Change in Income Distribution in the presence of reporting errors

Mathl. Comput.

Modelling Vol. 25, No. 7, pp. 33-42, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/97 517.00 + 0.00

Pergamon

PII:SO895-7177(97)00047-2

Change in Income Distribution in the Presence of Reporting Errors T.

PHAM-GIA* AND N. TURKKAN Applied Economics Research Group Universit6 de Moncton Moncton, New Brunswick, Canada ElA 3E9

(Received

Abstract-we

December

1996;

accepted

Februaq

1997)

presentsomeresultsin Income Distribution (ID) when both declared incomes and

errors in their reporting example is provided.

Keywords-Modelling, tion.

are in the beta or gamma family of statistical

Gamma,

Beta,

distributions.

A numerical

Lorenz curve, Gini index, Hypergeometric,

Appell func-

1. INTRODUCTION Declared statistical

incomes, as officially reported to a government agency, can be modeled according to some distributions, the most frequently used one being the lognormal (see [l]). However,

there are inherent complexities in the mathematical expression of the lognormal, and the beta and gamma distributions are also often used, due to the very versatile form of their shapes and In this article, we will focus on these two the convenience of their mathematical expressions. distributions as basic modeling tools, which would permit the derivation of several results that cannot be established otherwise. Declared incomes are subject to a variety of factors which make their reporting nonaccurate. From arithmetic errors to involuntary omission and deliberate under reporting of small amounts earned on a nonrecurrent basis, a substantial percentage of the real income can be unreported. However, we do not consider here revenues generated by the underground economy, the magnitude of which could be much higher. In the latter case, the revenues come from unreported income earned on a regular basis by legal activities or by any nonlegal activity. For the underground part of the economy, basic statistical methods have been used to study and evaluate its size. For example, De Leeuw [2] identifies industries where there are possible underground activities and uses different statistical techniques to evaluate them. For the error part however, few studies have been devoted to it and, mostly, they are limited to a rough estimate of its size by a fixed percentage. In this article, we will study income inequality in the presence of this error, distributed according to the two statistical models mentioned above. The real size of the error will not be of concern here since we focus on the methodology, which is applicable to any size. Our approach will be completely probabilistic, based on the addition of two random *Research supported by NSERC Grant 0GP9249. discussions and this article is written to his memory.

The authors

wish to thank Fl. Shorrock

for many helpful

Typeset 33

by A~-Tf$X

T. PHAM-CIA

34

AND

N. TURKKAN

variables, and will permit the consideration of different scenarios which, in turn, will allow us to evaluate the influence of the error distribution on the total income and have a unifying view of the problem. In Section 2, we will give the precise form of the density of X = Xi + Xs, where Xi and X2 are independent random variables belonging both to the general beta or the gamma family. In Section 3, application to the problem of incorporating the distribution error in the official income distribution is presented and related approaches proposed in the literature, on how to distribute the underground income, are discussed. As a numerical example, we give the analysis of the 1988 Canadian Income Distribution (ID) in the presence of reporting error, using both statistical models.

2. ADDITION

OF TWO

RANDOM

VARIABLES

In mathematical probability, the sum X = Xi + X2, of two independent random variables Xi and Xs, with respective distributions Fi(zi) and F~(Q), has as distribution

called the convolution of Fl and Fs, and denoted by F = Fl* Fz. When Xi and X2 are continuous, with densities fi(zr)

f(x) =

JW --m

fl(X

- YMYf

&A

and fz(xz),

-CO
the density of X is then

(1)

