Arthur CHARPENTIER - Welfare, Inequality and Poverty Poverty, in France Déjà en hausse de 0,5 point en 2009, le taux de pauvreté monétaire a augmenté en 2010 de 0,6 point pour atteindre 14,1%, soit son plus haut niveau depuis 1997. 8,6 millions de personnes vivaient en 2010 en-dessous du seuil de pauvreté monétaire (964 euros par mois). Elles n’étaient que 8,1 millions en 2009. Mais il y a pire : une personne pauvre sur deux vit avec moins de 781 euros par mois En 2010, le chômage a peu contribuéà l’augmentation de la pauvreté (les chômeurs représentent à peine 4% de l’accroissement du nombre des personnes pauvres). C’est du coté des inactifs qu’il faut plutôt se tourner : les retraités (11%), les adultes inactifs autres que les étudiants et les retraites (16%) - souvent les titulaires de minima sociaux - et les enfants. Les moins de 18 ans contribuent pour près des deux tiers (63%) à l’augmentation du nombre de personnes pauvres [...] 34
Arthur CHARPENTIER - Welfare, Inequality and Poverty Incomes in France See Houdré, Missègue & Seguin Inégalités de niveau de vie et pauvreté , 2012 35
Arthur CHARPENTIER - Welfare, Inequality and Poverty Incomes in France See Houdré, Missègue & Seguin Inégalités de niveau de vie et pauvreté , 2012 36
Arthur CHARPENTIER - Welfare, Inequality and Poverty Incomes in France See Houdré, Missègue & Seguin Inégalités de niveau de vie et pauvreté , 2012 37
Arthur CHARPENTIER - Welfare, Inequality and Poverty Incomes in France See Houdré, Missègue & Seguin Inégalités de niveau de vie et pauvreté , 2012 38
Arthur CHARPENTIER - Welfare, Inequality and Poverty Incomes in France See Houdré, Missègue & Seguin Inégalités de niveau de vie et pauvreté , 2012 39
Arthur CHARPENTIER - Welfare, Inequality and Poverty Incomes in France See Houdré, Missègue & Seguin Inégalités de niveau de vie et pauvreté , 2012 40
Arthur CHARPENTIER - Welfare, Inequality and Poverty Income ? See Statistics Canada Total Income , via Flachaire (2015). 41
Arthur CHARPENTIER - Welfare, Inequality and Poverty Income ? Micro vs macro Piketty Capital in the Twenty-First Century , 2014, 42
Arthur CHARPENTIER - Welfare, Inequality and Poverty Income ? Micro vs macro Piketty Capital in the Twenty-First Century , 2014, 43
Arthur CHARPENTIER - Welfare, Inequality and Poverty Income ? Micro vs macro To compare various household incomes • Oxford scale (OECD equivalent scale) ◦ 1.0 to the first adult ◦ 0.7 to each additional adult (aged 14, and more) ◦ 0.5 to each child • OECD-modified equivalent scale (late 90s by eurostat) ◦ 1.0 to the first adult ◦ 0.5 to each additional adult (aged 14, and more) ◦ 0.3 to each child • More recent OECD scale ◦ square root of household size 44
Arthur CHARPENTIER - Welfare, Inequality and Poverty Income ? Micro vs macro 45
Arthur CHARPENTIER - Welfare, Inequality and Poverty Income ? Tax Issues E.g. total taxes paid by total wage 46
Arthur CHARPENTIER - Welfare, Inequality and Poverty Income ? Tax Issues via Landais, Piketty & Saez Pour une révolution fiscale , 2011 47
Arthur CHARPENTIER - Welfare, Inequality and Poverty Income ? Tax Issues via Landais, Piketty & Saez Pour une révolution fiscale , 2011 48
