.2
0.3
1.1
0.7
6.5

2.09
3.71
5.24
6.21
6.46
6.14
5.51
4.78
4.09
3.50
3.03
2.68
2.42
2.24
2.13
2.06
2.03
2.02
2.04
2.06
2.07
2.07
2.05
2.00
1.93
1.82

3.22
4.03
4.67
5.09
5.26
5.22
5.03
4.73
4.39
4.03
3.68
3.36
3.07
2.81
2.59
2.40
2.24
2.10
1.98
1.88
1.79
1.71
1.64
1.57
1.50
1.44

27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51

0.6
0.9
0.2
1.2
0.3
2.4
0.2
0.4
0.5
1.0
0.4
0.3
0.7
1.7
0.1
0.5
0.2
0.9
0.1
0.5
0.1
0.6
0.3
0.3
0.3

1.69
1.54
1.38
1.22
1.07
0.93
0.81
0.70
0.62
0.55
0.50
0.47
0.47
0.44
0.44
0.45
0.48
0.50
0.53
0.54
0.53
0.47
0.38
0.26
0.06

1.37
1.30
1.24
1.17
1.10
1.03
0.95
0.88
0.81
0.75
0.69
0.63
0.58
0.53
0.49
0.46
0.43
0.40
0.39
0.38
0.37
0.38
0.39
0.41
0.44

218
411
467
1079
216
474
206
614
181
402
58
915
293
139
97
307
208
107
60
359
50
295
24
104
69
603

57
88
17
116
26
220
15
37
50
96
34
26
62
161
8
47
18
87
11
44
12
53
31
28
25

Observed samples are such that 218,000 persons reported one week of unemployment. Similarly,
411,000 persons reported 2 weeks of unemployment. The third column is histogram for the observed
samples of the second column. The fourth column is the maximum entropy density functions
estimated using Theorems A.1 and A.2 of the appendix and the "fth column is estimated using (2.3)
replacing the given moments with the sample moments. The size of the expansion series (J ) is 6.

the inequality measure of unemployment in each year to the headcount ratio
(the traditionally reported unemployment rate) and to the mean duration of
unemployment.
If the Gini coe$cient of unemployment duration exhibits di!erent behavior
over time vis-a`-vis the headcount ratio or in comparison to the mean duration
measure, then this result is of considerable signi"cance. Every unemployment
measure can be considered to be an indicator of economic well being. If di!erent
measures give con#icting signals of well being over time, then researchers and
policy makers alike should be aware of the multidimensional aspect of unemployment. We will say more on this below.

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

141

Fig. 1. ME estimated PDF from raw frequency.

In Table 1, weeks of unemployment are listed in the "rst column, the observed
numbers of corresponding weeks are reported in the second column, a histogram is listed (the proportion of persons belonging in that group) in the third
column, and the maximum entropy estimated density function estimated by
Theorems A.1 and A.2 (reported in the appendix) is listed in the fourth column.
The maximum entropy density function estimated by maximum likelihood
method is listed in the "fth column.
In Fig. 1, we plot the raw frequencies (third column of Table 1) and the
estimated densities by MOM and MLE (fourth column and "fth columns of
Table 1). We normalized the density function so that the area below the curve is
one.
To evaluate the degree of distortion produced by the digit preference e!ect, we
generate theoretical sample observations which we called the &pseudo'-true
samples from the maximum entropy estimated pdf by inverting the distribution
function
F(x8 )"i/nNx8 "F~1(i/n) for i"1, 2,2, n.
(4.1)
i
i
Therefore, the observed sample u can be decomposed into two parts, the
i
theoretical sample observation part and departure of it from the observed value.
u "x8 #f for i"1, 2,2, n.
(4.2)
i
i
i
Approximating the sample error moments of (2.8) as l( "(1/n)+n fl, the degree
i/1 i
l
of the digit preference in#uence can be evaluated using (2.5). In Table 2, we list

142

Table 2
Comparison of sample moments with "rst and second moment correction terms
fK

1
2
3
4
5
6
7
8
9
10
11
12

14.89
358.8
11090
393300
0.1519E#8
0.6202E#9
0.2634E#11
0.1151E#13
0.5142E#14
0.2334E#16
0.1074E#18
0.4992E#19

l(
0.0938
0.7503
0.1425
1.308
0.1856
3.090
0.0311
8.677
!1.378
27.27
!9.105
92.64

