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1 Space, Time and Regional Inequality

1 Space, Time and Regional Inequality

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444



S.J. Rey and X. Ye



of the data, assessing spatial models” (Haining and Wise 1997). ESDA is a powerful body of techniques to visualize spatial distributions and detect patterns of spatial

association (Anselin 1993), often revealing complex spatial phenomenon not identified otherwise (Le Gallo et al. 2003). Hence, the development of new methods of

ESDA has stimulated a number of research efforts (Anselin and Getis 1992; Longley

et al. 2001; Getis et al. 2004; Rey and Anselin 2006).

Both spatial and temporal attributes of data are important, but existing approaches

focus primarily on one of these attributes. For example, researchers have relied on

either spatial analysis or time series methods though regional inequality dynamics

has both temporal and spatial dimensions underlying empirical analysis (Rey 2004a).

It is clear that new methods are needed to truly integrate space and time. Goodchild (2004, 2006) suggests this is a major research priority for the processes that

define the Earth’s dynamics. To consider both dimensions jointly, requires extending EDA (and exploratory temporal data analysis) for space, and at the same time

incorporating time into ESDA (Rey et al. 2005).



2.2 Distribution Dynamics and Spatial Pattern Analysis

Barro and Sala-i-Martin (1991, 1992) and Sala-i-Martin (1996) discuss two types

of convergence in growth empirics: ¢ and “ convergence. The former reflects the

decline of the dispersion of income across the economic units over time; the latter indicates the negative partial correlation between the growth rate in income

over time and its initial level. Quah (1993) argues that these two empirical strategies might be misleading because of the arbitrary assumptions about the dynamics

as a whole. Distribution dynamics refer to the difference among the overall shape

characteristics of the regional income distribution and the evolution of these characteristics over time, as well as the amount of internal mixing or rank mobility taking

place within these same distributions. Quah (1996) comments that the distribution

dynamics empirics will lead to new theories on economic growth and convergence.

In response, a number of EDA techniques have been applied to regional income

distributions. Using Markov chain techniques, Quah documents the degree to which

this instability characterizes the data. Markov chains have been applied to study

steady-state trends (Magrini 1999), modality (Quah 1996) and rank mobility (Hammond and Thompson 2002). Stochastic kernels are considered as extensions of the

Markov chain to a continuous field. Bianchi (1997) employs Markov chain approach

in the analysis of modality and the application in the internal mixing is carried out

by Tsionas (2000).

Some recent work points out that the dominant focus in the empirical literature on

shape regularities may be masking some interesting patterns that are internal to those

distributions (Overman and Ioannides 2001; Ioannides and Overman 2004). Based

on a critical review of empirical approaches and methodological advances in spatial

econometrics and spatial statistics, Rey and Janikas (2005) highlight the important

roles of spatial dependence, spatial heterogeneity, and spatial scale in the analysis of regional income distribution dynamics. Rey (2001, 2004b) suggests a series

of spatial empirics for distributional dynamics, such as spatial Markov, regional



Comparative Spatial Dynamicsof Regional Systems



445



cohesion of rank mobility and spatial decomposition of rank dynamics. To characterize complex map patterns has always been a challenge for spatial analysis (Getis

and Boots 1978; Boots and Getis 1988; Okabe et al. 2000). Geometric indicators and

graphical depiction have been used to summarize spatial patterns, such as Weber’s

Triangle, the Gravity Model, and Central Place Theory, among others (Mu 2004).

Geometric criteria are also applied to identify spatial structure through a spatial

weight matrix (Anselin 1988; Getis and Aldstadt 2004; Aldstadt and Getis 2006).

For instance, Aldstadt and Getis (2006) demonstrate that spatial association varies

in distance/direction and clusters are irregular in shape. We suggest that these findings can be revisited with perspectives from computational geometry where methods

have been developed to meet fast algorithmic requirements for geometric computing

(Mulmuley 1994; O’Rourke, 1994; Chazelle 1995; Eppstein 2005). Recent progress

in statistical shape analysis (Goodall and Mardia 1999) reveals great potential for

studying shape variations at microscale such as human brains (Mardia and Dryden 1999) to the Voronoi polygons examination of the central place theory (Dryden

and Mardia 1998). As commented by Goodchild (2006), “. . . GIScience is applicable to varying degrees in any space, . . . such as the three-dimensional space of

the human brain, . . . At the same time, advances made in the study of other spaces

may be suitable sources of cross-fertilization in GIScience. Perhaps the next decade

will see a much greater degree of interaction between GIScience and the sciences of

other spaces, and much more productive collaboration.” While analytical cartography and computational geometry can generate in-depth visualization and summary

of location and spatial pattern, they largely ignore dynamic effects.

