Frank Cowell: TU Lisbon – Inequality & Poverty Inequality: Empirical Issues July 2006 Inequality and Poverty Measurement Technical University of Lisbon Frank Cowell

Frank Cowell: TU Lisbon – Inequality & Poverty Motivation Interested in sensitivity to extreme values for a number of reasons Interested in sensitivity to extreme values for a number of reasons Welfare properties of income distribution Welfare properties of income distribution Robustness in estimation Robustness in estimation Intrinsic interest in the very rich, the very poor. Intrinsic interest in the very rich, the very poor.

Frank Cowell: TU Lisbon – Inequality & Poverty 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).  not satisfactory for characterising general properties Welfare-theoretical approaches 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. Need a general and empirically applicable tool.

Frank Cowell: TU Lisbon – Inequality & Poverty Preliminaries A large class of inequality measures: A large class of inequality measures: Define two moments: Can be written as:

Frank Cowell: TU Lisbon – Inequality & Poverty The Influence Function Mixture distribution: Influence function: For the class of inequality measures: which yields:

Frank Cowell: TU Lisbon – Inequality & Poverty Some Standard Measures GE: Theil: MLD: Atkinson: Log var:

Frank Cowell: TU Lisbon – Inequality & Poverty …and their IFs GE: Theil: MLD: Atkinson: Log var:

Frank Cowell: TU Lisbon – Inequality & Poverty Special case The IF: The Gini coeff: where:

Frank Cowell: TU Lisbon – Inequality & Poverty Tail behaviour z   z z z z z zzzz z  0 [log z]  zzzz - log z - -  < 0  = 0  1  > 1 Log Var GiniGE

Frank Cowell: TU Lisbon – Inequality & Poverty Implications Generalised Entropy measures with  > 1 are very sensitive to high incomes in the data. Generalised Entropy measures with  > 1 are very sensitive to high incomes in the data. GE (  < 0) are very sensitive to low incomes 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 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. 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… So, let’s take a standard model…

Frank Cowell: TU Lisbon – Inequality & Poverty Singh-Maddala c = 1.2 c = 0.7 c = 1.7

Frank Cowell: TU Lisbon – Inequality & Poverty 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

Frank Cowell: TU Lisbon – Inequality & Poverty IFs based on S-M Gini

Frank Cowell: TU Lisbon – Inequality & Poverty IF using S-M: conclusions When z increases, IF increases faster with high values of . When z increases, IF increases faster with high values of . When z tends to 0, IF increases faster with small values of . When z tends to 0, IF increases faster with small values of . IF of Gini index increases slower than others but is larger for moderate values of z. 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. Comparison of the Gini index with GE or Log Variance does not lead to clear conclusions.

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

Frank Cowell: TU Lisbon – Inequality & Poverty 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 (  1) GE (0   1) is less sensitive if  is smaller Log Var is slightly more sensitive than Gini

Frank Cowell: TU Lisbon – Inequality & Poverty 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 (  1) GE (0   1) is less sensitive if  is larger Log Var is more sensitive than Gini

Frank Cowell: TU Lisbon – Inequality & Poverty Influential Observations Drop the i th 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

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

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

Frank Cowell: TU Lisbon – Inequality & Poverty High-leverage observations The term leaves open the question of whether such observations “belong” to the distribution 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. 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. Can use this performance to characterise the properties of inequality measures under certain conditions. Focus on the Error in Rejection Probability as a criterion. Focus on the Error in Rejection Probability as a criterion.

Frank Cowell: TU Lisbon – Inequality & Poverty Davidson-Flachaire (1) Even in very large samples the ERP of an asymptotic or bootstrap test based on the Theil index, can be significant 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. Tests are therefore not reliable. Three main possible causes : Three main possible causes : 1. Nonlinearity 2. Noise 3. Nature of the tails.

