Gini coefficient

Graphical representation of the Gini coefficient
(The area of the whole triangle is defined as 1, not 0.5)

The Gini coefficient is a measure of statistical dispersion most prominently used as a measure of inequality of income distribution or inequality of wealth distribution. It is defined as a ratio with values between 0 and 1: A low Gini coefficient indicates more equal income or wealth distribution, while a high Gini coefficient indicates more unequal distribution. 0 corresponds to perfect equality (everyone having exactly the same income) and 1 corresponds to perfect inequality (where one person has all the income, while everyone else has zero income). The Gini coefficient requires that no one have a negative net income or wealth. Worldwide, Gini coefficients range from approximately 0.232 in Denmark to 0.707 in Namibia. Most free market nations have a Gini coeffient between.25 and.50.

The Gini index is the Gini coefficient expressed as a percentage, thus Denmark's Gini index is 23.2% (Mathematically, this is equal to the Gini coefficient of 0.232, but the percentage sign is often omitted in the Gini index.)

The Gini coefficient was developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variability and Mutability" (Italian: Variabilità e mutabilità ).

Contents

Calculation

The Gini coefficient is defined as a ratio of the areas on the Lorenz curve diagram. If the area between the line of perfect equality and Lorenz curve is A, and the area under the Lorenz curve is B, then the Gini coefficient is A/(A+B). Since A+B = 0.5, the Gini coefficient, G = A/(0.5) = 2A = 1-2B. If the Lorenz curve is represented by the function Y = L(X), the value of B can be found with integration and:

G = 1 - 2\,\int_0^1 L(X) dX.

In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example:

G = \frac{1}{n}\left ( n+1 - 2 \left ( \frac{\Sigma_{i=1}^n \; (n+1-i)y_i}{\Sigma_{i=1}^n y_i} \right ) \right )
This may be simplified to:
G = \frac{2 \Sigma_{i=1}^n \; i y_i}{n \Sigma_{i=1}^n y_i} -\frac{n+1}{n}
G = 1 - \frac{\Sigma_{i=1}^n \; f(y_i)(S_{i-1}+S_i)}{S_n}
where
S_i = \Sigma_{j=1}^i \; f(y_j)\,y_j\, and S_0 = 0\,
G = 1 - \frac{1}{\mu}\int_0^\infty (1-F(y))^2dy = \frac{1}{\mu}\int_0^\infty F(y)(1-F(y))dy
G(S) = \frac{1}{n-1}\left (n+1 - 2 \left ( \frac{\Sigma_{i=1}^n \; (n+1-i)y_i}{\Sigma_{i=1}^n y_i}\right ) \right )
is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like, G, G(S) has a simpler form:
G(S) = 1 - \frac{2}{n-1}\left ( n - \frac{\Sigma_{i=1}^n \; iy_i}{\Sigma_{i=1}^n y_i}\right ) .

There does not exist a sample statistic that is in general an unbiased estimator of the population Gini coefficient, like the relative mean difference.

Sometimes the entire Lorenz curve is not known, and only values at certain intervals are given. In that case, the Gini coefficient can be approximated by using various techniques for interpolating the missing values of the Lorenz curve. If ( X k , Yk ) are the known points on the Lorenz curve, with the X k indexed in increasing order ( X k - 1 < X k ), so that:

If the Lorenz curve is approximated on each interval as a line between consecutive points, then the area B can be approximated with trapezoids and:

G_1 = 1 - \sum_{k=1}^{n} (X_{k} - X_{k-1}) (Y_{k} + Y_{k-1})

is the resulting approximation for G. More accurate results can be obtained using other methods to approximate the area B, such as approximating the Lorenz curve with a quadratic function across pairs of intervals, or building an appropriately smooth approximation to the underlying distribution function that matches the known data. If the population mean and boundary values for each interval are also known, these can also often be used to improve the accuracy of the approximation.

The Gini coefficient calculated from a sample is a statistic and its standard error, or confidence intervals for the population Gini coefficient, should be reported. These can be calculated using bootstrap techniques but those proposed have been mathematically complicated and computationally onerous even in an era of fast computers. Ogwang (2000) made the process more efficient by setting up a “trick regression model” in which the incomes in the sample are ranked with the lowest income being allocated rank 1. The model then expresses the rank (dependent variable) as the sum of a constant A and a normal error term whose variance is inversely proportional to yk;

k = A + \ N(0, s^{2}/y_k)

Ogwang showed that G can be expressed as a function of the weighted least squares estimate of the constant A and that this can be used to speed up the calculaton of the jackknife esimate for the standard error. Giles (2004) argued that the standard error of the estimate of A can be used to derive that of the estimate of G directly without using a jackknife at all. However it has since been argued that this is dependent on the model’s assumptions about the error distributions (Ogwang 2004) and the independence of error terms (Reza & Gastwirth 2006) and that these assumptions are often not valid for real data sets. It may therefore be better to stick with jackknife methods such as those proposed by Yitzhaki (1991) and Karagiannis and Kovacevic (2000). The debate continues.