First

First/fK

Second

Second/fK

!0.0938
!2.792
!101.0
!4160
!0.1844E#6
!0.8545E#7
!0.4072E#9
!0.1977E#11
!0.9719E#12
!0.4822E#14
!0.2408E#16
!0.1208E#18

!0.0063
!0.0078
!0.0091
!0.0106
!0.0121
!0.0138
!0.0154
!0.0172
!0.0189
!0.0207
!0.0224
!0.0242

0.0000
0.7504
!33.51
!1615
!0.8321E#5
!0.4427E#7
!0.2393E#9
!0.1303E#11
!0.7117E#12
0.3888E#14
!0.2122E#16
!0.1156E#18

0.0000
0.0021
!0.0030
!0.0041
!0.0055
!0.0071
!0.0091
!0.0113
!0.0138
!0.0167
!0.0198
!0.0232

The observed duration of unemployed week u is divided into true unobserved duration x and error e . If we de"ne true sample moment as
i
i
i
k( "(1/n)+n xk, observed sample moments as fK "(1/n)+n uk, and the sample error moments as l( "(1/n)+n el, then we can expand the true moment
k
i/1 i
k
i/1 i
l
i/1 i
in a series of observed sample moments and sample error moments as shown in (2.5):
k(k!1)
fK
l( #2
k( "fK !kfK
l( #
(k~2) 2
k
k
k~1 1
2
The &First' in the fourth column means the "rst right-hand side correction term of (2.5) !kfK
l( . Similar interpretation goes for the &Second'.
k~1 1

(2.5a)

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

Moments

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

143

observed sample moments f , sample error moments l , the "rst correction part
k
l
and the second correction part of (2.5). We also list the ratio of the "rst (as well
as second) correction part relative to the observed sample moments to see their
in#uence in the sample moment estimation. In all cases, the correction components appear to have little in#uence. There is a signi"cant digit preference e!ect
where reporting 4 weeks is more likely than 3 or 5 weeks. Ultimately this not
taken into account in the estimation, but expansions on Section 2.2 are used to
argue the e!ect is small, provided su$cient smoothing is done and there are no
genuine high frequencies in the data.
Now let us compare the performance of Ryu's MOM with that of MLE. For
our digit preference problem, we could not keep the size of the exponential series
very large because the digit preference e!ect is assumed to be an e!ect on the
higher ordered term in the series. Furthermore, only the lower-ordered sample
moments are assumed to be free of the digit preference e!ect and thus the
sampling error for using the sample moments, not the true moments, may not be
negligible for higher-ordered sample moments. In these respects, MLE looks to
be a better choice for this digit preference problem. However, Ryu's MOM
produced better performance in the goodness of "t than MLE. The use of
higher-ordered sample moments (J#1th,2, 2Jth moments) though not necessarily accurate seemed to be helpful in removing the digit preference e!ect. The
sample error moments l( "(1/n)+n ek stated in (2.8) are
i/1 i
k
Sample error moments comparison for two-parameter estimation methods
Order
1
2
3
4
5
6

MOM
0.0938
0.7503
0.1425
1.308
0.1856
3.090

MLE
0.1601
0.7475
0.3588
1.432
1.133
4.076

Order
7
8
9
10
11
12

MOM
0.0311
8.677
!1.378
27.27
!9.105
92.64

MLE
4.459
14.52
20.06
59.75
98.03
271.8

Note the sample error moments produced by MLE are a little bit larger than
those produced by MOM. Therefore, the digit preference e!ect stated in (2.5) is
less negligible for the ML estimated method.
In Fig. 2, we present a Boxplot consisting of the digit preference distortions f .
i
If everyone has a certain preference for an odd number or an even number, then
the di!erence between the observed sample and the theoretical sample values
should be con"ned between !1 and #1 with mean zero. However, in Fig. 2, we
see that people prefer to report 12, 26, 32 and 40 weeks. For example, those with
a true unemployment duration of 7, 8, 9, 10, and 11 weeks report upward biased
numbers so that they have a tendency to consider 12 weeks as a key level of
unemployment duration. Meanwhile, those with 13, 14, 15, 16, 17, 18, and 19
weeks report a downward biased number so that once again 12 weeks is a key

