inequality: empirical issues

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Inequality: Empirical Issues

July 2006 July 2006

Inequality and Poverty Measurement Inequality and Poverty Measurement

Technical University of LisbonTechnical University of Lisbon

Frank CowellFrank Cowellhttp://darp.lse.ac.uk/lisbon2006http://darp.lse.ac.uk/lisbon2006

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Introduction Focus on an issue common to most empirical applications in Focus on an issue common to most empirical applications in

distributional analysisdistributional analysis Sensitivity to extreme valuesSensitivity to extreme values Should be able to estimate inequality and other indices by using sample Should be able to estimate inequality and other indices by using sample

data.data. But, how do very low / very high observations affect estimates?But, how do very low / very high observations affect estimates?

References found in References found in Cowell, F. A. and Flachaire, E. (2002) "Sensitivity of Inequality Measures

to Extreme Values" Distributional Analysis Discussion Paper, 60, STICERD, LSE, Houghton St., London, WC2A 2AE.”

MotivationMotivation Interested in sensitivity to extreme values for a number of reasonsInterested in sensitivity to extreme values for a number of reasons Welfare properties of income distributionWelfare properties of income distribution Robustness in estimationRobustness in estimation Intrinsic interest in the very rich, the very poorIntrinsic interest in the very rich, the very poor ..

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Sensitivity? How to define a “sensitive” inequality measure? How to define a “sensitive” inequality measure? Ad hoc discussion of individual measures Ad hoc discussion of individual measures

empirical performance on actual data (Braulke 83). empirical performance on actual data (Braulke 83). not satisfactory for characterising general propertiesnot satisfactory for characterising general properties

Welfare-theoretical approachesWelfare-theoretical approaches focuses on transfer sensitivity (Shorrocks-Foster focuses on transfer sensitivity (Shorrocks-Foster

1987)1987) But does not provide a guide to the way measures But does not provide a guide to the way measures

may respond to extreme values. may respond to extreme values. Need a general and empirically applicable tool.Need a general and empirically applicable tool.

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Preliminaries

A large class of inequality A large class of inequality measures:measures:

• Define two moments:

• Can be written as:

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The Influence Function

• Mixture distribution:

• Influence function:

• For the class of inequality measures:

• which yields:

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Some Standard Measures

• GE:

• Theil:

• MLD:

• Atkinson:

• Log var:

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…and their IFs

• GE:

• Theil:

• MLD:

• Atkinson:

• Log var:

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Special case

• The IF:

• The Gini coeff:

• where:

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Tail behaviour

z z

zz

zz

zz

zz

zz

zz

z z 0 0

[log [log zz] ] zz

--

log log zz

--

--

< 0< 0

= 0= 0

11

> 1> 1

Log VarLog Var

GiniGini

GEGE

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Implications

Generalised Entropy measures with Generalised Entropy measures with > 1 are very > 1 are very sensitive to high incomes in the data.sensitive to high incomes in the data.

GE (GE ( < 0) are very sensitive to low incomes < 0) are very sensitive to low incomes

We can’t compare the speed of increase of the IF We can’t compare the speed of increase of the IF for different values of 0 < for different values of 0 < < 1 < 1

If we don’t know the income distribution, we can’t If we don’t know the income distribution, we can’t compare the IFs of different class of measures.compare the IFs of different class of measures.

So, let’s take a standard model…So, let’s take a standard model…

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Singh-Maddala

c = 1.2

c = 0.7

c = 1.7

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Using S-M to get the IFs

Use these to get true values of inequality measures.Use these to get true values of inequality measures. Obtained from the moments:Obtained from the moments:

• Take parameter values a=100, b=2.8, c=1.7

• Normalise the IFs • Use relative influence function

Good model of income distribution of German households

Good model of income distribution of German households

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Gini

Gini

Gini

Gini

IFs based on S-M

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IF using S-M: conclusions

When When zz increases, IF increases faster with high increases, IF increases faster with high values of values of . .

When When zz tends to 0, IF increases faster with small tends to 0, IF increases faster with small values of values of ..

IF of Gini index increases slower than others but IF of Gini index increases slower than others but is larger for moderate values of is larger for moderate values of zz. .

Comparison of the Gini index with GE or Log Comparison of the Gini index with GE or Log Variance does not lead to clear conclusions.Variance does not lead to clear conclusions.

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A simulation approach

Use a simulation study to evaluate the impact of a Use a simulation study to evaluate the impact of a contamination in extreme observations.contamination in extreme observations.

Simulate 100 samples of 200 observations from S-M Simulate 100 samples of 200 observations from S-M distribution. distribution.

Contaminate just one randomly chosen observation by Contaminate just one randomly chosen observation by multiplying it by 10.multiplying it by 10.