The gini coefficient can be calculated if you know the mean of a distribution, the number of people (or percentiles), and the income of each person (or percentile). Princeton development economist Angus Deaton (1997, 139) has simplified the Gini calculation to one easy formula:

G = \frac{N+1}{N-1}-\frac{2}{N(N-1)u}(\Sigma_{i=1}^n \; P_iX_i)

where u is mean income of the population, Pi is the income rank P of person i, with income X, such that the richest person receives a rank of 1 and the poorest a rank of N. This effectively gives higher weight to poorer people in the income distribution, which allows the Gini to meet the Transfer Principle.

Income Gini indices in the world

A complete listing is in list of countries by income equality; the article economic inequality discusses the social and policy aspects of income and asset inequality.

Gini coefficient, income distribution by country.

While most developed European nations tend to have Gini indices between 24 and 36, the United States' and Mexico's Gini indices are both above 40, indicating that the United States and Mexico have greater inequality. Using the Gini can help quantify differences in welfare and compensation policies and philosophies. However it should be borne in mind that the Gini coefficient can be misleading when used to make political comparisons between large and small countries (see criticisms section).

The Gini index for the entire world has been estimated by various parties to be between 56 and 66.[1][2]


Gini indices, income distribution over time for selected countries

Correlation with per-capita GDP

Poor countries (those with low per-capita GDP) generally have higher Gini indices, spread between 40 and 65, with extremes at 25 and 71, while rich countries generally have lower Gini indices (under 40). The lowest Gini coefficients (under 30) can be found in continental Europe. Overall, there is a clear negative correlation between Gini coefficient and GDP per capita, the U.S.A, Hongkong and Singapore being rich exceptions with high Gini coefficients.

In many of the former socialist countries and in-development capitalist countries (e.g., Brazil), the sizeable underground economy may hide income for many. In such a case, earning/wealth statistics over-represent certain income ranges (i.e., in lower-income regions), and may alter the Gini coefficient either up or down.

US income Gini indices over time

Gini indices for the United States at various times, according to the US Census Bureau:

Note: Estimates are not 100% accurate.

Advantages of Gini coefficient as a measure of inequality

Disadvantages of Gini coefficient as a measure of inequality

For this reason, the scores calculated for individual countries within the EU are difficult to compare with the score of the entire US: the overall value for the EU should be used in that case, 31.3[4], which is still much lower than the United States', 45.[5] Using decomposable inequality measures (e.g. the Theil index T converted by 1-{e^{-T}} into a inequality coefficient) averts such problems.

Problems in using the Gini coefficient

General problems of measurement

As one result of this criticism, in addition to or in competition with the Gini coefficient entropy measures are frequently used (e.g. the Theil Index and the index of Atkinson). These measures attempt to compare the distribution of resources by intelligent agents in the market with a maximum entropy random distribution, which would occur if these agents acted like non-intelligent particles in a closed system following the laws of statistical physics.

Credit risk

The Gini coefficient is also commonly used for the measurement of the discriminatory power of rating systems in credit risk management. Since Gini coefficient addresses wealth inequality it may be important to understand what a transformative asset is. Transformative assets increase the Gini coefficient as they provide a family or individual with a wealth advantage over most persons.

The discriminatory power refers to a credit risk model's ability to differentiate between defaulting and non-defaulting clients. The above formula G_1 may be used for the final model and also at individual model factor level, to quantify the discriminatory power of individual factors. In this case, negative Gini Coefficient values are possible. This is as a result of too many non defaulting clients falling into the lower points scale e.g. factor has a 10 point scale and 30% of non defaulting clients are being assigned the lowest points available e.g. 0 or negative points. This indicates that the factor is behaving in a counter-intuitive manner and would require further investigation at the model development stage.