144

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

Fig. 2. Boxplots of the departures due to digit preference.

duration level. Similar e!ects can be observed around other key duration weeks,
i.e., 26, 32, and 40 weeks. Therefore we can conclude that there is more than
a simple odd/even digit preference e!ect at work here.
4.2. Duration density from coarse sample observations
Our objective is to derive the whole distribution based upon "ve sample
observation points. We use (3.2) and the parameters are estimated by the
least-squares method. The performance of the estimated distribution can be
compared to the cumulative distribution based on the full sample data. In
Table 1, we have a list of full data, but let us use only the following "ve points.
Weeks
4
10
14
26
52

Cumulative %
z " 23.3
1
z " 45.8
1
z " 60.8
1
z " 85.3
1
z "100.0
1

We approximated the duration weeks D(z) with
log D(z)"a #a z#a z2#a z3#a z4#e
(3.2)
0
1
2
3
4
and the parameters were estimated by the least-squares method using "ve points
at D(z )"4, D(z )"10,2.
1
2

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

145

Table 3
Approximation of unemployment duration with an exponential polynomial series
Weeks

Freq

Cumulat.
dist

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

218
411
467
1079
216
474
206
614
181
402
58
915
293
139
97
307
208
107
60
359
50
295
24
104
69
603

2.3
6.7
11.8
23.3
25.6
30.7
32.9
39.5
41.5
45.8
46.4
56.2
59.3
60.8
61.9
65.2
67.4
68.5
69.2
73.0
73.6
76.7
77.0
78.1
78.9
85.3

Model
dist
6.9
14
19
23
27
31
35
38
42
46
50
53
58
61
64
67
70
73
75
77
79
81
82
83
85
86

Weeks

Freq

Cumulat.
dist

Model
dist

27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51

57
88
17
116
26
220
15
37
50
96
34
26
62
161
8
47
18
87
11
44
12
53
31
28
25

85.9
86.9
87.1
88.3
88.6
90.9
91.1
91.5
92.0
93.1
93.4
93.7
94.4
96.1
96.2
96.7
96.9
97.8
97.9
98.4
98.5
99.1
99.4
99.7
100.0

87
87
88
89
90
91
91
92
93
93
94
94
95
95
96
96
97
97
98
98
98
99
99
99
100

Observed samples are such that 218,000 persons reported one week of unemployment. The third
column is the distribution for the observed samples of the second column. The fourth column is the
estimated distribution using only "ve sample observation at 4, 10, 14, 26, and 52 weeks. We
approximated the duration weeks D(z) with
log D(z)"a #a z#a z2#a z3#a z4#e
0
1
2
3
4

(3.2b)

and the parameters were estimated by the least-squares method using "ve points at
D(z )"4, D(z )"10,2. The fourth column is estimated with z"D~1 (weeks).
1
2

In Table 3, we reported the observed distribution in the third column and
model estimated values in the fourth column. For the weeks of 1, 2, and 3, model
estimated values were larger than the observed values. Due to a small sample
problem, we could not pick up the detailed shape of the density near the origin.
However, for the remaining regions, its performance seemed to be very good.

