GY460 Techniques of Spatial Analysis

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GY460 Techniques of Spatial Analysis
Lecture 7 Measures of Inequality, Concentration
and Segregation

• Steve Gibbons

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Introduction

• Many situations where we want summary statistics
  that characterise the distribution of a
  characteristic across data units e.g.
• Number of industries in different regions
• Income across individuals
• Crime rates across wards
• Proportion in the population non-white in
  different wards
• This lecture discusses the use of these indices
  in relation to spatial patterns

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Descriptions of distributions
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Cumulative distribution function

• Basic statistical concept
• With a random variable that takes on discrete
  values, an estimate is

1
x F(x)
100 0.2
120 0.4
140 0.6
250 0.8
400 1
0.8
0.6
0.4
0.2
0
100
200
300
400
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Lorenz curve

• Commonly used to describe inequality (e.g.
  income)
• With a random variable that takes on discrete
  values, an estimate is

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Lorenz curve
L(x)
1
x F(x) L(x)
100 0.2 0.10
120 0.4 0.22
140 0.6 0.36
240 0.8 0.60
400 1 1.00
0.8
0.6
0.4
0.2
0
0.2
0.4
1
0.6
0.8
F(x)
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Segregation curve

• This is a variant of the Lorenz curve that is
  appropriate when considering inequality in
  proportions
• E.g. white/non-white
• Suppose we are interested in ethnic segregation.
  Should we consider whites or non whites?
• Lorenz curve gives different results
• Segregation curve base on comparing cumulative
  contribution of each unit (school, ward,
  district, firm etc.) to total white or non-white

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White Lorenz curve
L(w)
1
White F(w) S(w) L(w)
0.05 0.2 0.025 0.025
0.10 0.4 0.050 0.075
0.15 0.6 0.075 0.150
0.75 0.8 0.375 0.525
0.95 1 0.475 1.00
0.8
0.6
0.4
0.2
0
0.2
0.4
1
0.6
0.8
F(x)
Note here, sum(white) 2
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Non-white Lorenz curve
L(nw)
1
Non-White F(nw) S(nw) L(nw)
0.05 0.2 0.017 0.017
0.25 0.4 0.083 0.100
0.85 0.6 0.283 0.383
0.90 0.8 0.300 0.683
0.95 1 0.317 1.00
0.8
0.6
0.4
0.2
0
0.2
0.4
1
0.6
0.8
F(x)
Note here, sum(nonwhite) 3
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Segregation curve
L(nw)
1
Non-White L(nw) White L(w)
0.05 0.017 0.95 0.475
0.25 0.100 0.75 0.850
0.85 0.383 0.15 0.925
0.90 0.683 0.10 0.975
0.95 1.000 0.05 1.000
0.8
0.6
0.4
0.2
0
0.2
0.4
1
0.6
0.8
L(w)
Note units are ranked by nw here
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A smorgasbord of inequality indices
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Indices

• All the useful information about the
  distributions is contained in the
  Cumulative/Lorenz/Segregation curves plus the
  mean
• But useful to be able to summarize the features
  of the these distributions using single numbers
• Indices intended to rank distributions in study
  areas/periods according to the inequality
• Unfortunately no single index provides a complete
  summary

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Generalised entropy family

• Many commonly used indices have the same general
  form
• Indices of this form have the key properties of
  scale invariance and decomposability
• Sale invariance means that x and ?x give same
  index
• units of measurement or inflation dont matter
  for income inequality
• Decomposability means that index is a weighted
  sum of the indices for sub-groups of the
  population
• e.g. regions

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Coefficient of variation

• For beta 2, gives half-squared coefficient of
  variation
• So
• (where sample variance is the 1/n version )

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Herfindahl

• This is closely related to the Herfindahl index
• Which is often used to measure industrial
  concentration

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Theil index

• Another commonly used index is the Theil Index
• Which corresponds to the generalised entropy
  measure case when ? ? 1

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Additive decomposability

• Good thing about CV (squared), theil index and
  generalised entropy is that they can be
  decomposed into sub-groups
• E.g. suppose we have K regions with index Ik.
  Then the total inequality Itotal can be written
  as a sum of within region and between region
  indices
• Where wk is a region-specific weight which
  depends on the regional share of total x
• (In the generalised entropy case it can be shown
  that)

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Gini index

• The GINI isnt a member of the generalised
  entropy family
• GINI is twice area between the Lorenz curve and
  the 45 degree line (equality across data units)
• Computed in practice using (when units are same
  size)

x
100
120
140
240
400
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Gini index
0.5 x Gini
Lorenz curve
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Gini index for household incomes in Britain
Source Poverty and Inequality in Britain 2005,
IFS, London
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Segregation indices
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Indices for categorical variables

• Gini, generalised entropy family can be used when
  interest is on a categorical variable e.g.
• Black/white, industrial classification
• Though problem with asymmetry c.f. Lorenz curves
  for white/non-white shown earlier
• Various Segregation indices often used to
  describe distribution of categorical variables
• Measure inequality in one group relative to
  other group or total
• Benchmark is same proportion of each group in
  each data unit (e.g. regions)
• All have been re-invented many times

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Dissimilarity index

• Used for measuring distribution of some group j
  across units of aggregation i
• e.g.

