The World Distribution of Income (from Log-Normal Country Distributions)

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The World Distribution of Income (from Log-Normal
Country Distributions)

• Xavier Sala-i-Martin
• Columbia University
• June 2008

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Goal

• Estimate WDI
• Estimate Poverty Rates and Counts
• Estimate Income Inequality across the worlds
  citizens

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Data

• GDP Per capita (PPP-Adjusted).
• We usually use these data as the mean of each
  country/year distribution of income (for example,
  when we estimate growth regressions)
• Note I decompose China and India into Rural and
  Urban
• Use local surveys to get relative incomes of
  rural and urban
• Apply the ratio to PWT GDP and estimate per
  capita income in Rural and Urban and treat them
  as separate data points (as if they were
  different countries)
• Using GDP Per Capita we know

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GDP Per Capita Since 1970
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Annual Growth Rate of World Per Capita GDP
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ß-Non-Convergence 1970-2006
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s-Divergence (191 countries)
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Histogram Income Per Capita (countries)
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Adding Population Weights
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Back
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Population-Weighted ß-convergence (1970-2006)
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But NA Numbers do not show Personal Situation
Need Individual Income Distribution

• We can use Survey Data
• Problem
• Not available for every year
• Not available for every country
• Survey means do not coincide with NA means

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Surveys not available every year

• Can Interpolate Income Shares (they are slow
  moving animals)
• Regression
• Near-Observation
• Cubic Interpolation
• Others

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Missing Countries

• Can approximate using neighboring countries

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Method Step 1 Interpolate

• Break up our sample of countries into
  regions(World Bank region definitions).
• Interpolate the quintile shares for country-years
  with no data, according to the following scheme,
  and in the following order
• Group I countries with several years of
  distribution data
• We calculate quintile shares of years with no
  income distribution data that are WITHIN the
  range of the set of years with data by cubic
  spline interpolation of the quintile share time
  series for the country.
• We calculate quintile shares of years with no
  data that are OUTSIDE this range by assuming that
  the share of each quintile rises each year after
  the data time series ends by beta/2i, where i is
  the number of years after the series ends, and
  beta is the coefficient of the slope of the OLS
  regression of the data time series on a constant
  and on the year variable. This extrapolation
  adjustment ensures that 1) the trend in the
  evolution of each quintile share is maintained
  for the first few years after data ends, and 2)
  the shares eventually attain their all-time
  average values, which is the best extrapolation
  that we could make of them for years far outside
  the range of our sample.
• Group II countries with only one year of
  distribution data.
• We keep the single year of data, and impute the
  quintile shares for other years to have the same
  deviations from this year as does the average
  quintile share time series taken over all Group I
  countries in the given region, relative to the
  year for which we have data for the given
  country. Thus, we assume that the countrys
  inequality dynamics are the same as those of its
  region, but we use the single data point to
  determine the level of the countrys income
  distribution.
• Group III countries with no distribution data.
• We impute the average quintile share time series
  taken over all Group I countries in the given
  region.

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Step 2 Find the s of the lognormal distribution
using least squares
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Step 3 Compute the resulting normal
distributions, and the poverty and inequality
statistics
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Step 4 (to generate confidence intervals)
Generate a new data set of quintiles

• Having obtained our point estimates, we obtain
  our standard errors by reproducing our original
  set of income distribution data by drawing
  samples of the sample size given in the country
  information sheets for the WIDER database from
  each estimated lognormal distribution
  corresponding to a country-year with data,
  calculating the sample quintile shares for each
  of these samples, discarding the sample

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Step 5 Repeat steps 1 through 4 using the
original values of s and µ to generate samples in
step 4

• Repeat the steps 1 through 4 to generate a new
  set of poverty and inequality measures for each
  country-year and the world as a whole over the 34
  years. We repeat the procedure N (300) times.
  Note that we do not use our estimates to generate
  income shares for country-years with no data, but
  we obtain the data by the procedure described
  above in order to keep the data-generating
  process identical to the one we used to obtain
  point estimates. Note also that in all
  iterations, we generate our samples from the
  lognormals with parameters given by the point
  estimates we obtain from the true, rather than
  synthetic data.

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Step 6 Find the mean and the standard deviation
of poverty and inequality measures

• Note that we have as many observations of the
  poverty and inequality measures as we have
  iterations of step 5. For this paper we used
  N300. We can now estimate the mean and standard
  deviation of these observations. If our
  assumption about the nature of the sampling in
  the surveys as roughly i.i.d., our assumption
  that the country-year distributions are
  lognormal, and our assumption that the
  interpolation provides reasonable estimates of
  quintile shares for country-years with no data
  are all correct, the standard deviation of the
  estimates for the N iterations should converge to
  the population standard deviation of the
  (complicated) estimator that we use to obtain our
  point estimates.

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Results
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Poverty Rates
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Rates or Headcounts?

• Veil of Ignorance Would you Prefer your children
  to live in country A or B?
• (A) 1.000.000 people and 500.000 poor (poverty
  rate 50)
• (B) 2.000.000 people and 666.666 poor (poverty
  rate 33)
• If you prefer (A), try country (C)
• (C) 500.000 people and 499.999 poor.

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Poverty Counts
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Gini and Atkinson Index (coef1)
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Sen Index (Income(1-gini))
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Atkinson Welfare Level
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MLD and Theil
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MLD Decomposition (ttotal, wwithin, and
bbetween country inequality)
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Theil Decomposition (ttotal, wwithin, and
bbetween country inequality)
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Regional Analysis
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Sub Saharan Africa
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East Asia
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South Asia
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Latin America
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Middle East and North Africa
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Eastern Europe
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Former Soviet Union
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Counts (all regions, 1/day)
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Sensitivity of Functional form Poverty Rates
(1/day) with Kernel, Normal, Gamma, Adjusted
Normal, Weibull distributions
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Sensitivity of Functional form Gini (1/day)
with Kernel, Normal, Gamma, Weibull distributions
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Sensitivity of GDP Source Poverty Rates
(1/day) with PWT, WB, and Maddison
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Sensitivity of Source of GDP Gini with PWT, WB,
and Maddison
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Sensitivity of Interpolation Method Poverty
Rates 1/day with Nearest, Linear, Cubic and
Baseline
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Sensitivity of Interpolation Method Gini with
Nearest, Linear, Cubic and Baseline
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Preliminary Results on Confidence Intervals with
Lognormal Gini
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Preliminary Results on Confidence Intervals with
Lognormal MLD