146

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

Once we have estimated the whole distribution function, let us compare the
quintile shares to show how the total accumulated unemployed weeks are
distributed into the "rst 20 per cent of persons who reported unemployment,
second 20 per cent, 2
Observed
Model estimated
Q
0.0409
0.0237
1
Q
0.0862
0.0840
2
Q
0.1512
0.1605
3
Q
0.2662
0.2487
4
Q
0.4555
0.4831
5
Though we have used only "ve sample points, the estimated quintile values
are good approximations of the observed quintile values. At the tail areas,
Q and Q had more departures than did Q , Q , Q in the middle of the
1
5
2 3 4
distribution.
Once we have seen that (3.2) works well for the coarse observations, we can
apply this method to actual unemployment duration data collected by the
Bureau of Labor Statistics. In the estimation performed, we included only those
individuals who experienced unemployment for a portion of 52 weeks, but
excluded those who never worked or never experienced unemployment. We also
include one arti"cial sample point, so that 1 per cent of people in the sample are
considered to be unemployed for 0.3 of a week. In (3.2), we need D(z)"0 for
z"0, but this obviously is impossible for the logarithmic form given in (3.2).
The reason for including the arti"cial sample point is the following. For the
years 1966}1969, the performance of (3.2) was not satisfactory. These are &baby'
boom years and their distribution was quite di!erent from the distribution in
other years. Therefore, we applied (3.2) for the years 1958}1965, and found on
average 1 per cent of people were unemployed for 0.3 week. This condition is
imposed rather than attempting an impossible boundary condition, D(0)"0
over the problem period. We derived the duration density function based on
D(0.01)"0.3, D(z )"4, D(z )"10, D(z )"14, D(z )"26, and D(z )"52 for
1
2
3
4
5
1958}1994 and we report the quintile values in Table 4 for the entire population.
The Gini coe$cient is a convenient summary measure to describe the
distributive nature of a given distribution and it can be calculated from the mean
and higher moments of a given distribution. Nonetheless, a brief discussion is
probably necessary since the application here is not the usual one. When a Gini
coe$cient increases in value towards one, this indicates that the level of inequality has increased. How does one interpret this fact in the context of the
distribution of unemployment duration? Shorrocks (1992, 1993) argues that an
increase in the level of inequality in the unemployment duration distribution
means social welfare has decreased. This is because the Pigou}Dalton principle
applies. An increase of (say) one week of unemployment for one individual with
no change in anyone else's status will certainly make the level of social welfare

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

147

Table 4
Quintile values, Gini coe$cient, and average weeks of unemployment duration (total unemployment of men and women)
Year

Q

1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994

0.0196
0.0181
0.0185
0.0174
0.0186
0.0185
0.0170
0.0146
0.0124
0.0126
0.0109
0.0120
0.0169
0.0179
0.0159
0.0148
0.0164
0.0189
0.0187
0.0174
0.0173
0.0173
0.0199
0.0213
0.0252
0.0229
0.0202
0.0203
0.0215
0.0209
0.0201
0.0202
0.0210
0.0187
0.0199
0.0236
0.0273

1

Q

2

0.0827
0.0754
0.0769
0.0758
0.0790
0.0802
0.0725
0.0625
0.0508
0.0522
0.0435
0.0484
0.0723
0.0774
0.0687
0.0629
0.0693
0.0833
0.0811
0.0750
0.0743
0.0746
0.0856
0.0911
0.1008
0.0954
0.0866
0.0877
0.0919
0.0896
0.0861
0.0867
0.0903
0.0799
0.0845
0.0963
0.1050

Q

3

0.1522
0.1493
0.1497
0.1555
0.1543
0.1574
0.1520
0.1490
0.1364
0.1400
0.1304
0.1343
0.1530
0.1560
0.1515
0.1475
0.1494
0.1600
0.1567
0.1536
0.1539
0.1551
0.1581
0.1600
0.1518
0.1554
0.1575
0.1599
0.1599
0.1596
0.1581
0.1606
0.1616
0.1571
0.1574
0.1566
0.1511

Q

4

0.2570
0.2500
0.2512
0.2573
0.2509
0.2518
0.2516
0.2509
0.2490
0.2478
0.2485
0.2468
0.2480
0.2542
0.2567
0.2518
0.2500
0.2571
0.2554
0.2541
0.2495
0.2486
0.2520
0.2481
0.2460
0.2516
0.2540
0.2514
0.2491
0.2490
0.2488
0.2446
0.2456
0.2363
0.2373
0.2270
0.2170

Q

5

0.4884
0.5072
0.5038
0.4939
0.4972
0.4921
0.5070
0.5230
0.5513
0.5474
0.5667
0.5584
0.5099
0.4945
0.5071
0.5230
0.5148
0.4806
0.4881
0.4999
0.5049
0.5043
0.4844
0.4795
0.4762
0.4745
0.4816
0.4807
0.4776
0.4808
0.4868
0.4879
0.4814
0.5081
0.5010
0.4965
0.4996