Arthur CHARPENTIER - Welfare, Inequality and Poverty International Comparisons, Puchasing Power Parity See The Economist The Big Mac index , 2014 49
Arthur CHARPENTIER - Welfare, Inequality and Poverty International Comparisons, Puchasing Power Parity See The Economist The Big Mac index , 2014, via Flachaire 50
Arthur CHARPENTIER - Welfare, Inequality and Poverty International Comparisons, Puchasing Power Parity Piketty Capital in the Twenty-First Century , 2014, wealth, income, wage 51
Arthur CHARPENTIER - Welfare, Inequality and Poverty From Income and Wealth to Human Development The Human Development Index (HDI, see wikipedia ) is a composite statistic of life expectancy, education, and income indices used to rank countries into four tiers of human development. It was created by Indian economist Amartya Sen and Pakistani economist Mahbub ul Haq in 1990, and was published by the United Nations Development Programme. The HDI is a composite index at value between 0 (awful) and 1 (perfect) based on the mixing of three basic indices aiming at representing on an equal footing measures of helth, education and standard of living. 52
Arthur CHARPENTIER - Welfare, Inequality and Poverty HDI Computation, new method (2010) Published on 4 November 2010 (and updated on 10 June 2011), starting with the 2010 Human Development Report the HDI combines three dimensions : — A long and healthy life : Life expectancy at birth — An education index : Mean years of schooling and Expected years of schooling — A decent standard of living : GNI per capita (PPP US$) In its 2010 Human Development Report, the UNDP began using a new method of calculating the HDI. The following three indices are used. The idea is to define a x index as x − min ( x ) x index = max ( x ) − min ( x ) 1. Health, Life Expectancy Index (LEI) = LE − 20 85 − 20 where LE is Life Expectancy at birth 53
Arthur CHARPENTIER - Welfare, Inequality and Poverty HDI Computation, new method (2010) 2. Education, Education Index (EI) = MYSI + EYSI 2 2.1 Mean Years of Schooling Index (MYSI) = MYS 15 where MYS is the Mean years of schooling (Years that a 25-year-old person or older has spent in schools) 2.2 Expected Years of Schooling Index (EYSI) = EYS 18 EYS : Expected years of schooling (Years that a 5-year-old child will spend with his education in his whole life) 3. Standard of Living Income Index (II) = log(GNIpc) − log(100) log(75 , 000) − log(100) where GNIpc : Gross national income at purchasing power parity per capita Finally, the HDI is the geometric mean of the previous three normalized indices : √ 3 HDI = LEI · EI · II. 54
Arthur CHARPENTIER - Welfare, Inequality and Poverty Economic Well-Being See Osberg The Measurement of Economic Well-Being , 1985 and Osberg & Sharpe New Estimates of the Index of Economic Well-being , 2002 See also Jank & Owens Inequality in the United States , 2013, for stats and graphs about inequalities in the U.S., in terms of health, education, crime, etc. 55
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 56
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 57
