1. Inequality: Empirical Issues
Inequality and Poverty Measurement
Universitat Autònoma de Barcelona
Frank Cowell
July 2006

2. Introduction
Focus on an issue common to most empirical applications in distributional analysis
Sensitivity to extreme values
Should be able to estimate inequality and other indices by using sample data.
But, how do very low / very high observations affect estimates?
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.”
Motivation
Interested in sensitivity to extreme values for a number of reasons
Welfare properties of income distribution
Robustness in estimation
Intrinsic interest in the very rich, the very poor.

3. Sensitivity? How to define a “sensitive” inequality measure?
Ad hoc discussion of individual measures
empirical performance on actual data (Braulke 83).
not satisfactory for characterising general properties
Welfare-theoretical approaches
focuses on transfer sensitivity (Shorrocks-Foster 1987)
But does not provide a guide to the way measures may respond to extreme values.
Need a general and empirically applicable tool.

4. Preliminaries Define two moments:
A large class of inequality measures:
Can be written as:

5. The Influence Function
Mixture distribution:
Influence function:
For the class of inequality measures:
which yields:

6. Some Standard Measures
GE:
Theil:
MLD:
Atkinson:
Log var:

7. …and their IFs
GE:
Theil:
MLD:
Atkinson:
Log var:

8. Special case
The Gini coeff:
The IF:
where:

9. Tail behaviour z  0 [log z] 2 za - log z z   z za a < 0 a = 0
Log Var
Gini GE

10. Implications
Generalised Entropy measures with  > 1 are very sensitive to high incomes in the data.
GE ( < 0) are very sensitive to low incomes
We can’t compare the speed of increase of the IF for different values of 0 <  < 1
If we don’t know the income distribution, we can’t compare the IFs of different class of measures.
So, let’s take a standard model…

11. Singh-Maddala
c = 1.7
c = 1.2
c = 0.7

12. Using S-M to get the IFs Take parameter values a=100, b=2.8, c=1.7
Good model of income distribution of German households
Take parameter values a=100, b=2.8, c=1.7
Use these to get true values of inequality measures.
Obtained from the moments:
Normalise the IFs
Use relative influence function

13. IFs based on S-M
Gini
Gini
Gini
Gini

14. IF using S-M: conclusions
When z increases, IF increases faster with high values of a.
When z tends to 0, IF increases faster with small values of a.
IF of Gini index increases slower than others but is larger for moderate values of z.
Comparison of the Gini index with GE or Log Variance does not lead to clear conclusions.

15. A simulation approach
Use a simulation study to evaluate the impact of a contamination in extreme observations.
Simulate 100 samples of 200 observations from S-M distribution.
Contaminate just one randomly chosen observation by multiplying it by 10.
Contaminate just one randomly chosen observation by dividing it by 10.
Compute the quantity
Contaminated Distribution
Empirical Distribution

16. Contamination in high values
RC(I)
100 different samples sorted such that Gini realisations are increasing.
Gini is less affected by contamination than GE.
Impact on Log Var and GE (0<a1 is relatively small compared to GE (a<0) or GE (a>1)
GE (0 a1) is less sensitive if a is smaller
Log Var is slightly more sensitive than Gini

17. Contamination in low values
RC(I)
100 different samples sorted such that Gini realisations are increasing.
Gini is less affected by contamination than GE.
Impact on Log Var and GE (0<a1 is relatively small compared to GE (a<0) or GE (a>1)
GE (0 a1) is less sensitive if a is larger
Log Var is more sensitive than Gini

18. 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

19. Influential observations: summary
Observations in the middle of the sorted sample don’t affect estimates compared to smallest or highest observations.
Highest values are more influential than smallest values.
Highest value is very influential for GE (a = 2)
Its estimate should be modified by nearly if we remove it.
GE (a = –1) strongly influenced by the smallest observation.

20. Extreme values
An extreme value is not necessarily an error or some sort of contamination
Could be an observation belonging to the true distribution
Could convey important information.
Observation is extreme in the sense that its influence on the inequality measure estimate is important.
Call this a high-leverage observation.