This chapter hopes to contribute to the cross-fertilization of distribution dynamics

and spatial pattern analysis, through summarizing and comparing the geometry of

various spaces of regional economic growth.

In this regard, several interesting research questions are examined:

1. To what extent is economic growth associated with spatial context, dependence,

or heterogeneity?

2. Are regions with similar economic growth trends clustered?

3. How stable are certain spatial patterns (structures) over time? Are they clustered

over time?



3 Empirical Motivation: Regional Inequality in China

and the United States

Because of their growing importance in the world economic system, China and the

United States have been the center of numerous debates about economic growth

and regional convergence. Despite this rich empirical literature, comparative analysis of regional inequality dynamics between the two economies remains largely

unexplored, let alone the underlying geographical dimensions of regional growth

processes (Rey and Janikas 2005; Janikas 2007). Moreover, applications of comparative analysis between different economic systems are currently lacking an

inferential basis.



446



S.J. Rey and X. Ye



Fig. 1 Per capita incomes in the United States, 1978 and 1998



Regional inequality has generated lasting debates among the convergence, divergence, inverted-U, and Neo-Marxist uneven development schools (Pritchett 1997;

Fujita et al. 1999; Puga 1999; Tsionas 2000; Rey and Janikas 2005). The debate on

the trajectories and mechanisms of regional development has been focused over the

scope and consequences of regional policies and the extent and sources of regional

inequality (Sidaway and Simon 1990; Fan and Casetti 1994; Wei and Ye 2004;

Ye and Wei 2005), which is reflected in numerous empirical studies of specific

nations and continents (Rey and Janikas 2005). However, the findings are mixed

and sometimes conflicting (Ye and Wei 2005).

Many studies have been conducted on the US experience, and most of them conclude that regional convergence has been very strong, with two persistent regional

clusters: the Northeast-Mid Atlantic cluster of high income states and Southeast

cluster of low income states (Barro and Sala-i-Martin 1991, 1992; Fan and Casetti

1994; Bernard and Jones 1996; Vohra 1996; Rey and Montouri 1999; Tomljanovich

and Vogelsang 2002; Sommeiller 2007), as shown in Fig. 1.

Since the late 1970s, China has been undergoing economic reforms introducing market mechanisms and opening its economy to the outside world. The reform

process, however, was spatially uneven and has traditionally emphasized coastal

development (Lyons 1991; Lin 1997; Wei 2000, 2009; Benjamin et al. 2005; Wei

and Ye 2009), as shown in Fig. 2.2 Starting in the mid-1990s, the Chinese government began to make more efforts on development of poorer regions and reduction of

spatial inequalities through launching western development strategies and, recently,

providing incentives for developing rural areas. While some maintain that globalization and liberalization have brought wealth to transitional countries like China,

others argue that the transition in former socialist countries is characterized by

partial reform, path dependency, and geographical unevenness, and have recorded



2

While China’s official GDP statistics are sometimes regarded as of questionable quality

(Rawski 2001), the NSB (National Statistical Bureau of China) published adjusted GDP data to

deal with both overestimates and underestimates of provincial GDP data for the years before 2004

(Fan and Sun 2008). For a recent discussion justifying using per capita GDP as a valid and reliable

indicator of provincial economic development and well-being in China, see Fan and Sun (2008).



Comparative Spatial Dynamicsof Regional Systems



447



Fig. 2 Per capita incomes in China, 1978 and 1998



persistent or rising income gaps and spatial inequalities (Wei and Ye 2004; Ye and

Wei 2005).

Not only the presence of spatial dependence presents a challenge to the use

of statistical inference, but the partitioning of the economic units into either a

29-region system (29 land provinces in China) or 48-region system (48 states in

the United States) raises another concern very similar to the modifiable areal unit

problem (MAUP) (Openshaw and Alvanides 1999). Rey (2004a) finds that the

regional inequality decomposition fundamentally changes both quantitatively and

qualitatively when its spatial partition scheme (regionalization scheme) varies. It is

important to check whether the difference among regional systems is sensitive to

both how the observations are partitioned into each system and how they are spatially distributed within each system. However, this issue has been largely neglected

in previous comparative studies.

In the following sections, comparative space–time analysis of regional systems

will be conducted using the case study of China and the United States. The two

datasets are relative per capita income over the 1978–1998 period at the province

(China) and state (the United States) levels. The two data sets are comparable

regarding regional inequality because the states (United States) and provinces

(China) are self-contained and well-functioning units which form the theoretical

structure for spatial interaction models in spatial economy (Fan and Casetti 1994).