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 58
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 59
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 60
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 61
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 62
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 63
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 64
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 65
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 66
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 67
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 68
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 69
Arthur CHARPENTIER - Welfare, Inequality and Poverty Various Aspects of Inequalities in the U.S. Jank & Owens Inequality in the United States , 2013 70
Arthur CHARPENTIER - Welfare, Inequality and Poverty Modeling Income Distribution Let { x 1 , · · · , x n } denote some sample. Then n n � � x = 1 1 x i = nx i n i =1 i =1 This can be used when we have census data. ●● ● ● ● 1 load ( u r l ( " http : // freakonometrics . f r e e . f r / income_5 . RData" ) ) 2 income < − s o r t ( income ) 3 plot ( 1 : 5 , income ) 0 50000 100000 150000 200000 250000 income It is possible to use survey data. If π i denote the probability to be drawn, use weights 1 ω i ∝ nπ i 71
Arthur CHARPENTIER - Welfare, Inequality and Poverty The weighted average is then n � ω i x ω = ω x i i =1 where ω = � ω i . This is an unbaised estimator of the population mean. Sometime, data are obtained from stratified samples : before sampling, members of the population are groupes in homogeneous subgroupes (called a strata). Given S strata, such that the population in strata s is N s , then S � � N s N x s where x s = 1 x S = x i N s s =1 i ∈S s 72
Arthur CHARPENTIER - Welfare, Inequality and Poverty Statistical Tools Used to Describe the Distribution Consider a sample { x 1 , · · · , x n } . Usually, the order is not important. So let us order those values, x 1: n ≤ x 2: n ≤ · · · ≤ x n − 1: n ≤ x n : n ���� ���� min { x i } max { x i } As usual, assume that x i ’s were randomly drawn from an (unknown) distribution F . If F denotes the cumulative distribution function, F ( x ) = P ( X ≤ x ), one can prove that F ( x i : n ) = P ( X ≤ x i : n ) ∼ i n The quantile function is defined as the inverse of the cumulative distribution function F , Q ( u ) = F − 1 ( u ) or F ( Q ( u )) = P ( X ≤ Q ( u )) = u 73
Arthur CHARPENTIER - Welfare, Inequality and Poverty Lorenz curve Lorenz curve The empirical version of Lorenz curve is 1.0 0.8 � n, 1 i L = x j : n 0.6 nx L(p) j ≤ i ● 0.4 ● 0.2 1 > plot ( ( 0 : 5 ) / 5 , c (0 ,cumsum( income ) /sum( income ● ● ) ) ) 0.0 0.0 0.2 0.4 0.6 0.8 1.0 p 74
Arthur CHARPENTIER - Welfare, Inequality and Poverty Gini Coefficient A Gini coefficient is defined as the ratio of areas, A + B . It can be defined using order statistics as n � 2 i · x i : n − n + 1 1.0 ● G = n ( n − 1) x n − 1 0.8 i =1 0.6 L(p) 1 > n < − length ( income ) ● 0.4 A 2 > mu < − mean( income ) ● 0.2 B ● 3 > 2 ∗ sum ( ( 1 : n) ∗ s o r t ( income ) ) / (mu ∗ n ∗ (n − 1)) − (n ● 0.0 ● +1)/ (n − 1) 0.0 0.2 0.4 0.6 0.8 1.0 [ 1 ] 0.5800019 4 p 75