In economics, we usually have X > 0, and hence, proper integration bounds should be applied. Although equation (I) is relatively simple and can always be handled by numerical integration, usually, no closed form of f(z) is available. Then, no analytical study of f(z) is possible and few properties of f(z) can be established. For certain “stable families of distributions” f(z) is of the same type as fr(z) and fi(x), if both these two densities belong to that family. A typical example is the normal family. In this section, we consider the cases where both Xi and XZ belong to the general beta family, or the gamma family of distributions, which have been used ss income distribution models (see [3,4]). We will establish first, the closed form expression of the density of X = Xr + X2, from which several measures of income inequality can be derived, such as the Lorenz curve, the Gini index, and the Pietra ratio. As stated earlier, when other models are considered, for example, the lognormal for both Xi and X2, or two different families for Xr and Xs, X does not have a density in closed form. Then results must be obtained numerically, on a per case basis. The beta distribution defined on (0,l) is very much used in statistics. It has as density f(~; a,@ = zco-‘(l - s)“-lfB(a,@)O < z < 1, where B(a,P) = l?(o)I’(p)/I’(a + @) is the beta function (a > 0, p > 0). It has a wide variety of shapes and its general form on (O,a), denoted by X N beta (a; a,@), has as density f(z; ~,a,@ = z”-i(a - ~)~-‘/[u”+P-~B(c~,P)], 0 < 5 < a, and can be reduced to the preceding standard form by a change of variable. The general beta has 0 and a as end-points, and can be widely used as an approximation to any statistical distribution with the same end-points (see fs]). In economics, it has been used as a model for the underground income by Ryscavage [3] of the U.S. Bureau of the Census. Various questions related to the Lorenz curve of a beta distribution have been discussed by Pham-Gia and Turkkan [S]. C oncerning the sum of two independent general betas, we have the following original result.

Change in IncomeDistribution THEOREM 1. Let X1 w beta

(c; a,p)

Then X1 + X2 has its density

f(x)

of X2 w beta (d; y, 6), with 0 < c 2; d.

be independent

defined

35

on (0, c + d) by:

for 0 < 2 I c, f(x)

= Za+-+(C

x Fl

- “)P--1

(

7,1-h 1-P; a+Y; ;,

p+pr r >A 1

”

(x

(2)

c)

d-y

(a + r>r (6)r (P) (7 + 6) r (a + P) 1

for c < x 5 d, I

= (z - c)?--‘(d + c - x)&-l

(3) d’+6-’

and for d < x < c + d, f(z)

= (d + c - .)p+a-l(x

x Fl

- c)‘-’

(4)

x-(d+c) x-c

P,l-r,l-a;~+6;

(d+c)-z ’

cod7+6-lr(a)r(+$w+6)

C

Here, F1 (a, bl, b,; c; x, y) is the hypergeometric

r(a+tw(y+s)

function

in 2 variables

c;

Z,Y)

=

2

39 --7 m!

2 (c,

m==O n=O

U-L -i- n)

’

by

defined

(u,~+~)(~l,~)(~2,~) ~l(~,~l,~z;

1

y” n!

with (a, 0) = 1 and (a, n) = a(a -t- 1). . . (a + n - l), PROOF.

See the Appendix.

REMARK.

by Pham-Gia

When c = d = 1, we have the sum of two standard and Turkkan (see [7, p. 2641).

betas,

the density

of which is given

The Lorenz curve associated with X - beta (a, /?) can be shown to have the parametric equation w h ere I,(cr,p) is the incomplete beta ratio function, i.e., rZ(a,P) = (&(wP),U~ + l,P)), J: F-*(1 - t)@-‘/B(a,@), while the G ini index is 2B(2cu, ‘L,@/[aB(cr, j3)12, and the Pietra ratio i = 1,2, for their sum X = X1+-X2, is CY~-‘~~~/[(CY+/~>“+@B(a,j3)) (see IS]). If Xi N beta((ri,&), however, the above curve and measures have to be computed numerically. Let’s recall that the Gini index is also twice the area between the Lorenz curve and the diagonal, that the Pie&a ratio is the maximum of the verticai distance between them and that both measures are widely used in the study of income inequality (see [I]). They also have important applications in Reliability Theory as shown by Pham-Gia and Turkkan (81. On the infinite interval (O,oo), the two-parameter gamma (a,p) family of distributions can take a wide variety of shapes and provides excellent approximations to any distribution with nonbounded values. X N gamma (cY,~) has as density f(s;c~,@) = x”-le-Z’P/[PQr(a)], CC2 0 and CY > 0, p > 0. Its mean is cup and its variance a/.Y2. The Gini index is then I’(c~ c (1/2))/(,/%‘(tY + 1)) and the Pietra ratio (a/e)*/I’(cY -i- 1). The sum of two independent gamma variables, Xl N gamma (~1, ,/3x)and XZ N gamma (cy2, ,&), is another gamma if /31 = 02. Wowever, for 01 # pz, the density of X1 + X2 must be expressed in terms

of the confluent

THEOREM 2. If& f(x)

hypergeometric

function

@ as follows.