4 Comparative Spatial Dynamics

4.1 Inequality and Spatial Dependence

Many inequality measures have been introduced and discussed in the literature.

In regional inequality analysis, a popular measure is Theil’s inequality measure

(Theil 1967). Attention is first directed towards the relationship between regional



448



S.J. Rey and X. Ye



Fig. 3 Convergence and spatial independence in the United States and China



income inequality and spatial dependence over time using the global Theil and

Moran’s I (Fig. 3), which shed light on the debates between competing economic

growth theories and policies in these two distinct economic systems (Rey 2004a).

There is a U shape for regional inequality over time, with spatial clustering trends

in China while there is an obvious inverted U shape for spatial dependence with

relatively stable (or slightly inverted-U shape) regional inequality in the United

States.

In studies of regional income inequality, the decompositional property has been

exploited to investigate the extent to which global Theil is attributable to inequality between or within different partitions of the observational units. This approach

can provide a deeper understanding of global inequality (Rey 2004a). Two common regionalization schemes in China are Three Belts or Six Macro Regions

(Fig. 4 ). Three Belts are the eastern, central, and western economic belts while Six

Macro Regions refer to six main geographic regions (North-West, North, NorthEast, South-West, Central-South, East). There are four census regions in the United

States: Northeast, Midwest, South, and West. Eight BEA (Bureau of Economic

Analysis) Regions are New England, Mideast, Great Lakes, Plains, Southeast,

Southwest, Rocky Mountain, Far West (Fig. 5).

As revealed by Fig. 6, intra-regional inequality dominates the overall disparity in

China for most of the time regardless of the regionalization system while the dominance status in the United States will generally either be granted to inter-regional

(Eight BEA Regions) or intra-regional inequality (Four Census Regions). The interregional inequality share always grows in China while in the United States case this



Comparative Spatial Dynamicsof Regional Systems



449



Fig. 4 Regionalization system in China



Fig. 5 Regionalization system in the United States



0.70

0.60



Share



0.50

0.40

0.30

0.20



0.00

1978



Census4_US

D3_China



BEA8_US

D6_China



0.10



1980



1982



1984



1986



1988



1990



1992



Year



Fig. 6 Inter-regional inequality share in China and the United States



1994



1996



1998



450



S.J. Rey and X. Ye



component fluctuates substantially in the same time period. The choice of regionalization system matters in both systems. In China, the more aggregate regionalization

scheme (three belts) leads to a larger share of inter-regional inequality while in

the US case intra-regional inequality grows with the scale of the regionalization

scheme. China has witnessed a widening difference of the inter-regional inequality

shares between the two partition schemes over time while the United States has a

narrowing gap. The above studies have illuminated to some extent the spatial structure underlying the dynamics of regional inequality at various stages of economic

development.



4.2 Distance-Based Local Markov Transition

Local indicators of spatial autocorrelation (LISA) show a disaggregated view at the

nature of spatial dependence (Anselin 1995). We can embed these indicators in a

dynamic context by considering the movement of a given indicator in the scatter

plot over some time interval.

At a given time, t, the coordinate of each unit i ’s LISA is (yi;t , yli;t ) with:

yli;t D



n

X



wi;j yj;t



j D1



Given this, Di;t;t C1 is economic unit i ’s LISA transition from time t to t C 1,

measured by the segment length of [(yi;t 1 , yli;t 1 ), (yi;t , yli;t /].

In a similar vein to what is done in Markov models of income distributions, we

can discretize the values of the indicators to consider transitions across the classes

of a scatter plot over time. The four classes are High-Low (first quadrat), LowHigh (second quadrat), Low-Low (third quadrat) and High-Low (fourth quadrat).

Besides four types of intraclass transitions, 12 types of inter-class transitions can be

identified based on the four classes.

Because this discretization considers only class transitions it may treat transitions of different magnitudes as equal in constructing the LISA transition probability

matrix. We suggest using a threshold distance to address this issue. We can set the

threshold to be some value such as the average of all the transition distances on the

Moran scatter plot, which is on the conservative side. With this threshold, there are

two inter-class transitions in the left view of Fig. 7 and both of them move from



Fig. 7 Local Moran Markov

transition



Comparative Spatial Dynamicsof Regional Systems

Table 1 Local Moran

transition matrix in China

(ND/D)



Table 2 Local Moran

transition matrix in the

United States (ND/D)



451

HH



LH



LL



HL



HH

LH

LL

HL



82=82

3=2

0=0

1=1



0=0

47=48

1=1

0=0



0=0

1=1

397=397

2=1



0=0

0=0

2=2

44=45



HH



LH



LL



HL



HH

LH

LL

HL



223=228

6=3

0=0

3=2



9=6

141=146

5=2

0=0



0=0

9=7

356=362

8=6



6=4

0=0

7=4

187=190



Low-High section to Low-Low section on the Moran scatter plot. An inter-class

transition is significant only if its distance is larger than the threshold, otherwise the

transition is treated as an intra-class transition and will be considered to stay in the

original class, as shown on the right view of Fig. 7.