Contaminate just one randomly chosen observation by Contaminate just one randomly chosen observation by dividing it by 10.dividing it by 10.

Compute the quantityCompute the quantityEmpirical Distribution

Empirical Distribution

Contaminated Distribution

Contaminated Distribution

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Contamination in high values

100 different samples sorted such that Gini realisations are increasing.

RC(I)

• Gini is less affected by contamination than GE.

• Impact on Log Var and GE (1 is relatively small compared to GE (<0) or GE (>1)

• GE (0 1) is less sensitive if is smaller

• Log Var is slightly more sensitive than Gini

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Contamination in low values

100 different samples sorted such that Gini realisations are increasing.

RC(I)

• Gini is less affected by contamination than GE.

• Impact on Log Var and GE (1 is relatively small compared to GE (<0) or GE (>1)

• GE (0 1) is less sensitive if is larger

• Log Var is more sensitive than Gini

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Influential Observations•Drop the ith observation from the sample

•Call the resulting inequality estimate Î(i)

•Compare I(F) with Î(i)

•Use the statistic

•Take sorted sample of 5000

•Examine 10 from bottom, middle and top

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Influential observations: summary

Observations in the middle of the sorted sample don’t affect Observations in the middle of the sorted sample don’t affect estimates compared to smallest or highest observations. estimates compared to smallest or highest observations.

Highest values are more influential than smallest values. Highest values are more influential than smallest values. Highest value is very influential for GE (Highest value is very influential for GE (= 2) = 2) Its estimate should be modified by nearly 0.018 if we remove it. Its estimate should be modified by nearly 0.018 if we remove it. GE (GE (= = ––1) strongly influenced by the smallest observation.1) strongly influenced by the smallest observation.

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Extreme values

An extreme value is not necessarily an error or some An extreme value is not necessarily an error or some sort of contaminationsort of contamination

Could be an observation belonging to the true Could be an observation belonging to the true distribution distribution

Could convey important information.Could convey important information. Observation is extreme in the sense that its influence Observation is extreme in the sense that its influence

on the inequality measure estimate is important. on the inequality measure estimate is important. Call this a Call this a high-leveragehigh-leverage observation. observation.

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High-leverage observations

The term leaves open the question of whether such The term leaves open the question of whether such observations “belong” to the distributionobservations “belong” to the distribution

But they can have important consequences on the But they can have important consequences on the statistical performance of the measure.statistical performance of the measure.

Can use this performance to characterise the Can use this performance to characterise the properties of inequality measures under certain properties of inequality measures under certain conditions. conditions.

Focus on the Focus on the EError in rror in RRejection ejection PProbability as a robability as a criterion.criterion.

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Davidson-Flachaire (1) Even in very large samples the ERP of an Even in very large samples the ERP of an

asymptotic or bootstrap test based on the Theil asymptotic or bootstrap test based on the Theil index, can be significant index, can be significant

Tests are therefore not reliable. Tests are therefore not reliable. Three main possible causes : Three main possible causes :

1.1. NonlinearityNonlinearity

2.2. Noise Noise

3.3. Nature of the tails. Nature of the tails.

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Davidson-Flachaire (2) Three main possible causes : Three main possible causes :

1.1. Indices are nonlinear functions of sample moments. Induces Indices are nonlinear functions of sample moments. Induces biases and non-normality in estimates.biases and non-normality in estimates.

2.2. Estimates of the covariances of the sample moments used to Estimates of the covariances of the sample moments used to construct indices are often noisy. construct indices are often noisy.

3.3. Indices often sensitive to the exact nature of the tails. A Indices often sensitive to the exact nature of the tails. A bootstrap sample with nothing resampled from the tail can bootstrap sample with nothing resampled from the tail can have properties different from those of the population.have properties different from those of the population.

Simulation experiments show that case 3 is often Simulation experiments show that case 3 is often quantitatively the most important. quantitatively the most important.

Statistical performance should be better with MLD Statistical performance should be better with MLD and GE (0 < and GE (0 < < 1 ), than with Theil.< 1 ), than with Theil.

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Empirical methods

• The empirical distribution

• Inequality estimate

• Empirical moments



Indicator function

Indicator function

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Testing

• Test statistic

• Variance estimate

• For given value I0 test

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Bootstrap

To construct bootstrap test, resample To construct bootstrap test, resample from the original data. from the original data.