< &, then X = X1 + XZ has as density =

xal+aa-le-xl~l

J

a2;

a1

+

[ptyr

wz;

(l/P1 (al

+

1//32)x) a2)1

Here, $(a; c; x) = C~o[(a,n)l(c,n>l(xn/nl.) is called the confluent PROOF. See the Appendix.

z

>

0.

’

hypergeometric

function.

T. PHAM-CIA AND N. TURKKAN

36

0

5

10

15

20

x=x, +x2 Figure 1.

3. APPLICATION TO THE ERROR REPORTING AND NUMERICAL

IN INCOME EXAMPLE

Our modeling approach presented here will use the preceding theoretical results to obtain the distribution of the total income X, which is the reported income I plus error E, X = I+E. Hence, we will first find the distribution that would best fit the data of the official income distribution 1. This distribution fitting problem is well known in applied statistics and for the beta distribution a special approach has been devised by Pham-Gia and Turkkan IS]. Several scenarios concerning the error E and its distribution can now be considered. In general, this distribution is unknown, but if a finite interval of variation is considered, it can be approximated by a beta distribution. On (0, oo), it is the gamma distribution that would be used. Taking Xi = E and X2 = I, Xi +X, is the total income X, whose distributions are given by Theorem 1 or Theorem 2, depending on whether both Xi and X2 are beta or gamma distributed. By supposing Xi and Xs independent, when considering Xi +X2, an error of any size can be combined with any official income, subject only to the probabilities associated with each, as given by their probability models. As explained in Section 2, Theorems 1 and 2 will allow us to handle the problem very conveniently. Otherwise, complex numerical integration techniques must be used on an ad hoc basis since, to our knowledge, no explicit expression of the density of X is available when X1 and Xs are given other statistical distributions, usually adopted in income study. We will, in the numerical example that follows, use real data, as given by Statistics Canada [9], to illustrate the results obtained above. Section 3.1. The 1988 Canadian Income Distribution X2 has been considered by Pham-Gia and Turkkan IS] and, using the approach suggested by Kakwani and Podder, they found the best beta fit for this distribution, as beta (30; 2.1,12.1), where the cut-off point d = 30 (in ten thousands dollars). Its mean is (2.1/(2.1 + 12.1))0.30 = 4.4366, just slightly different from the official mean (computed from raw data) of 4.5329, and its standard deviation is a(Xs) = 2.7315.

Change

G = 0.297 G,= 0.345

in Income

Distribution

37

P = 0.214 P,= 0.249

0.25

0.50

1.00

0.75

w Figure

2.

The first hypothesis to formulate is about the size of X1, the error in reporting. If this size is supposed to be in the vicinity of 15% of the official one, the mean of Xi can be taken as 4.5329(0.15) = 0.6799. Similarly, its upper cut-off point c can be taken as (30)(0.15) = 4.5. Hence, the mean of the associated standard beta distribution is 0.6799/4.5 = 0.1511, and the relation between the coefficients Q: and p of Xi is (Y = (O.l511)(cu + /3) or ,L3= 5.61810. The second hypothesis is on the spread of Xi about its fixed mean 0.1511. This is measured by the standard deviation of Xi, 0(X,) = dm, w here Var (Xi) = (c2cy/3)/[(o+P)2(cy+p+l)]. In this case, a(Xi) = 1.6116/(1 + 6.6181cr)lj2. Taking (Y = 2, we have a(Xi) = 0.4271, a reasonable value of the error dispersion and _Xi N beta (4.5,2,11.2363). With Xr and Xz defined as above, by Theorem 1, we have the density of X defined by: for 0 < x 5 4.5, f(x)

= x3.1(4.5 - x)10.2362

(5) 2 _?30’ (x-4.5) >A

I(4.1) I’(l2.1) I? (11.23) 4.512.23(30)2.1 I? (14.2) r(13.23)