21. High-leverage observations
The term leaves open the question of whether such observations “belong” to the distribution
But they can have important consequences on the statistical performance of the measure.
Can use this performance to characterise the properties of inequality measures under certain conditions.
Focus on the Error in Rejection Probability as a criterion.

22. Davidson-Flachaire (1)
Even in very large samples the ERP of an asymptotic or bootstrap test based on the Theil index, can be significant
Tests are therefore not reliable.
Three main possible causes :
Nonlinearity
Noise
Nature of the tails.

23. Davidson-Flachaire (2)
Three main possible causes :
Indices are nonlinear functions of sample moments. Induces biases and non-normality in estimates.
Estimates of the covariances of the sample moments used to construct indices are often noisy.
Indices often sensitive to the exact nature of the tails. A bootstrap sample with nothing resampled from the tail can have properties different from those of the population.
Simulation experiments show that case 3 is often quantitatively the most important.
Statistical performance should be better with MLD and GE (0 < a < 1 ), than with Theil.

24. Empirical methods The empirical distribution Empirical moments
Indicator function
Empirical moments
Inequality estimate

25. Testing
Variance estimate
For given value I0 test
Test statistic

26. Bootstrap
To construct bootstrap test, resample from the original data.
Bootstrap inference should be superior
For bootstrap sample j, j = 1,…,B, a bootstrap statistic W*j is computed almost as W from the original data
But I0 in the numerator is replaced by the index Î estimated from the original data.
Then the bootstrap P-value is

27. 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:
N = observations
ERP of GE (a =2) is 0.11
Asymptotic test over-rejects the null hypothesis
The actual level is 16%, when the nominal level is 5%.

28. Error in Rejection Probability: B
ERPs of bootstrap tests.
Distortions are reduced for all measures
But ERP of GE (a = 2) is still very large even in large samples
ERPs of GE (a = 0.5, –1) is small only for large samples.
GE (a=0) (MLD) performs better than others. ERP is small for 500 or more observations.

29. More on ERP for GE 2 –1 0.5 1 N=100,000 N=50,000 a 0.0492 0.0113
What would happen in very large samples? 2 –1
0.5 1
N=100,000
N=50,000 a
0.0492
0.0113
0.0024
0.0054
0.0096
0.0415
0.0125
0.0043
0.0052
0.0096

30. 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.

31. Sensitivity: a broader perspective
Results so far are for a specific Singh-Maddala distribution.
It is realistic, but – obviously – special.
Consider alternative parameter values
Particular focus on behaviour in the upper tail
Consider alternative distributions
Use other familiar and “realistic” functional forms
Focus on lognormal and Pareto

32. Alternative distributions
First consider comparative contamination performance for alternative distributions, same inequality index
Use same diagrammatic tool as before
x-axis is the 100 different samples, sorted such inequality realizations are increasing
y-axis is RC(I) for the MLD index

33. Singh-Maddala Distribution function: Inequality found from:
c = 0.7 (“heavy” upper tail)
c = 1.2
c = 1.7

34. MLD Contamination S-M

35. Lognormal Distribution function: Inequality: s = 0.5 s = 0.7
s = 1.0 (“heavy” upper tail)

36. MLD Contamination: Lognormal

37. Pareto
a = 1.5 (“heavy” upper tail)
a = 2.0
a = 2.5

38. MLD Contamination Pareto

39. ERP at nominal 5%: MLD
Asymptotic tests
Bootstrap tests

40. ERP at nominal 5%: Theil
Asymptotic tests
Bootstrap tests

41. 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.

42. Why the Gini…? Why use the Gini coefficient? Obvious intuitive appeal
Sometimes suggested that Gini is less prone to the influence of outliers
Less sensitive to contamination in high incomes than GE indices.
But little to choose between…
the Gini coefficient and MLD
Gini and the logarithmic variance

43. The Bootstrap…? Does the bootstrap “get you out of trouble”?
bootstrap performs better than asymptotic methods,
but does it perform well enough?
In terms of the ERP, the bootstrap does well only for the Gini, MLD and logarithmic variance.
If we use a distribution with a heavy upper tail bootstrap performs poorly in the case of a = 0
even in large samples.