Gini

AVWEEKS

0.4561
0.4760
0.4721
0.4674
0.4659
0.4615
0.4793
0.5007
0.5319
0.5279
0.5503
0.5391
0.4814
0.4661
0.4835
0.5000
0.4876
0.4510
0.4580
0.4720
0.4760
0.4754
0.4507
0.4418
0.4285
0.4337
0.4479
0.4463
0.4398
0.4442
0.4517
0.4519
0.4339
0.4724
0.4628
0.4470
0.4390

11.07
9.58
9.85
10.11
9.82
9.89
9.28
8.27
7.52
7.39
7.02
7.21
8.83
9.91
9.42
8.49
8.90
10.76
10.45
9.75
9.16
9.04
10.51
10.58
11.92
11.78
10.91
10.60
10.81
10.54
10.19
9.93
10.11
8.42
8.98
9.02
9.00

decline. Thus, we assume that higher values of the Gini coe$cient are a worse
state of nature than lower values. An increase in the unemployment rate and/or
in the mean duration of unemployment have the same interpretation. From the
derived density, the Gini coe$cients are calculated and reported in the seventh
column of Table 4.

148

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

Table 5
Quintile values, Gini coe$cient, and average weeks of unemployment duration (unemployment of
men)
Year

Q

1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992

0.0231
0.0218
0.0218
0.0219
0.0222
0.0218
0.0200
0.0170
0.0149
0.0150
0.0126
0.0145
0.0199
0.0205
0.0191
0.0179
0.0193
0.0238
0.0222
0.0212
0.0215
0.0199
0.0247
0.0247
0.0306
0.0270
0.0245
0.0247
0.0260
0.0264
0.0231
0.0231
0.0239
0.0187
0.0217

1

Q

2

0.0935
0.0891
0.0894
0.0914
0.0921
0.0926
0.0840
0.0742
0.0631
0.0645
0.0534
0.0613
0.0842
0.0876
0.0819
0.0778
0.0816
0.0999
0.0925
0.0904
0.0905
0.0804
0.1009
0.1014
0.1114
0.1044
0.0944
0.1017
0.1045
0.1062
0.0962
0.0975
0.1000
0.0799
0.0901

Q

3

0.1505
0.1530
0.1534
0.1570
0.1567
0.1607
0.1558
0.1582
0.1489
0.1525
0.1450
0.1488
0.1578
0.1590
0.1571
0.1587
0.1559
0.1606
0.1558
0.1603
0.1585
0.1425
0.1564
0.1578
0.1412
0.1473
0.1533
0.1583
0.1555
0.1563
0.1582
0.1626
0.1613
0.1571
0.1565

Q

4

0.2494
0.2430
0.2437
0.2447
0.2427
0.2427
0.2442
0.2472
0.2447
0.2439
0.2471
0.2434
0.2413
0.2470
0.2479
0.2454
0.2445
0.2450
0.2491
0.2450
0.2407
0.2633
0.2411
0.2420
0.2351
0.2442
0.2465
0.2416
0.2401
0.2374
0.2430
0.2389
0.2410
0.2363
0.2334

Q

5

0.4835
0.4932
0.4917
0.4851
0.4863
0.4822
0.4960
0.5034
0.5283
0.5240
0.5418
0.5320
0.4969
0.4859
0.4940
0.5002
0.4986
0.4707
0.4804
0.4831
0.4888
0.4939
0.4768
0.4742
0.4817
0.4771
0.4762
0.4737
0.4739
0.4737
0.4794
0.4779
0.4738
0.5081
0.4983

Gini

AVWEEKS

0.4411
0.4522
0.4509
0.4448
0.4448
0.4419
0.4597
0.4754
0.5035
0.4994
0.5232
0.5080
0.4603
0.4496
0.4609
0.4695
0.4640
0.4267
0.4406
0.4448
0.4483
0.4615
0.4292
0.4271
0.4190
0.4248
0.4302
0.4265
0.4234
0.4214
0.4360
0.4340
0.4286
0.4724
0.4549

11.50
10.09
10.19
10.34
10.20
10.05
9.53
8.68
7.77
7.65
7.17
7.45
9.12
10.10
9.65
8.84
9.30
11.24
11.09
10.11
9.67
11.93
11.04
11.15
12.43
12.37
11.72
11.10
11.38
11.18
10.63
10.04
10.68
8.42
9.15