’

for 4.5 < 5 < 30, f(x)

= (x - 4.5)1.‘(34.5 - 2)ii.l x Fi

(6)

11.23, -1.1, -11.1; 13.23; ,,.,“‘”

x), c34pj5 ~) >A

3013.2w-.w~w r(14.2)

> ’

and for 30 < x < 34.5, f(x)

= (34.5 - x)zz.ss(x - 4.5)i.l 11.23,--1.1;-1;

(x-34

cx_4.5j

(7) 5) (34.5-x) ,

4.5

4 >/(

.

511.233013.2

(r(2)r(2.1)r(23.33))

r(l3.23)r(l4.2)

The density f( x ) is a smooth curve, as shown by Figure 1, where for x 2 15, we have f(x) practically zero.

T. PHAM-GIA AND N. TURKKAN

38

Table 1. Income Intervals

Observed Income Plus

Observed Income (in %)

0 to 0.50

Reporting

Error (in %)

As Officially

Fitting by

Fitting by

Beta

Gamma

Reported*

Beta (30; 2.1, 12.1,)

Gamma (2.75, 1.64)

Model**

Model’*

1.5

0.7

0.2

1

0.2

1

0.50 to 1.0

I

2.8

I

4.2

I

3.0

I

1.5

1

1.3

I

1

1.0 to 1.25

1

2.5

i

2.9

I

2.5

I

1.5

1

1.3

I

1 1

1.25 to 1.50

3.0

3.3

3.0

2.0

1.9

1.50 to 1.75

4.2

3.6

3.5

2.5

2.4

1.75 to 2.00

4.0

3.8

3.8

2.9

2.8

2.00 to 2.50 2.50 to 3.00

1 1

7.7 7.7

1

8.0

I

8.3

1

6.9

1

6.6

1

1

8.1

I

8.7

I

7.7

I

7.7

I

3.00 to 3.50

8.1

8.0

8.6

8.0

8.2

3.50 to 4.00

8.3

7.6

8.2

7.9

8.3

4.00 to 4.50

7.8

7.0

7.5

8.5

8.0

4.50 to 5.00

7.3

6.4

6.7

7.1

7.4

I I I

[

5.00 to 5.50

)

6.9

1

5.7

1

5.50 to 6.00

1

5.2

1

5.0

I

6.00 to 6.50

4.4

I

4.4

6.50 to 7.00

3.8

3.8

3.7

4.5

4.5

7.00 to 7.50

2.9

3.2

3.1

3.8

3.8

9.4

8.9

11.8

11.5

7.50 to 10.00 10.00 to 12.50

I I

13.5

12.4

3.1

12.50 to 15.00

0.8

15.00 and more

0.2

13.2

5.9

1

6.5

1

6.8

1

5.1

I

5.8

I

6.0

I

4.4

1

5.1

1

5.2

1

3.0

17.6

0.9

4.2

17.6

1.1

4.2

1.3

0.3

0.5

0.6

Mean

45,329

45,169

45,533

51,165

52,126

Median

40.430

39,165

39.770

45.600

46.000

Gini index Pietra ratio

I

not given not given

I I

0.338

I

0.326

I

0.297

I

0.291

I

0.244

I

0.233

1

0.214

1

0.238

1

*Source: Satistics Canada, No. 13-207 (Nov. 1989) **AS presented in this article.

The Lorenz curves for Xr and Xs, and their associated Gini index and Pietra ratios can be computed using formulas given in Section 3, but the Lorenz curve for X = Xr + Xz, must be computed numerically, using its parametric form L, = (J,” f(t) dt, (l/5.1165) JJ tf(t) dt), where f(t) is given by (5)-(7), and pxl + px, = 5.1165. The Gini index and Pietra ratio for X also have to be computed numerically and are given on Figure 2, where they are, respectively, denoted by G and P. The above precise expression of the density of X, also aliows us to have the distribution of the total income according to income intervals, as given in Table 1. Without expressions (5)-Q’), above it would not be possible to derive this distribution. The Lorenz curves for Xr, Xz, and X = Xr +Xz are shown in Figure 2. We can see that there has been a decrease in income inequality with the reporting error. The Gini index has decreased from 0.338 to 0.297, hence, by 13.80%.