We use these thresholds to construct Tables 1 and 2 which reveal that China

has more significant transitions in local Markov matrix. For instance, the 356/362

located in the LL–LL position of Table 2 indicates that 356 transitions are considered intra-class movements before the distance (ND) threshold is applied while six

more transitions will be treated as intra-class movements because their lengths are

shorter than the average movement (D). This is contrasted with the case of LH to HH

transitions where six original transitions occur, but three of these involve movements

that are shorter than the threshold distance and are therefore treated as intra-class

movements (LH–LH). We return to the use of the threshold based transitions in a

comparative analysis later in this chapter.



4.3 LISA Time Path

The LISA Time Path Plot takes a continuous view of these transition to illustrate the

pair-wise movement of an economic unit (observation)’s value and its spatial lag

over time (Rey et al. 2005). The path of observation i over time can be written as

[(yi;1 , yli;1 ),(yi;2 , yli;2 ),. . . ,(yi;T , yli;T /]. yi;t is per capita income of province/state

i at time t and yli;t is its spatial lag at time t. This graph is helpful in identifying the

stability levels of local growth across a given structural process on the Moran scatter

plot. Since individual aspects of the contemporaneous process can be dissected by

interval gaps, the length and tortuosity of the time path are summarized for each

economic unit, as follows:



452



S.J. Rey and X. Ye



PT 1

N

d .Li;t ; Li;t C1 /

€i D PN Pt D1

T 1

i D1

t D1 d .Li;t ; Li;t C1 /



(1)



where: Li;t is the location of economic unit i on the Moran scatter plot at time t,

which is (yi;t , yli;t ). d.Li;t , Li;t C1 / is the distance (movement) between the locations of economic unit i at time t and t C 1. N is the number of spatial units. If an

economic unit’s movement over time is longer than the average, €i will be larger

than 1, and vice versa.

PT 1

d.Li;t ; Li;t C1 /

i D t D1

(2)

d .Li;1 ; Li;T /

where i is the economic unit i ’s tortuosity on the Moran scatter plot over time.

A larger i indicates a more tortuous movement on the graph.

A scalar instability measure of dynamic LISA is:

N

ƒi D PN



i D1



i



(3)

i



where i is the standard deviation of economic unit i ’s interval segment lengths of

LISA time path.

Figure 8 contrasts the LISA time paths of all the provinces/states in China and

the United States at the same scale. It reveals that China has much more dispersed

spatial dynamics. These patterns can be furthered analyzed in several ways. Tables 3

and 4 report the three suggested indicators to capture the continuous nature of the

LISA time paths for each province and state. They are also mapped on Figs. 11 and

12. The lop left view is the geographical distribution of €i values (length); top right

view is for i (tortuosity); bottom left view displays ƒi (instability); and the bottom right view is for space–time integration ratio, which will be discussed in the

following section. China’s rich provinces (coastal) tend to be more dynamic (top

left view), more tortuous (top right view) and more stable (bottom left view) while

the Northeast-Mid Atlantic cluster of high income states are more dynamic, less tortuous and more instable reflected by these three types of values compared to the rest



Fig. 8 LISA time path (left: China; right: the United States)



