Session 3: Measuring Inequality
There are various approaches that have been used for the analysis of Inequality
\[\begin{aligned} y_i &\sim f(y) \rightarrow \int_{-\infty}^{z} f(y) dy = F(z) \\ F(0)&=0 \ \& \ F(\infty)=1 \\ F(Q_y(p)) &=p \rightarrow Q_y(p) = F^{-1}(p) \end{aligned} \]
The \(p_{th}\) quantile of \(y_i\) is the value \(Q_y(p)\) such that \(p\) percent of the population has income below \(Q_y(p)\).
Mean of Standard of Living:
\[\mu_y = E(y) = \int_{-\infty}^{\infty} y f(y) dy=\int_0^1Q(p)dp \]
Finally, the inequality measure can be written as:
\[I(y)=I(\mu_y, f(.)) = I(\mu_y, F(.)) \]
Density functions and histograms are used to visualize the distribution of income in the population.
They could be used to detect multimodality, skewness, etc
And could be used to compare distributions across groups.
Stata Commands
histogram varname [weight] [if]
kdensity varname [weight] [if]
set scheme white2
set linesize 255
color_style tableau
qui:frause oaxaca, clear
sum wt, meanonly
gen int wt2 = round(wt/r(min))
qui:two histogram lnwage [fw=wt2] ///
|| kdensity lnwage [w=wt2], ///
ysize(5) xsize(9) xtitle("Log Wages") ///
legend(order(1 "Histogram" 2 "Kernel Density") pos(6) col(2)) 
Perhaps the most popular tool to visualize income distribution.
This curve plots the cumulative share of income vs the cumulative share of population.
“Easy” to read:
Not easy to implement with negative and zero incomes.
Comparison across groups may be ambiguous.
its always increasing at an increasing rate respect to X%
Assume data is sorted by income
x-axis: Cum share of population
\[P_j = \frac{\sum_{i=1}^j w_i}{\sum_{i=1}^n w_i}\]
y-axis: Cum share of income
\[LC_j = \frac{\sum_{i=1}^j y_i w_i}{\sum_{i=1}^n y_iw_i}\]
frause oaxaca, clear
gen wage = exp(lnwage)
sort wage wt // sort by income and weight
// Estimate Totals for non missing data
egen twage = sum(wage * wt) if wage!=.
egen tpop = sum(wt) if wage!=.
// get cumulative shares
gen lc_i = sum( (wage*wt/twage) )*100 if wage!=.
gen p_i = sum( (wt/tpop) )*100 if wage!=.There are several measures of inequality. The most popular are:
Interquantile Range (IQR) (or normalizations) \[IQR(\#1,\#2) = Q(\#2) - Q(\#1) \]
Interquantile Share Ratio (Palma ratio (10/40)) \[ISR(\#1,\#2) = \frac{1-LC(\#2)}{LC(\#1)} \]
Coefficient of Variation (CV) \[CV = \frac{\sigma_y}{\mu_y} \]
\[LC(p) = \frac{\int_0^p Q_y(u)du}{\int_0^1 Q_y(u)du} = \frac{1}{\mu_y} \int_0^p Q_y(u)du \]
\[\frac{\partial LC(p)}{\partial p} = \frac{Q_y(p)}{\mu_y} \geq 0\]
\[Gini(y) = 2 \int_0^1 (p-LC(p)) dp\]
where \(p-LC(p)\) is the “loss” of income the Bottom \(p\) percent of the population experiences.
It is bounded between 0 (perfect Equality) and 1 (complete Inequality).
When Lorenz do not cross, Gini provides unambiguous ranking of inequality.
\[Gini(y) = \frac{2}{\mu_y} Cov(y_p,p)\]
Stata has plenty of commands that can be used to estimate Gini
search gini for few examplesfastgini (ssc install fastgini)ineqdeco (ssc install ineqdeco)sgini (ssc install sgini)rif (ssc install rif)
Gini coefficient for wage
-----------------------
Variable | v=2
-------------+---------
wage | 0.2460
-----------------------
Percentile ratios
----------------------------------------------------------
All obs | p90/p10 p90/p50 p10/p50 p75/p25
----------+-----------------------------------------------
| 3.154 1.694 0.537 1.771
----------------------------------------------------------
Generalized Entropy indices GE(a), where a = income difference
sensitivity parameter, and Gini coefficient
----------------------------------------------------------------------
All obs | GE(-1) GE(0) GE(1) GE(2) Gini
----------+-----------------------------------------------------------
| 0.23199 0.14240 0.12282 0.13398 0.26273
----------------------------------------------------------------------
Atkinson indices, A(e), where e > 0 is the inequality aversion parameter
----------------------------------------------
All obs | A(0.5) A(1) A(2)
----------+-----------------------------------
| 0.06292 0.13273 0.31693
----------------------------------------------There are other approaches that can be used to measure inequality.
\[I_A(y,\varepsilon) = 1 - \left( \frac{1}{N} \sum_{i=1}^N \left(\frac{y_i}{\mu_y}\right)^{1-\varepsilon} \right)^\frac{1}{1-\varepsilon} \]
where is a measure of inequality aversion.
\[I_{GE}(Y,\alpha)=\frac{1}{\alpha(1-\alpha)}\left[\frac{1}{N} \sum \left(\frac{y_i}{\mu_y}\right)^\alpha -1\right] \]
a note
mean command, or regressbootstrap gini=r(coeff): sgini wage
warning: sgini does not set e(sample), so no observations will be excluded from the resampling because of missing values or other reasons. To exclude observations, press Break, save the data, drop any observations that are to be excluded, and rerun
bootstrap.