Change in Income Distribution

39

‘.,T

X,-Gamma(l.20,0.56) X,-Gamma(2.75,1.64) 1.0 .z? E! X,

8 n.5 -

(iI $10,000) 0.0 0

I

I

I

5

IO

15

20

x=x,+x2 Figure 3. REMARKS.

(1) As

(2)

stated earlier, the beta form of the density of Xi depends on values given to cri and pi. The larger we choose cri, and then deduce /3i from the relation /3i = 5.6181~~1, the more concentrated about the fixed mean 0.6799 is that density, and the more sure we are about that mean as the point where most of the error is distributed. Also, the upper cut-off point, here chosen as 4.5, can vary, depending on our evaluation of the reasonable upper limit for the reporting error. However, there is practically no change to the general shape of the density of X for large variations of this upper limit about 4.5. Additional information on the reporting error can lead to a more accurate expression of Xi, but the general approach is unchanged. Then, a study similar to the one undertaken by Bishop

et al. [IO] can be made, but concerning

the total

income.

Section 3.2. If the gamma model is adopted, the gamma distribution fitted to the same ID is X:! py Gamma (2.7538,1.646) with mean 4.5327. It has a long right tail going to infinity but, practically, is zero for z > 15. Again, taking the reporting error Xr as 15% of the observed income, its mean is then 0.6799 and the two parameters of (~1 and ,& are related by the relation aiPi = 0.6799. Taking for example (~1 = 1.20, we have ,0i = 0.5666. X = Xi + X2 has its density given by Theorem 2. We have: f(z) = z2~g538e-Z~o~5sss~(2.7538; 3.9538; (l/O.5666 - l/1.646)2), 0 < z < co. The densities of Xi, X2, and X are given by Figure 3. The Lorenz curves before and after adding the reporting error are given by Figure 4 and again, we see a decrease in the Gini index from 0.326 to 0.291 or by 12.02%. We can see that the beta and gamma models give very similar results in this example. REMARK. Other methods could be used to distribute an error of a given magnitude, and distributed according to a certain distribution, to the population income distribution, by applying approaches similar to the ones used to study the underground economy. However, since they are based on raw data, they will require considerable work and, naturally, the access to these

T. PHAM-GIA AND N. TURKKAN

40

G = 0.291

G,= 0.465

P = 0.236 P,= 0.340

0.25

0.00

Figure 4.

for the underground raw data. For example, Ryscavage [3] considers several beta distributions economy, each with total area equal to a fixed percentage of the observed economy and having the same total variation range as the official distribution. Hence, for each income interval Iin, i, + I] the associated area of underground income can be computed, and is uniformly distributed to the raw data in the interval, resulting in a new distribution incorporating both observed and unobserved incomes. Income inequality measures are then computed for this distribution. Elteto and Vita [ll], on the other hand, subdivided the whole population into different subgroups, and to each of these subgroups assigned a different lognormal distribution for the associated underground economy. For each individual in a subgroup, a hidden income is now either created (denoted C-correction) or suplemented (S-correction), or both, using a random number from the uniform (0,l) distribution or from the above lognormal distributions. As can easily be seen, there is a very large degree of subjectivity

in either

method.

In the absence of complete information concerning Xi, we think that our modeling approach above offers a better distribution of the error to the reported income, in the sense that the “error income” in [i n, i, + l] is now not distributed exclusively to individuals interval (as in an approach similar to Ryscavage’s) nor determined Elteto and Vita’s method is followed).

in the same official income completely randomly (as if

Finally, a multiplicative model can also be considered, as in Krishnaji income Y is related to the total income X by the relation Y = RX, variable in [O, 1] associated with the error.

[12], where the reported with R being a random

4. CONCLUSION Modeling the total income is an effective way to study the effect of the reporting error on the officially reported income distribution. The addition of two random variables provides a logical approach to determining the distribution of that total income. Although numerical methods can always be used, for the beta and gamma cases, we have been able to study this total income analytically and derive the values of various income inequality measures in a precise way. Also, the

Change in Income Distribution approach

presented

problems,

here should

be widely

since the beta and gamma

applicable,

provide

41

for income

excellent

distribution

approximations

or other

to other

economic

distributions.