Arthur CHARPENTIER - Welfare, Inequality and Poverty Distribution Fitting Assume that we now have more observations, 1 > load ( u r l ( " http : // freakonometrics . f r e e . f r /income_500. RData" ) ) We can use some histogram to visualize the distribu- tion of the income Histogram of income 40 1 > summary( income ) Min . 1 st Qu. Median Mean 3rd Qu. 2 30 Max. Frequency 2191 23830 42750 77010 87430 20 3 2003000 10 4 > s o r t ( income ) [ 4 9 5 : 5 0 0 ] [ 1 ] 465354 489734 512231 539103 627292 5 0 2003241 0 500000 1000000 1500000 2000000 income 6 > h i s t ( income , breaks=seq (0 ,2005000 , by=5000) ) 76
Arthur CHARPENTIER - Welfare, Inequality and Poverty Distribution Fitting Because of the dispersion, look at the histogram of the logarithm of the data Histogram of log(income, 10) 1 > h i s t ( log ( income , 1 0 ) , breaks=seq ( 3 , 6 . 5 , length =51) ) 40 2 > boxplot ( income , h o r i z o n t a l= TRUE, log=" x " ) 30 Frequency 20 10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 log(income, 10) 2e+03 1e+04 5e+04 2e+05 1e+06 77
Arthur CHARPENTIER - Welfare, Inequality and Poverty Distribution Fitting 1.0 The cumulative distribution function (on the log of 0.8 the income) Cumulated Probabilities 0.6 1 > u < − s o r t ( income ) 0.4 2 > v < − ( 1 : 5 0 0 ) /500 0.2 3 > plot (u , v , type=" s " , log=" x " ) 0.0 2e+03 1e+04 5e+04 2e+05 1e+06 Income (log scale) 78
Arthur CHARPENTIER - Welfare, Inequality and Poverty Distribution Fitting 1e+06 2e+05 If we invert that graph, we have the quantile function Income (log scale) 5e+04 1 > plot (v , u , type=" s " , c o l=" red " , log=" y " ) 1e+04 2e+03 0.0 0.2 0.4 0.6 0.8 1.0 Probabilities 79
Arthur CHARPENTIER - Welfare, Inequality and Poverty Distribution Fitting 1.0 ● 0.8 On that dataset, Lorenz curve is 0.6 L(p) 1 > plot ( ( 0 : 5 0 0 ) / 500 , c (0 ,cumsum( income ) /sum( 0.4 income ) ) ) 0.2 0.0 ● 0.0 0.2 0.4 0.6 0.8 1.0 p 80
Arthur CHARPENTIER - Welfare, Inequality and Poverty Distribution and Confidence Intervals There are two techniques to get the distribution of an estimator � θ , — a parametric one, based on some assumptions on the underlying distribution, — a nonparametric one, based on sampling techniques � � µ, σ 2 If X i ’s have a N ( µ, σ 2 ) distribution, then X ∼ N n But sometimes, distribution can only be obtained as an approximation, because of asymptotic properties. � � µ, σ 2 From the central limit theorem, X → N as n → ∞ . n In the nonparametric case, the idea is to generate pseudo-samples of size n , by resampling from the original distribution. 81
Arthur CHARPENTIER - Welfare, Inequality and Poverty Bootstraping Consider a sample x = { x 1 , · · · , x n } . At step b = 1 , 2 , · · · , B , generate a pseudo sample x b by sampling (with replacement) within sample x . Then compute any statistic � θ ( x b ) 1 > boot < − function ( sample , f , b=500){ 2 + F < − rep (NA, b) 3 + n < − length ( sample ) 4 + f o r ( i in 1 : b) { 5 + idx < − sample ( 1 : n , s i z e=n , r e p l a c e= TRUE) 6 + F[ i ] < − f ( sample [ idx ] ) } 7 + return (F) } 82
