Income inequality typically declines with the length of time taken into account for measurement. This note derives an exact analytical relationship between the accounting period and inequality as measured by the Gini index. The formal relationship is similar to the decomposition of the coefficient of variation. The methodology is illustrated with panel data on urban wages from Mexico. It is found that the effect of the accounting period on inequality is sensitive to the properties of the Gini correlations between the periodical incomes. Reporting this type of correlation enables the evaluation of the impact of the length of the accounting period on inequality.

Quentin Wodon

Shlomo Yitzhaki

English

Research Papers in Economics

Inequality and the accounting period 
Quentin Wodon Shlomo Yitzhaki
World Bank Hebrew University
Abstract
Income inequality typically declines with the length of time taken into account for
measurement. This note derives an exact analytical relationship between the accounting
period and inequality as measured by the Gini index. The formal relationship is similar to the
decomposition of the coefficient of variation. The methodology is illustrated with panel data
on urban wages from Mexico. It is found that the effect of the accounting period on
inequality is sensitive to the properties of the Gini correlations between the periodical
incomes. Reporting this type of correlation enables the evaluation of the impact of the length
of the accounting period on inequality.
This paper was funded by a research grant from the World Bank. The views expressed here are those of the authors and need not
reflect those of the World Bank.
Citation: Wodon, Quentin and Shlomo Yitzhaki, (2003) "Inequality and the accounting period." Economics Bulletin, Vol. 4,
No. 36 pp. 1−8
Submitted: September 12, 2003.  Accepted: December 6, 2003.
URL: http://www.economicsbulletin.com/2003/volume4/EB−03D30005A.pdf1.  Introduction 
It is well known that income inequality typically declines with the length of time taken 
into account for measurement (e.g., Creedy 1979, 1991).  Recent evidence from Gibson, Huang 
and Rozelle (2001) demonstrates that inequality in urban China is low relative to other countries 
in large part because the accounting period is based on a full year of income, while in many other 
countries, surveys record household income for a month or less. They estimate that if the 
accounting period in China would be one month, then inequality would be between 17 and 69 
percent higher. Similar findings have been observed for the United States and Germany by 
Burkhauser and Poupore (1997). Other statistics may also be sensitive to the accounting period. 
For example, Behrman and Taubman (1989) find that the estimated inter-generational correlation 
of parental income and off-springs is 0.58 when ten years of earnings are used, compared to 0.37 
for a single year (see also Bowles and Gintis 2002, for additional evidence on the impact of the 
accounting period.) 
In this note, using information on the Gini correlation of income between different time 
periods, we derive an exact analytical relationship between inequality and the length of the 
accounting period for the Gini index of inequality.  The methodology is illustrated with panel 
data on urban wages from Mexico.  While we focus our discussion on the accounting period, our 
decomposition could have many other useful applications since it is very similar to the 
decomposition of the widely used coefficient of variation, and thereby simple to implement.   
 
2. Methodology 
 
This section derives the relationship between the Gini index of inequality for the 
weighted average of several time periods and the Gini inequality indices for each period taken 
separately. To simplify notation, we restrict the proof to two periods. The extension to many 
periods is immediate. Let (Y1,Y2) be drawn from a bi-variate continuous distribution, where Yi is 
the income distribution in period i. Let Y0 = b1Y1 + b2 Y2, where bi > 0 (i=1,2) is a constant. The 
value of the constant determines whether we are dealing with the sum of the incomes, or the 
average income, possibly weighted. For example, if b1= b2= 0.5 then Y0 represents the straight 
average income measured over two periods. Denoting by F(Yi) the cumulative distribution and µi 
the expected income, the Gini coefficient (Lerman and Yitzhaki 1984) is: 
          
Gi = 2 cov(Yi,F(Yi))/ µi  .                                                                 (1) 
 
Denote by 
)) F(Y , COV(Y
)) F(Y , COV(Y
Γ
i i
j i
ij = , i, j = 0, 1, 2, the Gini correlation between incomes 
from periods Yi and Yj, or by extension between income from one period and average income. 
As discussed in Schechtman and Yitzhaki (1987, 1999), the properties of the Gini correlations 
are a mixture of Pearson’s and Spearman’s correlation coefficients. In particular, Γij is bounded 
by minus one and one, but Γij is not necessarily equal to Γji. Define also Di0 = Γi0 - Γ0i, for i=1,2 
(here, the Gini correlations are taken between the income in each of the two periods and the 
average income over time), and ai = bi (µi/µ0), where µi > 0.  
 