Bootstrap results Number of obs = 1,647
Replications = 50
Command: sgini wage
gini: r(coeff)
------------------------------------------------------------------------------
| Observed Bootstrap Normal-based
| coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
gini | .2460329 .0063639 38.66 0.000 .2335599 .2585059
------------------------------------------------------------------------------rifhdreg wage , rif(gini)
Linear regression Number of obs = 1,434
F(0, 1433) = 0.00
Prob > F = .
R-squared = 0.0000
Root MSE = .24613
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
_cons | .2460329 .0064995 37.85 0.000 .2332833 .2587825
------------------------------------------------------------------------------
Distributional Statistic: gini
Sample Mean RIF gini : .24603rifhdreg wage ibn.female [pw=wt], rif(gini) over(female) noconstant
Linear regression Number of obs = 1,434
F(2, 1432) = 640.56
Prob > F = 0.0000
R-squared = 0.5056
Root MSE = .25631
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
female |
0 | .2377884 .0084103 28.27 0.000 .2212907 .2542862
1 | .2820059 .0128488 21.95 0.000 .2568015 .3072103
------------------------------------------------------------------------------
Distributional Statistic: gini
Sample Mean RIF gini : .25805rifhdreg wage i.female [pw=wt], rif(gini) over(female)
Linear regression Number of obs = 1,434
F(1, 1432) = 8.29
Prob > F = 0.0040
R-squared = 0.0073
Root MSE = .25631
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
1.female | .0442175 .0153565 2.88 0.004 .0140938 .0743412
_cons | .2377884 .0084103 28.27 0.000 .2212907 .2542862
------------------------------------------------------------------------------
Distributional Statistic: gini
Sample Mean RIF gini : .25805Setup: \[\begin{aligned} Y &= y_1 + y_2 + \cdots + y_n \\ I_V(Y) &= V(Y) = Cov(Y,Y) \\ &= Cov(Y,y_1) + Cov(Y,y_2) + \cdots \\ \end{aligned} \]
Covariance, however could be rewritten as: \(Cov(Y,y_k)=\rho_k \sigma_Y \sigma_{y_k}\), thus
\[\begin{aligned} V(y) = \sigma^2_Y &= \rho_1 \sigma_Y \sigma_{y_1} + \rho_2 \sigma_Y \sigma_{y_2} + \cdots \\ \sigma_Y&= \rho_1 \sigma_{y_1} + \rho_2 \sigma_{y_2} + \cdots \\ \end{aligned} \]
Finally, if we divide all by \(\mu_y\) and multiply by \(\mu_{yk}/\mu_{yk}\), we get:
\[\begin{aligned} \frac{\sigma_Y}{\mu_y} = CV(y) &= \rho_1 \frac{1}{\mu_{y}} \frac{\mu_{y1}}{\mu_{y1}}\sigma_{y_1} + \rho_2 \frac{1}{\mu_{y}} \frac{\mu_{y2}}{\mu_{y2}} \sigma_{y_2} + \cdots \\ &= \rho_1 sh_1 CV(y_1) + \rho_2 sh_2 CV(y_2) + \cdots \\ \end{aligned} \]
Statassc install ineqfac
frause limew1972.dta, clear
capture: ssc install ineqfac
ineqfac basemi inchomewealth incnonhomewealth govtr_n pubcon tx_n valhp [aw=hhwgt]
Inequality decomposition by factor components (Shorrocks' method)
-------------------------------------------------------------------------------
Factor | 100*s_f S_f 100*m_f/m rho_f CV_f CV_f/CV(Total)
--------------------+----------------------------------------------------------
basemi | 29.115 0.284 44.434 0.711 0.900 0.922
inchomewealth | 1.872 0.018 4.302 0.318 1.335 1.367
incnonhomewealth | 43.136 0.421 7.905 0.754 7.065 7.235
govtr_n | -0.979 -0.010 5.868 -0.107 1.528 1.565
pubcon | 2.091 0.020 8.303 0.238 1.035 1.060
tx_n | 15.629 0.153 11.462 0.754 1.767 1.809
valhp | 9.136 0.089 17.726 0.502 1.003 1.027
--------------------+----------------------------------------------------------
Total | 100.000 0.977 100.000 1.000 0.977 1.000
-------------------------------------------------------------------------------
Note: s_f is the proportionate contribution of factor f
to inequality of Total, where ...
s_f = rho_f*sd(f)/sd(Total)
= rho_f*[m(factor_f)/m(totvar)]*[CV(factor_f)/CV(totvar)].
