Parametric Approaches to Welfare Measurement. Background.
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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.
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 yt=yt-1(1+et) where et is considered i.i.d. and E(et) = growth rate with V(et) small relative to 1.
Gibrats Law (Gibrat (1930),(1931)): Yti = μi + Yt-1i + Uti implies for process of life length T, Y ~ N(μT,σ2T) i.e. y is non-stationary
Kalecki’s modification (Kalecki(1945)): Yti = μi + βiYt-1i + Uti where |βi| < 1; a stationary version of Gibrat’s Law. Y ~ N(μ/(1-β),σ2/(1-β2)).
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 Ei be the number of observations expected in the i’th interval and let Oi be the number of observations actually observed in the i’th interval i=1,..,K then ΣKi=1(Oi-Ei)2/Ei ~ χ2(K-1-h) where h is the number of estimated parameters needed to calculate the Ei.
Kolmogorov- Smirnov Test (see previous lecture)
Hall Yatchew Expected Squared Difference Test.(see previous lecture).
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