In Tables 5 and 6, we repeat the analysis done in Table 4 using the unemployment data for men only and women only. We note that the average number of
unemployed weeks for women is shorter than that of men. This suggests that
women have a higher chance on average of "nding new jobs. Since women have
a higher Gini coe$cient with a lower Q and higher Q values, the burden of
1
5
unemployment is borne by a relatively small portion of women so that those

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

149

Table 6
Quintile values, Gini coe$cient, and average weeks of unemployment duration (unemployment of
women)
Year

Q

1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992

0.0146
0.0132
0.0140
0.0112
0.0140
0.0143
0.0131
0.0118
0.0095
0.0098
0.0091
0.0093
0.0134
0.0147
0.0122
0.0120
0.0135
0.0149
0.0156
0.0142
0.0137
0.0138
0.0158
0.0182
0.0202
0.0189
0.0166
0.0164
0.0175
0.0164
0.0171
0.0179
0.0172
0.0187
0.0177

1

Q

2

0.0609
0.0519
0.0557
0.0496
0.0585
0.0613
0.0552
0.0466
0.0354
0.0370
0.0326
0.0331
0.0561
0.0635
0.0518
0.0474
0.0552
0.0643
0.0677
0.0586
0.0575
0.0588
0.0683
0.0787
0.0843
0.0812
0.0711
0.0716
0.0766
0.0706
0.0735
0.0725
0.0741
0.0799
0.0768

Q

3

0.1397
0.1293
0.1321
0.1415
0.1408
0.1449
0.1406
0.1295
0.1148
0.1184
0.1099
0.1100
0.1414
0.1480
0.1385
0.1288
0.1371
0.1471
0.1512
0.1383
0.1421
0.1452
0.1505
0.1570
0.1517
0.1548
0.1509
0.1531
0.1564
0.1512
0.1540
0.1446
0.1535
0.1571
0.1570

Q

4

0.2647
0.2531
0.2577
0.2768
0.2592
0.2635
0.2610
0.2505
0.2506
0.2485
0.2458
0.2457
0.2553
0.2627
0.2665
0.2542
0.2534
0.2655
0.2591
0.2595
0.2552
0.2550
0.2610
0.2522
0.2559
0.2582
0.2571
0.2589
0.2563
0.2567
0.2496
0.2452
0.2534
0.2363
0.2422

Q

5

0.5200
0.5525
0.5405
0.5209
0.5275
0.5161
0.5300
0.5616
0.5897
0.5863
0.6026
0.6019
0.5339
0.5112
0.5310
0.5577
0.5407
0.5083
0.5064
0.5295
0.5315
0.5272
0.5053
0.4939
0.4880
0.4870
0.5044
0.5000
0.4932
0.5051
0.5058
0.5198
0.5018
0.5081
0.5062

Gini

AVWEEKS

0.4998
0.5305
0.5183
0.5137
0.5075
0.4973
0.5129
0.5429
0.5739
0.5701
0.5854
0.5844
0.5145
0.4918
0.5173
0.5392
0.5196
0.4890
0.4842
0.5085
0.5114
0.5074
0.4828
0.4642
0.4541
0.4571
0.4791
0.4759
0.4662
0.4801
0.4775
0.4870
0.4742
0.4724
0.4745

10.01
8.45
9.14
9.62
9.07
9.61
8.86
7.59
7.17
7.01
6.84
6.87
8.39
9.62
9.08
8.04
8.35
10.07
9.58
9.26
8.51
8.46
9.79
9.79
11.14
10.89
9.77
9.90
10.04
9.65
9.08
9.07
9.58
8.42
8.74

women belonging to Q and Q will have real di$culty in "nding new jobs while
4
5
those women belonging to Q and Q su!er only shorter spells of number of
1
2
unemployed weeks. In comparison, those men belonging to Q and Q will su!er
1
2
longer spells of number of unemployed weeks relative to those of women
because the average number of unemployed weeks of men are longer than those
of women and Q and Q values of men are larger than the Q and Q for
1
2
1
2
women.