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Dissimilarity index

• Dissimilarity ranges between 0 (all units the
  same) and 1 (units are either all group j or zero
  group j) e.g.

b w
800 800
600 600
400 400
200 200
b w
1000 0
1000 0
0 1000
0 1000
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Dissimilarity index

• Indicates the proportions of one group that would
  have to re-locate to generate no segregation

b w
200 800
400 600
600 400
800 200
b w
800 800
600 600
400 400
200 200
200
600
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Dissimilarity index

• One problem is that it isnt scale invariant,
  i.e. sensitive if there are proportional changes
  in one group

b w
100 900
200 800
300 700
400 600
b w
200 800
400 600
600 400
800 200
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Segregation index

• Same purpose all thats different is that the
  comparison with total numbers in unit i, not
  numbers that are not in the j group
• e.g.
• The Krugman index is just 2 x this, using
  employment or GDP
• Sepcialisation of place i i as geographical
  units, j as industries
• Concentration of industry j j as geographical
  units, i as industries

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Segregation/Krugman index

• Not sensitive to proportional changes in the
  group of interest

b All
100 1000
200 1000
300 1000
400 1000
b All
200 1000
400 1000
600 1000
800 1000
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Segregation/Krugman index

• But upper bound varies with total proportion in
  group
• It is (1 - proportion in group j) D

b All
1000 1000
1000 1000
0 1000
0 1000
b All
2000 2000
2000 2000
0 1000
0 1000
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Isolation index

• Measures the probability that random minority
  group member (e.g. black) shares a unit with
  another minority member rather sensitive to
  overall share

b w
250 750
250 750
250 750
250 750
b w
250 750
0 1000
0 1000
0 1000
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Isolation index

• Modified by Cutler, Glaeser, Vigdor (Journal of
  Political Economy 1999) to allow for overall
  minority group size divide by the maximum value
  to scale between 0-1

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Isolation index

• The CGV version

b w
250 750
250 750
250 750
250 750
b w
250 750
0 1000
0 1000
0 1000
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Spatial indices

• All the indices discussed measure inequality
  between data units so are spatial only if the
  data units are regions, districts or other
  spatial units!
• No measure here of how data is distributed within
  units
• E.g. all poor residents live in one part of the
  district
• Or whether there are spatial patterns across
  units
• e.g. all the majority poor districts next to each
  other
• Some indices try to take account of these factors
• See Massey and Denton (1988) or White (1983), The
  Measurement of Spatial Segregation, AJS, 88
  1008-1019
• Echinique and Fryer (2005), On the Measurement of
  Segregation, NBER W11258

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Example applications of segregation indices
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Ethnic segregation indices in English secondary
schools
Source Burgess and Wilson 2003
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Ethnic segregation in US cities
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Ethnic segregation in US cities
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US segregation and black white test gap
Source Vigdor and Ludwig 2007, NBER Working
Paper W12988
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Segregation indices are descriptive!

• Remember that segregation indices are descriptive
  statistics!
• Usual rules apply about inferring causality
• See Hoxby (2000) on reading list for example of
  attempt to use similar indices for causal
  analysis
• Uses numbers of rivers in US metropolitan areas
  as instrument for market fragmentation in
  schooling

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Industrial concentration using aggregated data
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Another segregation index

• Variation on a theme square the difference
  rather than take absolute difference
• I.e. its the squared difference between the
  contribution of unit i to total of j and
  contribution of i to overall total (or other
  comparison group)
• Can be used measuring concentration due to
  agglomeration forces?
• Ellison and Glaeser (1997) develop this index

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Another segregation index

• The G index
• Sometimes called Gini though Gini here is (by
  one calculation) 0.23

b All L(b)
100 1000 0.1
200 1000 0.3
300 1000 0.6
400 1000 1.0
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The Ellison and Glaeser Index

• But not possible to distinguish industrial
  concentration caused by market concentration (a
  few large plants) from agglomerative forces (many
  small plants co-located)
• E G (Journal of Political Economy 1997) correct
  the index to allow for this
• Requires plant-level Herfindahl for industry j Hj

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US 446/449 industries more concentrated than
expected. State-level data
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Industrial location

• See the further readings on the list
• Holmes, T And J. Stevens (2004) The Spatial
  Distribution Of Economic Activates In North
  America Handbook Of Urban And Regional Economics,
  Volume 4, Jacques Thisse And Vernon Henderson
  (Eds.)
• Combes, P. P. And H. G. Overman (2004) The
  Spatial Distribution Of Economic Activities In
  The EU Handbook Of Urban And Regional Economics,
  Volume 4, Jacques Thisse And Vernon Henderson
  (Eds.)

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References

• Cutler, DM, Glaeser, EL and Vidgor, JL (1999),
  The rise and decline of the American ghetto,
  Journal of Political Economy, 107(3) 455-506
• Burgess, S and D. Wilson (2003) Ethnic
  Segregation in Englands Schools, CMPO Working
  Paper 03/086
• Ellison, G. and E. Glaeser (1997) Geographic
  Concentration in US Manufacturing Industries A
  Dartboard Approach, Journal of Political Economy
  105 (5) 889-927