Poverty Estimation in Small Areas Agne Bikauskaite European Conference on Quality in Official Statistics (Q2014) Vienna, 3-5 June 2014.

Poverty Estimation in Small Areas

Agne Bikauskaite

European Conference on Quality in Official Statistics (Q2014)Vienna, 3-5 June 2014

Data:

• The population size N = 3000

• The sample size n = 300

• Number of mutually exclusive strata H = 7

• The income of individuals (yh1, ..., yhN)

• The auxiliary information (x1i, ..., xji, ..., xJi)

• 1000 simple random samples

Income distribution

Data:

• The population size N = 3000

• The sample size n = 300

• Number of mutually exclusive strata H = 7

• The income of individuals (yh1, ..., yhN)

• The auxiliary information (x1i, ..., xji, ..., xJi)

• 1000 simple random samples

Strata size

hN hnNumber of strata Population size Strata size

1 496 50

2 333 33

3 177 18

4 119 12

5 92 9

6 794 79

7 989 99

Total 3000 300

Stratified Sampling

• The sample design probability when element i belongs to stratum h is

• The sampling weight for selected person i from the h stratum is

;h

hih N

n

h

h

ihih n

Nw

1

Estimated parameters

• The Average Income

• The Poverty Line

• The Headcount Index

• The Poverty Gap Index

The Average Income

• The average income in strata h is

• The average income estimate is

n

iii

h

h ywN 1ˆ1̂

hN

ii

hh yN 1

1

The Poverty Line

• The Poverty Line is defined as 60 per cent of the median equivalent disposable income

• The Poverty Line estimate is

Mz %60

Mz ˆ%60ˆ

The Headcount Index

• The headcount index is defined as the number of persons below the poverty line divided by the population number

• The Headcount index estimate is

N

izyi

IN

P1

)(0

1

n

iziyi IwN

P1

)ˆ(0 ˆ1ˆ

The Poverty Gap

• The poverty gap G is defined as an amount of difference between poverty line and income value y of ith person living in poverty or social exclusion

• The poverty gap estimate

zyii iIyzG

zyiii iIwyzG

The Poverty Gap Index

• The poverty gap index is a proportion of the poverty gap and the poverty line

• The poverty gap index estimate is

zy

N

i

iq

i

i

iI

z

yz

Nz

G

NP

11

1

11

.ˆ

ˆˆ1ˆ

ˆ1

1 zy

n

ii

ii

Iwz

yz

NP

Population

What is Small Area?

Sampling in Small Areas

Direct and Indirect Estimates

• Direct Estimates:– Not using auxiliary information– Using auxiliary information from the same

area

• Indirect Estimates:– Using auxiliary information from adjacent

areas

Simulated Estimation Methods

• The Horvitz-Thompson (HT)

• The Generalised Regression (GREG)

• The Synthetic (S)

The Absolute Relative Bias

• The Absolute Relative Bias (ARB) assessed the accuracy of the estimates

K

k h

hh

KARB

1

ˆ1

The Horvitz-Thompson estimator

• The sum estimate is

i

n

ii

n

i i

i ywy

t

11

ˆ

The ARB of the average income estimates

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised Regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population -0.06447544 0.098310539 -0.08398375

1 -0.3211974 -0.31518106 -0.34310121

2 -0.02643092 -0.014056 -0.06902109

3 0.465571393 0.551799055 0.403882282

4 -0.81562095 -0.88208503 -0.65375062

5 0.485715332 0.510841272 0.492216146

6 -0.1417938 -0.13401672 -0.14913289

7 0.079252793 0.090945055 0.188597999

The ARB of the headcount index estimates

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population 0.36396329 0.147665664 0.152247869

1 -3.51959494 -3.7958481 -3.8266288

2 1.468493151 1.192029888 1.003491015

3 4.644761905 4.757144543 5.058255185

4 2.859782609 2.601086957 2.877924901

5 -2.80634921 -2.90370419 -1.68042706

6 -0.63675717 -0.78252971 -0.8622043

7 1.344097079 1.068357786 1.298860988

ARB of the poverty gap index estimate

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population -0.1594528 -0.35543944 -0.41065525

1 -1.37126072 -1.59705543 -1.57592157

2 -0.9619282 -1.23793553 -1.64985282

3 -0.49766038 -0.6453229 -0.69178013

4 -1.21749012 -1.2989069 -1.45047831

5 -0.73358855 -0.95628486 0.02299011

6 0.702989553 0.477610719 0.357964671

7 0.19625962 0.02379632 0.278143276

The GREG estimator

• The sum estimate

J

jXXjyGREGy jjttBtt

1,

ˆˆˆˆ

The ARB of the average income estimates

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised Regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population -0.06447544 0.098310539 -0.08398375