Bootstrap inference should be superiorBootstrap inference should be superior For bootstrap sample For bootstrap sample jj, , jj = 1,…, = 1,…,BB, a bootstrap , a bootstrap

statistic statistic WW**jj is computed almost as is computed almost as WW from the from the

original dataoriginal data But But II00 in the numerator is replaced by the index in the numerator is replaced by the index ÎÎ

estimated from the original data.estimated from the original data. Then the bootstrap Then the bootstrap PP-value is-value is

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Error in Rejection Probability: A

• ERPs of asymptotic tests at the nominal level 0.05

• Difference between the actual and nominal probabilities of rejection

• Example:o N = 2 000 observations o ERP of GE ( =2) is 0.11o Asymptotic test over-rejects the

null hypothesis o The actual level is 16%, when

the nominal level is 5%.

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Error in Rejection Probability: B

• ERPs of bootstrap tests. • Distortions are reduced for

all measures • But ERP of GE ( = 2) is still

very large even in large samples

• ERPs of GE ( = 0.5, –1) is small only for large samples.

• GE (=0) (MLD) performs better than others. ERP is small for 500 or more observations.

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More on ERP for GE

What would happen in very large samples?

0.04150.0415

0.01250.0125

0.00430.0043

0.00520.0052

0.00960.0096

0.04920.0492

0.01130.0113

0.00240.0024

0.00540.0054

0.00960.0096

22

––11

00

0.50.5

11

NN=100,000=100,000NN=50,000=50,000

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ERP: conclusions

• Rate of convergence to zero of ERP of asymptotic tests is very slow.

• Same applies to bootstrap

• Tests based on GE measures can be unreliable even in large samples.

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Sensitivity: a broader perspective

Results so far are for a specific Singh-Maddala Results so far are for a specific Singh-Maddala distribution. distribution.

It is realistic, but – obviously – special.It is realistic, but – obviously – special. Consider alternative parameter valuesConsider alternative parameter values

Particular focus on behaviour in the upper tailParticular focus on behaviour in the upper tail Consider alternative distributionsConsider alternative distributions

Use other familiar and “realistic” functional Use other familiar and “realistic” functional formsforms

Focus on lognormal and ParetoFocus on lognormal and Pareto

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Alternative distributions

First consider comparative contamination First consider comparative contamination performance for alternative distributions, performance for alternative distributions, same inequality indexsame inequality index

Use same diagrammatic tool as beforeUse same diagrammatic tool as before

xx-axis is the 100 different samples, sorted -axis is the 100 different samples, sorted such inequality realizations are increasing such inequality realizations are increasing

yy-axis is RC(-axis is RC(II) for the MLD index ) for the MLD index

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Singh-Maddala

c = 1.2

c = 0.7 (“heavy” upper tail)

c = 1.7

• Inequality found from:

• Distribution function:

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MLD Contamination S-M

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Lognormal

= 0.7 = 1.0 (“heavy” upper tail)

= 0.5

• Inequality:

• Distribution function:

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MLD Contamination: Lognormal

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Pareto

= 2.0 = 2.5

= 1.5 (“heavy” upper tail)

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MLD Contamination Pareto

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ERP at nominal 5%: MLD• Asymptotic tests

• Bootstrap tests

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ERP at nominal 5%: Theil• Asymptotic tests

• Bootstrap tests

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Comparing Distributions• Bootstrap tests usually improve numerical

performance.

• MLD is more sensitive to contamination in high incomes when the underlying distribution upper tail is heavy.

• ERP of an asymptotic and bootstrap test based on the MLD or Theil index is more significant when the underlying distribution upper tail is heavy.

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Why the Gini…? Why use the Gini coefficient? Why use the Gini coefficient?

Obvious intuitive appeal Obvious intuitive appeal

Sometimes suggested that Gini is less prone to the Sometimes suggested that Gini is less prone to the influence of outliers influence of outliers

Less sensitive to contamination in high incomes than Less sensitive to contamination in high incomes than GE indices. GE indices.

But little to choose between…But little to choose between…

the Gini coefficient and MLD the Gini coefficient and MLD

Gini and the logarithmic varianceGini and the logarithmic variance

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The Bootstrap…? Does the bootstrap “get you out of trouble”?Does the bootstrap “get you out of trouble”?

bootstrap performs better than asymptotic methods,bootstrap performs better than asymptotic methods,

but does it perform well enough? but does it perform well enough?

In terms of the ERP, the bootstrap does well only In terms of the ERP, the bootstrap does well only for the Gini, MLD and logarithmic variance.for the Gini, MLD and logarithmic variance.

If we use a distribution with a heavy upper tail If we use a distribution with a heavy upper tail bootstrap performs poorly in the case of bootstrap performs poorly in the case of = = 0 0

even in large samples.even in large samples.

inequality: empirical issues

Documents

different values

high values of

moderate values of

small values

way measures

extreme observations

sensitive inequality

gini realisations