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Measuring Inequality
A practical workshopOn theory and technique
San Jose, Costa RicaAugust 4 -5, 2004
Panel Session on:
Mechanics to Calculate the Theil
and
Use of Programs and Macros
byJames K. Galbraith and Enrique Garcilazo
The University of Texas Inequality Project
http://utip.gov.utexas.edu
Session 3
Outline
1. Two hierarchical level grouping2. Data Collection3. Between Group Theil4. Within – Group Theil 5. Programs and Macros6. Working with Output – Analysis
Two Level Hierarchical grouping
Data typically given by geographical units and several sectors within each unit
The between group component The within group component
1
1 1 1
1 1
ln ln
n
i Yn n kip ip ii i i
n ni i p i ij i
i i ii i
Y YY Y YY Y
TY Y n n
Y n n
Collecting data
Variables: – #1 Compensation earned – wages– #2 Number of people earning –
employees
Two sets of data– Equal number of observations for each
data set– Making sure there are no zeros– Arrange data into a yearly matrix of
columns and rows (columns are geographical units, rows sectors)
Data Management
Sectors = {a, b, c} Geographical units = {1, 2, 3} Check:
– matching # observations between both matrixes
– non zero observations
Year kemployment compensation
a b c a b c1 12 23 34 4
Cleaning The Data
Compare both data-bases and look for missing observations– count function in Excel– Compare # of observations in rows and
columns
Take out observations equal to zero If the database is time series, make
sure that the number of industries included in the analysis is consistent through time
Calculating The Between Theil
The between component can be expressed by two equivalent equations:
– First equation – more intuitive – Second equation – easier to calculate
n
i ii
in
iiiB Y
y
Y
ynnT
1 1
ln
n
in
iii
n
iYi
n
ii
iB
nn
YY
Y
YT
1
1
1
1
ln
Calculating The Between Theil
– Add across rows and columns
Year kemployment compensation
a b c sum a b c sum1 12 23 34 4
sum sum
Calculating The Between Theil
0048.0181
45
750
190log*
750
1901
BT
Year k employment compensationj b/w i Theil b/w j Theil a b c sum a b c sum component component
i 1 10 15 20 45 1 40 65 85 190 0.0048 -0.042692 11 8 22 41 2 30 66 88 184 0.0196 0.0554153 12 14 21 47 3 35 67 90 192 -0.0036 -0.003134 9 16 23 48 4 33 60 91 184 -0.0191 0.009601
sum 42 53 86 181 sum 138 258 354 750 0.0016
04269.0181
42
750
138log*
750
138
BaT
4
1
4
sec1
B region Bii
B tor Bjj
T T
T T
Calculating The Between Theil
Using natural logs instead of base 10 Interpreting the between components
– Positive component means the group is above the mean
– Negative component means it is below the mean
Make sure the sum of all components (between Theil) is positive and inside a reasonable range
Between Theil used as estimate of total Theil
Calculating The Within Theil
1
lnk
ip ip iw
p i ij i
Y Y YT
Y n n
1
ni
w wi
YT T
Y
unweighted within Theil
weighted within Theil w
w
T
T
Calculating The Within Theil
1.First: Calculate the unweighted within Theil in each group (among p individuals/groups)
Calculate the Theil by sector or by region The unweighted Theil is the total Theil for the
group in the second hierarchical level
1
lnk
ip ip iwi
p i ij i
Y Y YT
Y n n
Calculating The Within Theil
2. Second - multiply the unweighted within Theil component by the income weight of each group
3. Third – take the sum of all the individual components to obtain the weighted within Theil
1
ni
w wii
YT T
Y
Calculating The Within Theil
.
011.045
10
190
40log*
190
40*
iaT
****icibiai TTTT
00044.00029.0089.)011.0(*1 T
00011.00044.0*2533.01/ inWT
Year kemployment compensationj Unweigh. w/in Th
a b c sum a b c sum Theil Weights compi 1 10 15 20 45 1 40 65 85 190 0.00044 0.2533 0.00011
2 11 8 22 41 2 30 66 88 184 0.08215 0.2453 0.02015
3 12 14 21 47 3 35 67 90 192 0.01629 0.256 0.00417
4 9 16 23 48 4 33 60 91 184 0.0005 0.2453 0.00012
sum 42 53 86 181 sum 138 258 354 750 0.02456
-0.0114 0.0089 0.0029
-0.0812 0.2184 -0.055
-0.0614 0.0552 0.0225
-0.008 -0.0072 0.0156
Output
For a given year obtain:– Total Theil– Between group Theil– Weighted within group Theil
Interpreting the individual between Theil components
The unweighted within group Theil component interpreted as the total Theil in the 2nd hierarchical level
Programs and Macros
Excel Spreadsheet Stata program SAS options
For more information:
The University of Texas Inequality Project
http://utip.gov.utexas.edu
Type “Inequality” into Google to find us on the Web