  1Proposition: 
 
(a) The following identity holds: 
      G                              (2)  ). Γ (Γ G G a a G a G a ]G G D a G D [a 21 12 2 1 2 1
2
2
2
2
2
1
2
1 0 2 20 2 1 10 1
2
0 + + + = + −
 
(b) Provided that Di0 =0, for i=1,2, and  Γ12 = Γ21  = Γ, then:  
 
           G
2
0 = a1
2G1
2 + a2
2 G2
2 + 2a1a2G1G2Γ.                                                                            (3) 
 
Proof: The proof is in the appendix.  
 
The extension of equations (2) and (3) to k periods is trivial. Let   , and a ∑ = =
k
1 i i i 0 Y b Y i = 
bi µi / µ0, then  
                                                             (4)  . Γ G G a a G a G D a G G
1 ij i
ij j i j i
k
1 i
2
i
2
i
k
1 i
i i0 i 0
2
0 ∑∑ ∑ ∑
=≠ = =
+ = −
 
If Di0 =0, for i=1, …, k and  Γij = Γji, then:  
                                                                                   (5)  . Γ G G a a 2 G a G
1j i
ij j i j i
k
1 i
2
i
2
i
2
0 ∑∑ ∑ + =
=< =
k
i
 
Equation (5) is identical in its structure to the decomposition of the variance and the 
coefficient of variation.  For it to hold, the Gini correlations between each pair of variables Y0, 
Y1, and Y2 must be equal. Schechtman and Yitzhaki (1987) show that a sufficient condition for 
Γij = Γji is that the variables are exchangeable up to a linear transformation. Examples of such 
distributions are the multinormal and the multivariate lognormal, provided that σi = σj, where σ is 
the logarithmic standard deviation.  If the Gini correlations between pairs of variables are not 
equal, we need to use equation (4), where each “violation” of the equality of the Gini correlations 
is captured by an additional term in the decomposition (hence, we can treat each violation 
separately and evaluate its effect on the decomposition; in particular we can see whether the 
violation tends to increase or decrease overall inequality).  Since ai < 1 for all i, it is easy to see 
that the shorter the accounting period is, the higher the inequality will be.  
Note that lower values for Γij and Γji will yield a larger decrease in the Gini index of 
inequality over several periods of time, as compared to the average level of inequality for the 
various periods taken separately. The magnitude of the (Gini) correlations between incomes in 
different time periods is thus a key factor in determining the impact of the length of the 
accounting period on measured inequality.   
Note also that it is not straightforward to assess a priori what values the Gini  correlations 
will take.  The length of the various periods taken  into account may affect the value of the Gini 
correlations in several ways.  First, we would assume that with observations corresponding to 
longer periods of time, there will be less noise in the data, so that the longer the period the larger 
will be the denominator in the expression of the correlation (as would the variance under less 
noisy data), and therefore the absolute value of the Gini correlations between longer periods will 
be higher.  On the other hand, the same reduction in noise may also increase the absolute value 
of the numerator, since less noise will tend to increase the covariance between 
  2income in one period and the rank of the individual in the distribution of income in the other 
period. The net effect of these two factors related to the impact of noise in the data is difficult to 
determine a priori. Independently of the issue of noise in the data, when working with longer 
 periods of time, employment mobility over time (with some workers gaining and some loosing) 
could also lead to lower correlations between two income observations.  This is because our 
guess is that the impact of employment mobility would tend to be larger when the time periods 
under review are themselves longer.   In general, experiments with different data sets are 
needed to enable us to get a better understanding of the issues at hand, and a better feeling for the 
values that Gini correlations may take in various settings. 
 