References: The Analytics of risk model validation

See also

  • List of countries by income equality
  • List of countries by Human Development Index
  • Human Poverty Index
  • Pareto distribution
  • ROC analysis
  • Social welfare provision
  • Welfare economics
  • Atkinson index
  • Theil index
  • Spreadsheet computations
  • Robin Hood index
  • Suits index
  • Income inequality metrics
  • Wealth condensation

References

  1. ↑ Bob Sutcliffe (April 2007), Postscript to the article ‘World inequality and globalization’ (Oxford Review of Economic Policy, Spring 2004), http://siteresources.worldbank.org/INTDECINEQ/Resources/PSBSutcliffe.pdf, retrieved on 2007-12-13 
  2. ↑ United Nations Development Programme
  3. ↑ Note that the calculation of the index for the United States was changed in 1992, resulting in an upwards shift of about 2.
  4. ↑ European Union, CIA World Factbook, https://www.cia.gov/library/publications/the-world-factbook/geos/ee.html, retrieved on 2007-12-13 
  5. ↑ United States, CIA World Factbook, https://www.cia.gov/library/publications/the-world-factbook/geos/us.html, retrieved on 2007-12-13 
  6. ↑ Friedman, David D.
  7. ↑ (Data from the Statistics Sweden.)

Further reading

  • Amiel, Y.; Cowell, F.A. (1999). Thinking about Inequality. Cambridge. 
  • Anand, Sudhir (1983). Inequality and Poverty in Malaysia. New York: Oxford University Press. 
  • Brown, Malcolm (1994). "Using Gini-Style Indices to Evaluate the Spatial Patterns of Health Practitioners: Theoretical Considerations and an Application Based on Alberta Data". Social Science Medicine 38: 1243–1256. doi:10.1016/0277-9536(94)90189-9. 
  • Chakravarty, S. R. (1990). Ethical Social Index Numbers. New York: Springer-Verlag. 
  • Deaton, Angus (1997). Analysis of Household Surveys. Baltimore MD: Johns Hopkins University Press. 
  • Dixon, PM, Weiner J., Mitchell-Olds T, Woodley R. (1987). "Bootstrapping the Gini coefficient of inequality". Ecology 68: 1548–1551. doi:10.2307/1939238. 
  • Dorfman, Robert (1979). "A Formula for the Gini Coefficient". The Review of Economics and Statistics 61: 146–149. doi:10.2307/1924845. 
  • Gastwirth, Joseph L. (1972). "The Estimation of the Lorenz Curve and Gini Index". The Review of Economics and Statistics 54: 306–316. doi:10.2307/1937992. 
  • Giles, David (2004). "Calculating a Standard Error for the Gini Coefficient: Some Further Results". Oxford Bulletin of Economics and Statistics 66: 425–433. doi:10.1111/j.1468-0084.2004.00086.x. 
  • Gini, Corrado (1912). "Variabilità e mutabilità" Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Veschi (1955).
  • Gini, Corrado (1921). "Measurement of Inequality and Incomes". The Economic Journal 31: 124–126. doi:10.2307/2223319. 
  • Karagiannis, E. and Kovacevic, M. (2000). "A Method to Calculate the Jackknife Variance Estimator for the Gini Coefficient". Oxford Bulletin of Economics and Statistics 62: 119–122. doi:10.1111/1468-0084.00163. 
  • Mills, Jeffrey A.; Zandvakili, Sourushe (1997). "Statistical Inference via Bootstrapping for Measures of Inequality". Journal of Applied Econometrics 12: 133–150. doi:10.1002/(SICI)1099-1255(199703)12:2<133::AID-JAE433>3.0.CO;2-H. 
  • Modarres, Reza and Gastwirth, Joseph L. (2006). "A Cautionary Note on Estimating the Standard Error of the Gini Index of Inequality". Oxford Bulletin of Economics and Statistics 68: 385–390. doi:10.1111/j.1468-0084.2006.00167.x. 
  • Morgan, James (1962). "The Anatomy of Income Distribution". The Review of Economics and Statistics 44: 270–283. doi:10.2307/1926398. 
  • Ogwang, Tomson (2000). "A Convenient Method of Computing the Gini Index and its Standard Error". Oxford Bulletin of Economics and Statistics 62: 123–129. doi:10.1111/1468-0084.00164. 
  • Ogwang, Tomson (2004). "Calculating a Standard Error for the Gini Coefficient: Some Further Results: Reply". Oxford Bulletin of Economics and Statistics 66: 435–437. doi:10.1111/j.1468-0084.2004.00087.x. 
  • Xu, Kuan (January, 2004). "How Has the Literature on Gini's Index Evolved in the Past 80 Years?". Department of Economics, Dalhousie University. Retrieved on 2006-06-01. The Chinese version of this paper appears in Xu, Kuan (2003). "How Has the Literature on Gini's Index Evolved in the Past 80 Years?". China Economic Quarterly 2: 757–778. 
  • Yitzhaki, S. (1991). "Calculating Jackknife Variance Estimators for Parameters of the Gini Method". Journal of Business and Economic Statistics 9: 235–239. doi:10.2307/1391792. 

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