1 -0.3211974 -0.31518106 -0.34310121

2 -0.02643092 -0.014056 -0.06902109

3 0.465571393 0.551799055 0.403882282

4 -0.81562095 -0.88208503 -0.65375062

5 0.485715332 0.510841272 0.492216146

6 -0.1417938 -0.13401672 -0.14913289

7 0.079252793 0.090945055 0.188597999

The ARB of the headcount index estimates

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population 0.36396329 0.147665664 0.152247869

1 -3.51959494 -3.7958481 -3.8266288

2 1.468493151 1.192029888 1.003491015

3 4.644761905 4.757144543 5.058255185

4 2.859782609 2.601086957 2.877924901

5 -2.80634921 -2.90370419 -1.68042706

6 -0.63675717 -0.78252971 -0.8622043

7 1.344097079 1.068357786 1.298860988

ARB of the poverty gap index estimate

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population -0.1594528 -0.35543944 -0.41065525

1 -1.37126072 -1.59705543 -1.57592157

2 -0.9619282 -1.23793553 -1.64985282

3 -0.49766038 -0.6453229 -0.69178013

4 -1.21749012 -1.2989069 -1.45047831

5 -0.73358855 -0.95628486 0.02299011

6 0.702989553 0.477610719 0.357964671

7 0.19625962 0.02379632 0.278143276

The Synethetic estimator

• The sum estimate is

K

khkyhk

tyh Nt

0

sin ˆˆ

The ARB of the average income estimates

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised Regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population -0.06447544 0.098310539 -0.08398375

1 -0.3211974 -0.31518106 -0.34310121

2 -0.02643092 -0.014056 -0.06902109

3 0.465571393 0.551799055 0.403882282

4 -0.81562095 -0.88208503 -0.65375062

5 0.485715332 0.510841272 0.492216146

6 -0.1417938 -0.13401672 -0.14913289

7 0.079252793 0.090945055 0.188597999

The ARB of the headcount index estimates

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population 0.36396329 0.147665664 0.152247869

1 -3.51959494 -3.7958481 -3.8266288

2 1.468493151 1.192029888 1.003491015

3 4.644761905 4.757144543 5.058255185

4 2.859782609 2.601086957 2.877924901

5 -2.80634921 -2.90370419 -1.68042706

6 -0.63675717 -0.78252971 -0.8622043

7 1.344097079 1.068357786 1.298860988

ARB of the poverty gap index estimate

StratumHorvitz-Thompson

estimate’s ARB (%)Generalised regression

estimate’s ARB (%)Synthetic estimate’s

ARB (%)

Population -0.1594528 -0.35543944 -0.41065525

1 -1.37126072 -1.59705543 -1.57592157

2 -0.9619282 -1.23793553 -1.64985282

3 -0.49766038 -0.6453229 -0.69178013

4 -1.21749012 -1.2989069 -1.45047831

5 -0.73358855 -0.95628486 0.02299011

6 0.702989553 0.477610719 0.357964671

7 0.19625962 0.02379632 0.278143276

The mean estimate’s variance

,)ˆˆ(1000

1ˆ

1000

1

2

i

iD

The Jack-Knife method

• The Jack-Knife method’s idea is to divide stratified sample into mutually exclusive subgroups.

• The modified sampling weights

stratum. tobelongselement when ,1

subgroup, and stratum tobelongselement when,0

stratum, tobelongnot doeselement hen w ,

thth

thth

thth

hiwn

nkhi

hiw

w

ii

i

thi

hki

The Jack-Knife variance estimator

• Then the Jack-Knife variance estimator of estimated parameter is

2

11

ˆˆ1ˆˆ

hK

khkhk

H

h h

hhkJACK K

KD

hK

khk

hhk K 1

ˆ1ˆ

Conclusions:Poverty parameters estimation

• Different estimation methods for large and for small areas

• The Synthetic method for poverty estimation in small areas

• If auxiliary information from adjacent areas is not available then the most appropriate estimation method is Horvitz-Thompson

Conclusions:Variances estimation of the

estimated parameters

• Large ARBs

• The best results of estimation are given by the Horvitz-Thompson method

• Applying Jack-Knife method precision of the estimates increases when the group size is extremely small