Comparative Spatial Dynamicsof Regional Systems

Table 3 Spatial dynamics in China

Province

Length

Tourtuosity

AH

BJ

FJ

GS

GD

GX

GZ

HEB

HL

HEN

HUB

HUN

NM

JS

JX

JL

NX

QH

SN

SD

SH

SX

SC

TJ

XJ

XZ

YN

ZJ



0:65

2:17

1:05

1:08

1:18

0:45

0:41

0:58

0:86

0:76

0:63

0:35

0:91

1:32

0:55

1:12

1:12

1:58

0:81

0:77

1:92

0:86

0:36

1:68

0:82

2:13

0:51

1:20



0:83

1:91

1:23

0:83

1:32

0:48

0:43

0:65

1:10

0:50

0:58

0:41

0:76

1:41

0:67

1:09

0:97

1:10

0:67

0:75

2:45

0:73

0:46

1:79

1:04

1:67

0:64

1:48



Instability

0:68

3:06

0:44

0:79

0:48

0:78

0:92

1:88

0:58

1:89

1:84

0:67

0:97

0:55

0:51

1:01

0:77

0:57

1:33

0:52

0:84

1:05

0:73

1:28

0:83

1:40

1:22

0:49



453



Spatial

joins

6

2

3

6

4

4

4

7

2

6

6

6

8

4

6

3

3

4

7

4

2

4

8

2

3

4

4

5



Similar

dynamics

1

0

2

4

1

1

3

1

0

0

1

3

5

2

0

2

3

3

4

2

0

2

5

0

0

2

1

2



ST integration

0:17

0:0

0:67

0:67

0:25

0:25

0:75

0:14

0:00

0:00

0:17

0:50

0:63

0:50

0:00

0:67

1:00

0:75

0:57

0:50

0:00

0:50

0:63

0:00

0:00

0:20

0:25

0:40



of their systems. As mentioned above, a longer movement on the Moran scatter plot

suggests a more mobile local spatial dependence over time. A more tortuous path

indicates a more fluctuating local spatial dependence evolution in direction while a

large variance among the segments of LISA time path demonstrates a more fluctuating local spatial dependence evolution. The maximum and minimum of €i are 2.17

(Beijing) and 0.35 (Hunan) in China, 2.78 (North Dakota) and 0.48 (Alabama) in

the United States. The maximum and minimum of i are 2.45 (Shanghai) and 0.41

(Hunan) in China, 5.69 (Arkansas) and 0.31 (North Carolina) in the Untied States.

The maximum and minimum of ƒi are 3.06 (Beijing) and 0.44 (Fujian) in China,

3.45 (North Dakota) and 0.34 (Alabama) in the United States.



454



S.J. Rey and X. Ye



Table 4 Spatial dynamics in the United States

State



Length



Tourtuosity



Instability



Spatial joins

joins



Similar dynamics



ST integration

dynamics



AL

AZ

AR

CA

CO

CT

DE

FL

GA

ID

IL

IN

IO

KA

KN

LO

ME

MD

MA

MI

MN

MS

MO

MT

NE

NV

NH

NJ

NM

NY

NC

ND

OH

OK

OR

PA

RI

SC

SD

TN

TX

UT

VT

VA

WA

WV

WI

WY



0:48

0:79

0:73

0:84

1:02

1:48

1:08

0:83

0:63

1:10

0:59

0:63

1:20

0:83

0:57

1:02

1:53

0:84

1:37

1:08

1:43

0:58

0:62

1:71

1:12

0:77

1:54

1:03

0:86

1:03

0:64

2:78

0:51

1:16

0:81

0:62

1:42

0:67

1:65

0:60

1:02

0:80

1:15

0:69

1:03

0:64

0:70

1:77



0:49

0:92

5:69

0:39

0:73

0:42

1:12

0:74

0:33

0:58

0:84

0:71

0:73

1:45

4:43

3:17

0:66

1:53

0:40

0:75

0:99

1:10

1:42

0:60

1:18

0:46

0:57

0:53

1:05

0:46

0:31

1:21

0:54

0:97

0:50

1:39

0:43

0:35

1:08

0:50

1:64

0:70

0:48

0:63

0:85

0:62

0:82

0:53



0:34

0:67

0:70

1:06

0:73

1:51

0:82

0:74

0:67

0:91

0:57

0:71

1:23

0:80

0:54

1:19

1:28

0:74

1:34

1:03

1:37

0:49

0:53

1:01

1:10

1:16

1:37

1:07

0:69

1:03

0:75

3:45

0:63

1:55

1:13

0:45

1:46

0:75

1:89

0:51

1:36

0:61

1:04

0:66

0:90

0:60

0:72

2:10



4

5

6

3

7

3

3

2

9

12

5

4

6

4

7

3

1

4

5

3

4

4

8

4

6

5

3

3

5

5

4

3

5

6

4

6

2

2

6

8

4

6

3

5

2

5

4

6



2

3

2

1

3

3

3

0

4

6

4

4

3

1

3

1

1

3

5

3

0

2

0

3

2

3

3

3

5

5

4

1

3

4

3

4

2

2

1

6

3

6

3

3

2

2

3

3



0:50

0:60

0:33

0:33

0:43

1:00

1:00

0:00

0:44

0:50

0:80

1:00

0:50

0:25

0:43

0:33

1:00

0:75

1:00

1:00

0:00

0:50

0:00

0:75

0:33

0:60

1:00

1:00

1:00

1:00

1:00

0:33

0:60

0:67

0:75

0:67

1:00

1:00

0:17

0:75

0:75

1:00

1:00

0:60

1:00

0:40

0:75

0:50



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