3. Illustration 
 
We use data from Mexico’s 1996 National Urban Employment Survey, a panel of 
individuals living in 16 different metropolitan areas. The individuals are interviewed on a 
quarterly basis for five quarters.  Table I gives the various statistical results.  The first part of the 
table gives the Gini indices of wage inequality for each quarter among males aged 15 to 65 (we 
have 6262 observations in the sample).  Wages for the five quarters are converted into real terms 
using the Consumer Price Index.  The quarterly Gini indices vary from 0.5672 to 0.6153.  Giving 
equal weight to the five periods, the income shares per quarter vary from 19.30 percent to 20.99 
percent.  The second part of the table gives the matrix of pair-wise correlations Γij between the 
various periods. As expected, the farther the periods, the lower the correlations, with the 
exception that for most quarters, the correlation with the quarter four periods later (or earlier) 
increases a bit, which suggests mild seasonality in earnings.  Another interesting property is that 
for most periods, there are differences between Γij and Γji, which means that the marginal 
distributions are not exchangeable. This implies that the shape of the marginal distribution 
change in a non-linear way.
1 
The third and fourth parts of the table provides the pair-wise correlations Γi0 and Γ0i.  The 
correlations with aggregate income tends to be higher than the correlations with another period, 
which is reasonable since the aggregate income includes as a component the periodical income. 
The larger the number of quarters included in average income, the lower the correlations are, 
since each period of time represents a smaller share of overall income.   
The last part of the table provides the decomposition from equation (4) applied to the 
various accounting periods, from two to five quarters.  Figure 1 shows how the Gini index of 
inequality is reduced by taking more periods in consideration.  Specifically, the Gini index for 
average income between quarters one and two is 0.5855.  For the first three quarters, it is 0.5711.  
For the first four periods, it is 0.5551.  For all five quarters, it is 0.5475. 
It is interesting to note, that all the terms reflecting the deviation of the Gini correlations of 
each period with the overall distribution are positive, which according to equation (4) imply that 
they increase the Gini of the overall period. It is not clear whether this property is typical to other 
data sets, or whether one should expect all the terms to have the same sign.  
With five quarters, a total of 48 correlations must be computed in order to implement the 
decomposition in equation (4). In the variance-like decomposition of equation (5), the number of 
correlations required is reduced to 20 for five quarters.  How good an approximation is equation 
                                                 
1 If the only change that occurs between the periodical distributions is a linear shift, then the Gini correlations will 
be equal. Further research is needed to develop a statistical test to show whether the difference between Gini 
correlation coefficients is significant.    
  3(5)?  Table 1 can be used to show that using equation (5) would have yielded a Gini for the five 
periods of 0.5388, which accounts for 98.4 percent of the actual Gini obtained with equation (4).  
Equation (5) accounts for an even larger share of the actual Gini for fewer periods. Hence, it is a 
useful first order approximation for evaluating the impact of the accounting period on the Gini.  
 
4.   Conclusion 
 
The Gini index of inequality for a sum of random variables can be decomposed in a way 
that resembles the decomposition of the variance, plus an additional term, which reflects the 
deviation of the underlying distributions from exchangeability up to a linear transformation. To 
be able to make quantitative inferences on the effect of the accounting period on the Gini 
coefficient, we should evaluate the Gini correlation for different types of variables (households 
or individual, monthly or quarterly etc.). If we find the magnitude of those correlations, and if 
they are relatively stable over time (and possibly over countries), we may be able to predict the 
impact of the accounting period on inequality in quite general settings.   
 
  4Appendix: Proof of equations (2) and (3). 
 
For simplicity, we define ai = bi (µi/µ0), and provide the decomposition for G0 = 2COV[a1Y1+ 
a2Y2, F(Y0)].  This normalization enables us to work with variables with unit means, but it does 
not affect the generality of the proof.  Using the properties of the covariance we can write:   
 
  G0 =2COV[a1Y1+ a2Y2, F(Y0)] = a1 2COV[Y1, F(Y0)] + a2 2COV[Y2, F(Y0))]                   (A.1) 
       = a1 Γ10G1 + a2 Γ20 G2  . 
 
Define the identity: 
                                       Γi0 =  Γ0i + Di0  for i=1,2,                                                                  (A.2) 
 
where Di0 is the difference between the two Gini correlations defined between Y0 and Yi . Using 
(A.1) and (A.2), we get: 
 
           G0 = a1 (Γ01+ D10) G1 + a2 (Γ02 + D20) G2.   
 
Rearranging terms: 
 
           G0  - a1 D10G1 - a2D20G2 = a1 Γ01 G1 + a2 Γ02 G2.                                            
 
Using the properties of the covariance: 
 
          
.
G
Γ G G a
))} F(Y , cov(Y )) F(Y , cov(Y {a
)) F(Y , cov(Y
1
)) F(Y , cov(Y
)) F(Y , cov(Y
Γ
0
21 2 2 1 1
1 2 2 1 1 1
0 0 0 0
1 0
1 0
a
a
+
=
= + = =
    
Writing Γ02 in a similar manner, we get equation (2): 
 
). Γ (Γ G G a a G a G a
) G a G Γ (a G a ) Γ G a G (a G a ]G G D a G D [a G
21 12 2 1 2 1
2
2
2
2
2
1
2
1
2 2 1 12 1 2 2 21 2 2 1 1 1 1 0 2 20 2 1 10 1
2
0
+ + + =
+ + + = + −
 
 
Assuming equality of the Gini correlation coefficients between Y0 and Y1 sets D10 =0.  A similar 
assumption for Y2 and Y0 sets D20 =0. The assumption Γ=Γ12=Γ21 completes the proof of (3).   
  5References 
 
Behrman, J. R. and P. Taubman (1989) “Is Schooling ‘Mostly in the Genes’?” Journal of 
Political Economy 97: 1425-46.  
 