Frank Cowell: TU Lisbon – Inequality & Poverty Davidson-Flachaire (2) Three main possible causes : Three main possible causes : 1. Indices are nonlinear functions of sample moments. Induces biases and nonnormality in estimates. 2. Estimates of the covariances of the sample moments used to construct indices are often noisy. 3. 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. Simulation experiments show that case 3 is often quantitatively the most important. Statistical performance should be better with MLD and GE (0 <  < 1 ), than with Theil. Statistical performance should be better with MLD and GE (0 <  < 1 ), than with Theil.

Frank Cowell: TU Lisbon – Inequality & Poverty Empirical methods The empirical distribution Inequality estimate Empirical moments Empirical Distribution Indicator function

Frank Cowell: TU Lisbon – Inequality & Poverty Testing Test statistic Variance estimate For given value I 0 test

Frank Cowell: TU Lisbon – Inequality & Poverty Bootstrap To construct bootstrap test, resample from the original data. To construct bootstrap test, resample from the original data. Bootstrap inference should be superior 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 For bootstrap sample j, j = 1,…,B, a bootstrap statistic W * j is computed almost as W from the original data But I 0 in the numerator is replaced by the index Î estimated from the original data. But I 0 in the numerator is replaced by the index Î estimated from the original data. Then the bootstrap P-value is Then the bootstrap P-value is

Frank Cowell: TU Lisbon – Inequality & Poverty 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 oN = observations o oERP of GE (  =2) is 0.11 o oAsymptotic test over-rejects the null hypothesis o oThe actual level is 16%, when the nominal level is 5%.

Frank Cowell: TU Lisbon – Inequality & Poverty 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.

Frank Cowell: TU Lisbon – Inequality & Poverty More on ERP for GE What would happen in very large samples? –1–1–1– N=100,000 N=50,000 

Frank Cowell: TU Lisbon – Inequality & Poverty 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.

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

Frank Cowell: TU Lisbon – Inequality & Poverty Alternative distributions First consider comparative contamination performance for alternative distributions, same inequality index First consider comparative contamination performance for alternative distributions, same inequality index Use same diagrammatic tool as before Use same diagrammatic tool as before x-axis is the 100 different samples, sorted such inequality realizations are increasing x-axis is the 100 different samples, sorted such inequality realizations are increasing y-axis is RC(I) for the MLD index y-axis is RC(I) for the MLD index

Frank Cowell: TU Lisbon – Inequality & Poverty Singh-Maddala c = 1.2 c = 0.7 (“heavy” upper tail) c = 1.7 Inequality found from: Distribution function:

Frank Cowell: TU Lisbon – Inequality & Poverty Contamination S-M

Frank Cowell: TU Lisbon – Inequality & Poverty Lognormal  = 0.7  = 1.0 (“heavy” upper tail)  = 0.5 Inequality: Distribution function:

Frank Cowell: TU Lisbon – Inequality & Poverty Contamination: Lognormal

Frank Cowell: TU Lisbon – Inequality & Poverty Pareto  = 2.0  = 2.5  = 1.5 (“heavy” upper tail)

Frank Cowell: TU Lisbon – Inequality & Poverty MLD Contamination Pareto

Frank Cowell: TU Lisbon – Inequality & Poverty ERP at nominal 5%: MLD Asymptotic tests Bootstrap tests

Frank Cowell: TU Lisbon – Inequality & Poverty ERP at nominal 5%: Theil Asymptotic tests Bootstrap tests

Frank Cowell: TU Lisbon – Inequality & Poverty 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.

Frank Cowell: TU Lisbon – Inequality & Poverty Why the Gini…? Why use the Gini coefficient? 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. Less sensitive to contamination in high incomes than GE indices. But little to choose between… But little to choose between…  the Gini coefficient and MLD  Gini and the logarithmic variance

Frank Cowell: TU Lisbon – Inequality & Poverty 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? In terms of the ERP, the bootstrap does well only for the Gini, MLD and logarithmic variance. 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  = 0 If we use a distribution with a heavy upper tail bootstrap performs poorly in the case of  = 0  even in large samples.