Arthur CHARPENTIER - Welfare, Inequality and Poverty Bootstraping Let us generate 10,000 bootstraped sample, and com- pute Gini index on those 15 1 >boot_g i n i < − boot ( income , gini ,1 e4 ) To visualize the distribution of the index 10 Density 1 > h i s t ( boot_gini , p r o b a b i l i t y= TRUE) 5 2 > u < − seq ( . 4 , . 7 , length =251) 3 > v < − dnorm(u , mean( boot_g i n i ) , sd ( boot_g i n i ) 0 ) 0.45 0.50 0.55 0.60 4 > l i n e s (u , v , c o l=" red " , l t y =2) boot_gini 83
Arthur CHARPENTIER - Welfare, Inequality and Poverty Continuous Versions The empirical cumulative distribution function n � F n ( x ) = 1 � 1 ( x i ≤ x ) n i =1 Observe that F n ( x j : n ) = j � n If F is absolutely continuous, � x f ( t ) dt i.e. f ( x ) = dF ( x ) F ( x ) = . dx 0 Then � b P ( x ∈ [ a, b ]) = f ( t ) dt = F ( b ) − F ( a ) . a 84
Arthur CHARPENTIER - Welfare, Inequality and Poverty Continuous Versions One can define quantiles as x = Q ( p ) = F − 1 ( p ) The expected value is � ∞ � ∞ � 1 µ = xf ( x ) dx = [1 − F ( x )] dx = Q ( p ) dp. 0 0 0 We can compute the average standard of living of the group below z . This is equivalent to the expectation of a truncated distribution. � � � z � ∞ 1 1 − F ( x ) µ − z = xf ( x ) dx = fx F ( z ) F ( z ) 0 0 85
Arthur CHARPENTIER - Welfare, Inequality and Poverty Continuous Versions Lorenz curve is p �→ L ( p ) with � Q ( p ) L ( p ) = 1 xf ( x ) dx µ 0 Gastwirth (1971) proved that � p � p 0 Q ( u ) du L ( p ) = 1 Q ( u ) du = � 1 µ 0 Q ( u ) du 0 The numerator sums the incomes of the bottom p proportion of the population. The denominator sums the incomes of all the population. L is a [0 , 1] → [0 , 1] function, continuous if F is continuous. Observe that L is increasing, since dL ( p ) = Q ( p ) dp µ Further, L is convex 86
Arthur CHARPENTIER - Welfare, Inequality and Poverty The sample case � i � i � j =1 x j : n � n L = n j =1 x j : n The points { i/n, L ( i/n ) } are then linearly interpolated to complete the corresponding Lorenz curve. The continuous distribution case � F − 1 ( p ) � p ydF ( y ) 1 0 F − 1 ( u ) du � ∞ L ( p ) = = E ( X ) ydF ( y ) 0 0 with p ∈ (0 , 1). Let L be a continuous function on [0 , 1], then L is a Lorenz curve if and only if L (0) = 0 , L (1) = 1 , L ′ (0 + ) ≥ 0 and L ′′ ( p ) ≥ 0 on [0 , 1] . 87
Arthur CHARPENTIER - Welfare, Inequality and Poverty From Lorenz to Bonferroni The Bonferroni curve is B ( p ) = L ( p ) p and the Bonferroni index is � 1 BI = 1 − B ( p ) dp. 0 Define i � P i = i n and Q i = 1 x j nx j =1 then � P i − Q i � n − 1 � 1 B = n − 1 P i i =1 88
Arthur CHARPENTIER - Welfare, Inequality and Poverty Gini and Pietra indices The Gini index is defined as twice the area between the egalitarian line and the Lorenz curve � 1 � 1 [ p − L ( p )] dp = 1 − 2 G = 2 L ( p ) dp 0 0 which can also be writen � ∞ 1 [1 − F ( x )] 2 dx 1 − E ( X ) 0 Pietra index is defined as the maximal vertical deviation between the Lorenz curve and the egalitarian line p ∈ (0 , 1) { p − L ( p ) } = E ( | X − E ( X ) | ) P = max 2 E ( X ) if F is strictly increasing (the maximum is reached in p ⋆ = F ( E ( X ))) 89