150

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

Table 7
Pearson and Spearman rank correlation coe$cients
Year

GINI (G)

AVWEEKS (=)

UN RATE (;)

1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994

15
26
23
21
19
17
28
33
35
34
37
36
29
20
30
32
31
12
16
22
27
25
11
5
1
2
10
8
4
7
13
14
6
24
18
9
3

35
17
20
25
19
21
15
5
4
3
1
2
8
22
16
7
9
32
27
18
14
13
28
30
37
36
34
31
33
29
26
23
24
6
10
12
11

26
11
14
23
12
17
8
5
4
3
2
1
6
19
15
7
16
35
34
29
20
18
28
33
37
36
32
30
27
21
13
9
10
24
31
25
22

Rank (G.=.)"!0.8112, Rank (G.;.)"!0.7409, Rank (=.;.)"0.7267. The Null hypothesis of
no association is rejected for all three cases because Prob (R'0.478)"0.005 if n"30.

In Table 7 we present the Pearson and Spearman rank correlation coe$cients. In the second column, we report the rank of the Gini coe$cient. The
smallest Gini is ranked number 1. We note the severe recession in 1982 so that
unemployment is experienced by more people. It is interesting to note that

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

151

Fig. 3. Av unemployed weeks vs. unemployment Gini coe$cient.

unemployment is shared by many people and the distribution has less inequality
relative to other years. In contrast, 1968 is a boom year with the smallest average
unemployed weeks and possesses the largest Gini coe$cient. Relatively few
people su!ered unemployment for short duration but the burden of unemployment is borne by those few people who were unemployed relatively long time. In
the third and fourth column, we report the rank of average unemployment
weeks and unemployment rate. These two values move together so that they
have smaller numbers (lower ranks) during the boom years and larger numbers
(upper ranks) during the recession years. The Spearman rank correlation number between the Gini and Average Weeks is !0.8112, and thus we reject the
null hypothesis of no association between the Gini and Average Weeks at the
signi"cance level of 0.005. Similarly, the Spearman rank correlation number
between the Gini and Unemployment Rate is !0.7409 and between Average
Weeks and Unemployment Rate is 0.7267.
In Fig. 3, we plotted average unemployed weeks versus unemployment Gini
coe$cients. As indicated in Table 7, smaller average weeks are combined with
larger Gini coe$cients.
Table 8 shows the relationship between the change in the Gini coe$cient and
the change in the mean duration of unemployment and the head count ratio
over time. Recalling Table 4, we see that from (say) 1965 to 1966 the Gini
coe$cient increased from 0.5007 to 0.5319 while the mean duration of unemployment fell from 8.27 to 7.52 weeks. In Table 8, this is re#ected in a &#' in the
second column and a &!' in the third column. If the Gini coe$cient increases in
value for a year to year change and the mean duration and head count ratio

152

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

Table 8
Comovement of Gini, AV weeks, and unemployment rate
Year

GINI

AVWEEKS

UNEMPL

1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994

#
!
!
!
!
#
#
#
!
#
!
!
!
#
#
!
!
#
#
#
!
!
!
!
#
#
!
!
#
#
#
!
#
!
!
!

!
#
#
!
#
!
!
!
!
!
#
#
#
!
!
#
#
!
!
!
!
#
#
#
!
!
!
#
!
!
!
#
!
#
#
!

!
#
#
!
#
!
!
!
#
!
!
#
#
!
!
#
#
!
!
!
!
#
#
#
!
!
!
!
!
!
!
#
#
#
!
!

decrease in value, one could conclude there is an unambiguous increase in social
welfare. For example, from 1958 to 1959, the Gini value increases (indicating
more inequality in the distribution of unemployment) but the various unemployment ratios fell too. How would a policy maker interpret this result?

H.K. Ryu, D.J. Slottje / Journal of Econometrics 95 (2000) 131}156

153

Obviously, the 1958}1959 changes re#ect less mean unemployment duration but
more hardship for a small number of the population. This group is
likely the hardcore unemployed and might well deserve special emphasis
in the implementation of training programs. This holds for 1958}1994 except
for the years of 1961}1962, 1966}1967, 1978}1979, 1984}1985, and 1993}1994.
A policymaker or government agency that focused only on the head count
ratio or the mean duration of unemployment would have concluded that
welfare had increased. The Gini coe$cient sent a di!erent message, a certain
portion of the population was hit disproportionately hard. The point is that
additional information from examining the Gini coe$cient of unemployment
duration can be relevant in targeting certain groups of the population for policy
initiative