APPENDIX In the following proofs, we will make use of the two basic results in integral representation hypergeometric functions (see [13]). Let the first Appeli function F’r(a, br, b,; c; x, y) be defined by the double series

Fl(%

h,

z y> =L 2 2 (a, n + m)(br,m)(k, 7 (c, m -t-n) m=O n=O

62; c;

of

n) x” Y/” --, m! n!

where a, br, b2, and c are real or complex with c # negative integer, (a,O) = 1 and (a,n) a(a -6 1). . . (u + n - 1). This series converges for 1~1: 19(/j< 1. If Re (a) and Re (c - u) are positive, we then have

r (4 r(alr(c--af

1

J

g-1

(I -

x~L)-~~ (1 -

(I-

~~~~~~~

grzljwb2 du = ~~(a,~~,b2;

c; z,yf.

=

(8)

0

Similarly, we define the conhuent hypergeometric function and for Re (a) > 0 and Re (c - a) > 0, we have

as @(a; c; z) = Cr=,

((a, n)/(c,

n))

(z”/n!),

4(%c; x) =

1

I- Cc) r(a)r(C_u)

J

o

--ztp-I

(1

e

_

t)c-a-l

&

PROOF OF THEOREM 1. Let X1 N beta(c; have f(;cr) = x~-‘(~-~~)~-‘/[c”~~-“B(~,P)],

cy, p) and X2 N beta(d; y,6) be independent. 0 < 21 < c and g(Lcz) ~*“~-1(~-“2)s-1/[d’+s-1

B(G)], 0 < x2 < of. Let X = XI + X2 and

- ~~)g(~2~ d x2, 0 < 3: cc c + d.

J:f N(cQ) dzz/A, values

e(z)

where H(q)

= low f(z = (z - ~)*-r(c

- z + zzfa-rz~-r(d

- az)‘-r,

for 11 and 12, depending on values of 5 and A = c*fP-ldr+G-lB(ct,

Hence,

We

9(z)

=

with appropriate

P)B(y, 6).

1. ForO
Setting u = Q/E,

we have

f fx> x7 ‘(x)= dTB(y,6)

l

vy_l

(1

_

Jo

x7Yf (x> B(y,CY)F1 = drB(y,6)

v)cY-l (1 - y)‘-’

-/,1-6,1-p;

(1 - v$--)‘-I

dv

a-t-f;

where f(.) is density of XI above. Using (8), expression (2) follows easily. 2. For c < x 5 d, Q(x) = J:_JT(z~) &2/A. Changing to u = (c-zr+~)/c and simplifying, we have CQ+13-l

9 (x) =

(X-c)+(d+c-zf6-1 A

’&-’ J (I-g-l 0

T. PHAM-CIA

42

Hence,

AND

N. TURKKAN

using (8),

~ (x) = (x -

cy

(d + c -

xp

B(y, c5)dy+6-1 We obtain

FI Al-r,l-4

a+&

2,

d+z_x

(3) after simplification.

3. For d < x 5 dfc, we have

Q(x)

= s;“_, H(x2)

dxz/A.

= (d+c-x)P+6-‘c”-1(x-c)Y-1

\k(x)

to u = (c - 2 + xz)/((d

Changing

s @++xu a-1 ’

A

+ c) - x),

&1

(1 -

,)6_’

0

U

1_

du 7

C

and hence, (d + c - x)‘+~-~ Q(x)

=

cfldr+s-lB x FI

P,l

(x - c)‘-1

(a, 0) I3 (y, 6)

-y,l

--a;

P+S;

B (P,6) x-(d+c)

which leads to (4).

x-c

(d+c)-x '

C

, > I

PROOF OF THEOREM 2. The proof of this theorem can be done in several ways. A direct proof using the integral representation of the confluent hypergeometric function 4(a; c; z) is given by Pham-Gia and Turkkan [14].

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