S_f = s_f*CV(Total). rho_f = corr(f,Total).
m_f = mean(f). sd(f) = std.dev. of f. CV_f = sd(f)/m_f.\[\begin{aligned} Gini(y) &= \frac{2}{\mu_y} Cov(y,F(y)) = \frac{2}{\mu_y} Cov\left(\sum y_k,F(y)\right) =\frac{2}{\mu_y} \sum Cov\left( y_k,F(y)\right) \end{aligned} \]
\[\begin{aligned} Gini(y) &= \sum \frac{2}{\mu_y} \times Cov( y_k,F(y)) \times \frac{\mu_{yk}}{\mu_{yk}} \times \frac{Cov( y_k,F(y_k))}{Cov ( y_k,F(y_k))} \\ &=\sum \frac{Cov ( y_k,F(y) )}{Cov ( y_k,F(y_k) )} \times \frac{2Cov( y_k,F(y_k)}{\mu_{yk}} \times \frac{\mu_{yk}}{\mu_y} \\ &= \sum R_k \times G_k \times sh_k \end{aligned} \]
\[\begin{aligned} Gini(y)&= \sum R_k \times G_k \times sh_k \end{aligned} \]
StataYou probably already have it installed
ssc install sgini
Gini coefficient for basemi, inchomewealth, incnonhomewealth, govtr_n, pubcon, tx_n, valhp
Note: inchomewealth has 690 negative observations (used in calculations).
Note: incnonhomewealth has 10359 negative observations (used in calculations).
-----------------------
Variable | v=2
-------------+---------
basemi | 0.4787
inchomewea~h | 0.6161
incnonhome~h | 0.9503
govtr_n | 0.7099
pubcon | 0.4913
tx_n | 0.6036
valhp | 0.5021
-----------------------
Decomposition by source:
TOTAL = basemi + inchomewealth + incnonhomewealth + govtr_n + pubcon + tx_n + valhp
Parameter: v=2
--------------------------------------------------------------------------------
| Share Coeff. Corr. Conc. Contri. %Contri. Elasticity
Variable | s g r c=g*r s*g*r s*g*r/G s*g*r/G-s
-------------+------------------------------------------------------------------
basemi | 0.4443 0.4787 0.8870 0.4246 0.1887 0.4868 0.0425
inchomewea~h | 0.0430 0.6161 0.4617 0.2845 0.0122 0.0316 -0.0114
incnonhome~h | 0.0790 0.9503 0.7724 0.7340 0.0580 0.1497 0.0707
govtr_n | 0.0587 0.7099 -0.2298 -0.1631 -0.0096 -0.0247 -0.0834
pubcon | 0.0830 0.4913 0.4593 0.2257 0.0187 0.0483 -0.0347
tx_n | 0.1146 0.6036 0.8776 0.5297 0.0607 0.1567 0.0420
valhp | 0.1773 0.5021 0.6601 0.3314 0.0587 0.1516 -0.0257
-------------+------------------------------------------------------------------
TOTAL | 1.0000 0.3876 1.0000 0.3876 0.3876 1.0000 0.0000
--------------------------------------------------------------------------------Decomposition of the GINI coefficient by groups Milanovic and Yitzhaki (2002)
The method: To decompose the GINI by groups, one can use the following:
\[\begin{aligned} Gini(y) &= \sum_{k=1}^K s_k O_k Gini(y_k) + Gini_{bw} \end{aligned} \]
\[Gini_{bw} = \frac{2}{\mu_y} Cov(\mu_i,\bar F_i)\]
Overlapping \(O_k\) measures to what extend the distribution of income in group \(k\) overlaps with the distribution of income in other groups.
If there is no overlapping, then \(O_k=p_i\) (the population share of group k, and incomes are fully stratified).
Otherwise, this adjustment factors ensures that the sum of the Gini of each group is equal to the Gini of the total population + Between Gini.
Statassc install anogi (Tom Masterson is one of the authors)
ssc install moremata (needed for anogi)
Analysis of Gini
--------------------------------------------------
| Coef. %
--------------------------+-----------------------
Overall Gini | .3875623 100.00
|
G_wo = sum s_i*G_i*O_i | .3405469 87.87
G_b | .0470154 12.13
|
IG = sum s_i*G_i | .3657639 94.38
IGO = sum s_i*G_i(O_i-1) | -.025217 -6.51
BGp = G_bp | .1317698 34.00
BGO = G_b - G_bp | -.0847544 -21.87
--------------------------+-----------------------
Mean of limew | 20403.6
N. of obs | 44872
N. of subgroups | 4
--------------------------------------------------
Detailed statistics for subgroups
-------------------------------------------------------------------------------------------
| N p mean s G O F
-------------+-----------------------------------------------------------------------------
1 | 18244.58 .4065916 15466.07 .3081992 .3844549 .9986868 .3963075
2 | 14552.19 .3243045 21019.76 .3340981 .338758 .9239628 .5363091
3 | 5609.305 .1250068 22705.72 .1391112 .3640736 .9435821 .5546081
4 | 6465.924 .1440971 30951.73 .2185916 .3817625 .8370495 .6634934
-------------------------------------------------------------------------------------------Until next week!