Arthur CHARPENTIER - Welfare, Inequality and Poverty Examples E.g. consider the uniform distribution F ( x ) = min { 1 , x − a b − a 1 ( x ≥ a ) } Then L ( p ) = 2 ap + ( b − a ) 2 p 2 a + b and Gini index is b − a G = 3( a + b ) E.g. consider a Pareto distribution, � x 0 � α F ( x ) = 1 − , x ≥ x 0 , x with shape parameter α > 0. Then x 0 F − 1 ( u ) = 1 (1 − u ) α 90
Arthur CHARPENTIER - Welfare, Inequality and Poverty and L ( p ) = 1 − [1 − p ] 1 − 1 α p ∈ (0 , 1) . and Gini index is 1 G = 2 α − 1 while Pietra index is, if α > 1 P = ( α − 1) α − 1 α α E.g. consider the lognormal distribution, � log x − µ � F ( x ) = Φ σ then L ( p ) = Φ(Φ − 1 ( p ) − σ ) p ∈ (0 , 1) . and Gini index is � σ � √ − 1 G = 2Φ 2 91
Arthur CHARPENTIER - Welfare, Inequality and Poverty Fitting a Distribution The standard technique is based on maximum likelihood estimation, provided by 1 > l i b r a r y (MASS) 2 > f i t d i s t r ( income , " lognormal " ) meanlog sdlog 3 10.72264538 1.01091329 4 ( 0.04520942) ( 0.03196789) 5 For other distribution (such as the Gamma distribution), we might have to rescale 1 > ( f i t_g < − f i t d i s t r ( income/1e2 , "gamma" ) ) shape rate 2 1.0812757769 0.0014040438 3 (0.0473722529) (0.0000544185) 4 5 > ( f i t_ln < − f i t d i s t r ( income/1e2 , " lognormal " ) ) meanlog sdlog 6 6.11747519 1.01091329 7 (0.04520942) (0.03196789) 8 92
Arthur CHARPENTIER - Welfare, Inequality and Poverty Fitting a Distribution We can compare the densities 1 > u=seq (0 ,2 e5 , length =251) 1.0 2 > h i s t ( income , breaks=seq (0 ,2005000 , by=5000) , c o l=rgb ( 0 , 0 , 1 , . 5 ) , border=" white " , xlim=c 0.8 Cumulated Probabilities (0 ,2 e5 ) , p r o b a b i l i t y= TRUE) 0.6 3 > v_g < − dgamma(u/1e2 , f i t_g$ estimate [ 1 ] , f i t 0.4 _g$ estimate [ 2 ] ) /1e2 4 > v_ln < − dlnorm (u/1e2 , f i t_ln $ estimate [ 1 ] , 0.2 Gamma f i t_ln $ estimate [ 2 ] ) /1e2 0.0 Log Normal 5 > l i n e s (u , v_g , c o l=" red " , l t y =2) 0 50000 100000 150000 200000 Income 6 > l i n e s (u , v_ln , c o l=rgb ( 1 , 0 , 0 , . 4 ) ) 93
Arthur CHARPENTIER - Welfare, Inequality and Poverty Fitting a Distribution or the cumuluative distributions 1 x < − s o r t ( income ) Gamma 1.5e−05 Log Normal 2 y < − ( 1 : 5 0 0 ) /500 3 plot (x , y , type=" s " , c o l=" black " ) 1.0e−05 4 v_g < − pgamma(u/1e2 , f i t_g$ estimate [ 1 ] , f i t_g Density $ estimate [ 2 ] ) 5.0e−06 5 v_ln < − plnorm (u/1e2 , f i t_ln $ estimate [ 1 ] , f i t _ln $ estimate [ 2 ] ) 0.0e+00 l i n e s (u , v_g , c o l=" red " , l t y =2) 6 0 50000 100000 150000 200000 l i n e s (u , v_ln , c o l=rgb ( 1 , 0 , 0 , . 4 ) ) 7 income One might consider the parametric version of Lorenz curve, to confirm the goodness of fit, e.g. a lognormal distribution with σ = 1 since 1 > f i t d i s t r ( income , " lognormal " ) meanlog sdlog 2 10.72264538 1.01091329 3 94
Arthur CHARPENTIER - Welfare, Inequality and Poverty Fitting a Distribution We can use functions of R 1 l i b r a r y ( ineq ) 2 Lc . sim < − Lc ( income ) 1.0 ● 3 plot ( 0 : 1 , 0 : 1 , xlab="p" , ylab="L(p) " , c o l=" white 0.8 " ) 0.6 4 polygon ( c (0 ,1 ,1 ,0) , c (0 ,0 ,1 ,0) , c o l=rgb L(p) ( 0 , 0 , 1 , . 1 ) , border= NA) 0.4 5 polygon ( Lc . sim$p , Lc . sim$L , c o l=rgb ( 0 , 0 , 1 , . 3 ) , 0.2 border= NA) 0.0 ● l i n e s ( Lc . sim ) 6 0.0 0.2 0.4 0.6 0.8 1.0 7 segments (0 ,0 ,1 ,1) p l i n e s ( Lc . lognorm , parameter =1, l t y =2) 8 95
