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Parametric Approaches to Welfare Measurement. Background Up until now our examination of welfare has been essentially non-parametric in a statistical.

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Presentation on theme: "Parametric Approaches to Welfare Measurement. Background Up until now our examination of welfare has been essentially non-parametric in a statistical."— Presentation transcript:

1 Parametric Approaches to Welfare Measurement

2 Background Up until now our examination of welfare has been essentially non-parametric in a statistical sense, we have not specified or estimated the parametric structure of the size distributions of income rather we have compared distributions via the empirical distributions functions directly. When we know the parametric structure of distributions some welfare implications can be inferred directly. (e.g. x~N(10,10), y~N(10+ε,10) → y FOD x and x~N(10,10), y~N(10,10-ε) → y SOD x). (There also exist explicit formulae for many of the inequality indices). There is an old and more recent literature which considers specifying and estimating size distributions of income and making such comparisons.

3 [A] Processes that Generate Log Normal Y under CLT arguments. Letting Y=ln(y=Income) yields what is called the Law of Proportionate Effects. Used most frequently in a time series context from the notion that y t =y t-1 (1+e t ) where e t is considered i.i.d. and E(e t ) = growth rate with V(e t ) small relative to 1. Gibrats Law (Gibrat (1930),(1931)): Y ti = μ i + Y t-1i + U ti implies for process of life length T, Y ~ N(μT,σ 2 T) i.e. y is non-stationary Kalecki’s modification (Kalecki(1945)): Y ti = μ i + β i Y t-1i + U ti where |β i | < 1; a stationary version of Gibrat’s Law. Y ~ N(μ/(1-β),σ 2 /(1-β 2 )).

4 Models of the Size Distribution of Income (Y=ln(per capita income) [B] Processes that generate “Pareto” type distributions under CLT arguments (Pareto (1897)) Gabaix (1999); Y(i,t)= μ(i) + Y(i,t-1) + u(i,t) (with Y(i,t) bounded from below) Champernowne (1953) Markov Chain Process where f(y t )=Mf(y t-1 ) with M a lower triangular transition matrix Double Pareto (Reed(2001)) Y(i,t)= μ(i) + Y(i,t) + u(i,t) (“t” governed by an exponential process)

5 The “Latest”

6 Models of the Size Distribution of Income (Y=ln(per capita income) [C] Ad Hoc Parametric Generalizations Mandelbrot(1960) Singh and Maddala (1976) McDonald (1984) Houthakker (2002)

7 Tests for verifying the model specification. Pearson’s Goodness of Fit Test. Partition the range of x into K mutually exclusive and exhaustive intervals then for a sample of size n let E i be the number of observations expected in the i’th interval and let O i be the number of observations actually observed in the i’th interval i=1,..,K then Σ K i=1 (O i -E i ) 2 /E i ~ χ 2 (K-1-h) where h is the number of estimated parameters needed to calculate the E i. Kolmogorov- Smirnov Test (see previous lecture) Hall Yatchew Expected Squared Difference Test.(see previous lecture).

8 Poverty rates under Gibrat’s Law For an absolute poverty line x* lim T->∞ Φ([(ln(x*/x 0 )-T(μ+0.5σ 2 ))/(σ√T)]) Growth exceeding -0.5σ 2 implies a poverty rate of 0 in the limit, for growth less than -0.5σ 2 the poverty rate would be 1. For a relative poverty line (0.6 of median income) the poverty rate would be Φ([ln(0.6)/(σ√T)]) which increases with time reaching.5 at infinity.

9 Poverty and Inequality Under Pareto’s Law Poverty and inequality measures would be constant over time The Gini for a Pareto distribution is 1/(2θ-1) which is 1 when the shape coefficient is one because in this case the Pareto distribution has no moments or an infinite mean. If the poverty frontier is a real lower boundary below which no-one is allowed to fall, the income distribution would end up as Pareto – hence a very strong test of the efficacy of such a frontier.

10 The Gini under Gibrat’s Law Gini may be written as: 2F(exp(ln(x 0 )+T(μ+0.5σ 2 ))| exp(ln(x 0 )+T(μ+0.5σ 2 )),Tσ 2 ) – 1 where F(z | θ, γ ) is the log normal distribution function with mean and variance θ, γ respectively This will tend to zero as T => ∞ when μ < -0.5σ 2 and will tend to 1 otherwise, note particularly for zero growth Gini will tend to 1.

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12 Mixture Distributions

13 Polarization and Mixtures If we think of societal income distributions as mixtures then we can analyze the progress of rich and poor groups as distinct entities. Use the Trapezoidal Index as a measure of relative poverty. 0.5(w p f p (x mp )+(1-w p )f r (x mr ))(x mr -x mp )

14 Parametric Approaches to Mixture Distributions. To facilitate modeling one can fit distributions to the data and track the fitted sub distributions. To illustrate these issues data on per capita GDP for 47 African countries together with their populations were drawn from the World Bank African Development Indicators CD-ROM for the years 1985, 1990, 1995, 2000, 2005 were used

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18 Comments Growth rate in the mean and variance greater in the un-weighted sample than in the weighted sample Much more evidence for Log normality than for Pareto.

19 Mobility in Africa

20 Tests of the Mixtures

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