Burkhauser, R. V. and J. G. Poupore (1997) “A Cross-National Comparison of Permanent 
Inequality in the United States and Germany” Review of Economics and Statistics 79: 10-
17 
 
Bowles, S. and H. Gintis (2002) “Intergenerational Inequality” Journal of Economic 
Perspectives, forthcoming.  
 
Lerman, R. and S. Yitzhaki (1984) “A Note on the Calculation and Interpretation of the Gini 
Index”  Economics Letters 15: 363-68. 
 
Creedy, J. (1979) “The Inequality of Earning and the Accounting Period” Scottish Journal of 
Political Economy 26: 89-96. 
 
Creedy, J. (1991). “Lifetime Earning and Inequality,” Economic Record 67: 46-58.    
 
Gibson, J., Huang, J. and Rozelle, S. (2001) “Why is Income Inequality so Low in China 
Compared to Other Countries? The Effect of Household Survey Method” Economics 
Letters 71: 329-333.  
 
Schechtman, E. and S. Yitzhaki (1987) “A Measure of Association Based on Gini's Mean 
Difference” Communications in Statistics: Theory and Methods A16: 207-231. 
 
Schechtman, E. and S. Yitzhaki (1999) “On The Proper Bounds of The Gini Correlation” 
Economics Letters 63: 133-138.  
  6Table I: Gini index of inequality and accounting period, Mexico 1996 
  Gini indices of inequality (Gi) and income shares (ai) 
 1
st quarter  2
nd quarter  3
rd quarter  4
th quarter  5
th quarter 
Gini  index  0.6153  0.6027 0.5967 0.5672  0.5789 
Income share 
(five  quarters)  0.2037  0.1930 0.1997 0.1937  0.2099 
  Gini correlations matrix (Γij) 
 1
st quarter  2
nd quarter  3
rd quarter  4
th quarter  5
th quarter 
1
st  quarter  1.0000  0.7908 0.7669 0.7044  0.7200 
2
nd  quarter  0.8267  1.0000 0.8480 0.7462  0.7700 
3
rd  quarter  0.7703  0.8143 1.0000 0.7796  0.8064 
4
th  quarter  0.7961  0.8024 0.8158 1.0000  0.8082 
5
th  quarter  0.7833  0.7938 0.7851 0.7722  1.0000 
  Gini correlations with aggregate income (Γi0)  
 
One 
quarter 
Two 
 quarters 
Three 
 quarters 
Four 
quarters 
Five  
quarters 
1
st  quarter  1.0000  0.9662 0.9416 0.9322  0.9232 
2
nd  quarter  -  0.9556 0.9454 0.9371  0.9309 
3
rd quarter  -  -  0.9448  0.9392  0.9318 
4
th quarter  -  -  -  0.9178  0.9136 
5
th  quarter  -  - - - 0.9224 
  Gini correlations with aggregate income((Γ0i) 
 1
st quarter  2
nd quarter  3
rd quarter  4
th quarter  5
th quarter 
One  quarter  1.0000  - - -   
Two quarters  0.9537  0.9276  -  -   
Three quarters  0.9199  0.9175  0.9216  -   
Four  quarters  0.9137  0.9134 0.9200 0.8616  - 
Five  quarters  0.9058  0.9079 0.9113 0.8615  0.8881 
  0
2
0 G 
∑
=
k
1 i
i i0 i 0 G D a G
G   
∑
=
k
1 i
2
i
2
iG a ∑∑
=≠ 1 i ij
ij j i j i Γ G G a a  
One  quarter  0.6153  - - -  - 
Two  quarters  0.5855  0.3428 0.0071 0.1858  0.1499 
Three  quarters  0.5711  0.3261 0.0084 0.1221  0.1956 
For  quarters  0.5551  0.3082 0.0095 0.0889  0.2097 
All  quarters 0.5475  0.2998 0.0094 0.0703  0.2201 
Source: Authors’ estimation from ENEU data. 
 
  7  8
Figure 1: Accounting period and wage inequality in Mexico, 1996
0.5475
All quarters
0.6153 (quarter 1)
0.5551 (quarters 1-4)
0.5855 (quarters 1-2)
0.5711 (quarters 1-3)
0.500
0.520
0.540
0.560
0.580
0.600
0.620
0.640
12345
Accounting period (number of quarters)
G
i
n
i
 
i
n
d
e
x
 
o
f
 
i
n
e
q
u
a
l
i
t
y
 
  Source: Authors’ estimation from ENEU data. 

Inequality and the accounting period

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