Arthur CHARPENTIER - Welfare, Inequality and Poverty Standard Parametric Distribution For those distributions, we mention the R names in the gamlss package. Inference can be done using f i t < − gamlss ( y~ 1 , family= LNO) 1 • log normal 1 2 π e − (ln x − µ )2 √ f ( x ) = , x ≥ 0 2 σ 2 xσ with mean e µ + σ 2 / 2 , median e µ , and variance ( e σ 2 − 1) e 2 µ + σ 2 1 LNO(mu. l i n k = " i d e n t i t y " , sigma . l i n k = " log " ) 2 dLNO(x , mu = 1 , sigma = 0.1 , nu = 0 , log = FALSE) • gamma f ( x ) = x 1 /σ 2 − 1 exp[ − x/ ( σ 2 µ )] , x ≥ 0 ( σ 2 µ ) 1 /σ 2 Γ(1 /σ 2 ) 96
Arthur CHARPENTIER - Welfare, Inequality and Poverty with mean µ and variance σ 2 1 GA(mu. l i n k = " log " , sigma . l i n k =" log " ) 2 dGA(x , mu = 1 , sigma = 1 , log = FALSE) • Pareto f ( x ) = α x α m x α +1 for x ≥ x m with cumulated distribution � x m � α F ( x ) = 1 − for x ≥ x m x x 2 αx m m α with mean ( α − 1) if α > 1, and variance ( α − 1) 2 ( α − 2) if α > 2. 1 PARETO2(mu. l i n k = " log " , sigma . l i n k = " log " ) 2 dPARETO2(x , mu = 1 , sigma = 0 .5 , log = FALSE) 97
Arthur CHARPENTIER - Welfare, Inequality and Poverty Larger Families • GB1 - generalized Beta type 1 f ( x ) = | a | x ap − 1 (1 − ( x/b ) a ) q − 1 , 0 < x a < b a b ap B ( p, q ) where b , p , and q are positive 1 GB1(mu. l i n k = " l o g i t " , sigma . l i n k = " l o g i t " , nu . l i n k = " log " , tau . l i n k = " log " ) 2 dGB1(x , mu = 0 .5 , sigma = 0 .4 , nu = 1 , tau = 1 , log = FALSE) The GB1 family includes the generalized gamma(GG), and Pareto as special cases. • GB2 - generalized Beta type 2 | a | x ap − 1 f ( x ) = b ap B ( p, q )(1 + ( x/b ) a ) p + q 98
Arthur CHARPENTIER - Welfare, Inequality and Poverty 1 GB2(mu. l i n k = " log " , sigma . l i n k = " i d e n t i t y " , nu . l i n k = " log " , tau . l i n k = " log " ) 4 dGB2(x , mu = 1 , sigma = 1 , nu = 1 , tau = 0 .5 , log = FALSE) The GB2 nests common distributions such as the generalized gamma (GG), Burr, lognormal, Weibull, Gamma, Rayleigh, Chi-square, Exponential, and the log-logistic. • Generalized Gamma f ( x ) = ( p/a d ) x d − 1 e − ( x/a ) p , Γ( d/p ) 99
Arthur CHARPENTIER - Welfare, Inequality and Poverty Dealing with Binned Data 1 > load ( u r l ( " http : // freakonometrics . f r e e . f r /income_binned . RData" ) ) 2 > head ( income_binned ) low high number mean std_e r r 3 4 1 0 4999 95 3606 964 5 2 5000 9999 267 7686 1439 6 3 10000 14999 373 12505 1471 7 4 15000 19999 350 17408 1368 8 5 20000 24999 329 22558 1428 9 6 25000 29999 337 27584 1520 10 > t a i l ( income_binned ) low high number mean std_e r r 11 12 46 225000 229999 10 228374 1197 13 47 230000 234999 13 232920 1370 14 48 235000 239999 11 236341 1157 15 49 240000 244999 14 242359 1474 16 50 245000 249999 11 247782 1487 17 51 250000 I n f 228 395459 189032 100
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