JEL: C46, R11. GeoDa Center for Geospatial Analysis and Computation, School of Geographical Sciences and Urban Planning, Arizona State University ...
Visualizing Regional Income Distribution Dynamics S.J. Rey, A.T. Murray, L. Anselin 2010 Working Paper Number 14
Noname manuscript No. (will be inserted by the editor)
Visualizing regional income distribution dynamics
Sergio J. Rey · Alan T. Murray · Luc Anselin
the date of receipt and acceptance should be inserted later
Abstract This paper introduces a new approach to the analysis of regional income distribution dynamics. Drawing on recent advances in geovisualization, we suggest a spatially explicit view of income mobility. Based on the integration of a dynamic local indicator of spatial association (LISA) together with directional statistics, this framework provides new insights on the role of spatial dependence in regional income growth and change. These new approaches are illustrated in a case study of state level incomes in the U.S. over the 1969-2008 period. Key Words: ESTDA, spatial dynamics, regional convergence. JEL: C46, R11
GeoDa Center for Geospatial Analysis and Computation, School of Geographical Sciences and Urban Planning, Arizona State University
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1 Introduction
An emerging strand of the literature on regional convergence has focused on new exploratory approaches to the analysis of income dynamics. In part this is in response to the dissatisfaction with the overly restrictive nature of existing regional growth theories that are largely at odds with the rich set of spatial dynamics encountered in empirical work, although some point to the potential for synergies between confirmatory and exploratory approaches as a way forward (Rey and Le Gallo, 2009). This exploratory turn in convergence research has made heavy use of a number of novel visualization techniques. These include stochastic kernels (L´ opez-Bazo et al., 1998; Magrini, 1999; Basile and Gress, 2005), quantile contours (Trede, 1998), high density region box plots (Fischer and Stumpner, 2008), and stochastic dominance (Carrington, 2006), among others. Although different in form and implementation, each of these methods involves a comparison of the regional income distribution at two points in time, with particular focus on changes in the overall morphology of the distribution (e.g., σ-convergence and polarization) as well as the question of mobility or the movement of individual regional economies within the income distribution over a period of time. While these views of income distribution dynamics can be illuminating, they are silent on the underlying spatial dimensions of the changes in income. More specifically, they treat each economy as an individual observational unit and ignore the potential for income growth and change to be spatially de-
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pendent. In this paper we introduce several new visualization based methods that are designed to address this limitation. Drawing on recent advances in geovisualization we suggest a spatially explicit view of regional income mobility. Based on the integration of a dynamic local indicator of spatial association (LISA) together with directional statistics, these new methods provide insights on the role of spatial dependence in regional income growth and change. We also pair these graphical methods together with computationally based inferential methods to evaluate several different hypotheses about regional income dynamics. These new approaches are used in a case study of state level incomes in the U.S. over the 1969-2008 period.
2 Visualization of regional income dynamics
Our approach to exploring the role of space in the evolution of regional income distributions draws on recent work in geovisualization together with previous work on exploratory spatial data analysis. Our point of departure is the framework for visualizing movement object patterns recently introduced by Murray et al. (2010). This method considers the spatial change in an object location over time, represented as a movement vector, and couples map based displays together with a set of statistics based on standardized movement vectors to support the exploration of patterns. We extend this approach to the case of regional income distribution dynamics by integrating versions of local indicators of spatial association (LISA) introduced by Anselin (1995) to redefine our movement vectors. Given a set
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of n regional economies, the LISA for location i at time period t is measured as:
Li,t =
zi,t
P
Pj
wi,j zj,t
(1)
2 i zi,t
where zi,t is the per-capita income for economy i at time t (expressed in deviations from the mean) and wi,j is an element from a spatial weights matrix expressing whether i and j are neighbors.1 Our visualization of spatial income dynamics is based on a decomposition of the LISA at each point in time. More specifically, we position each economy in a Moran Scatter Plot using the coordinates (ri,t , yi,t /
Pn
i=1
P
j
wi,j rj,t ), where ri,t =
yi,t , and yi,t is per capita income for region i in time period t.
Figure 1(a) contains the Moran Scatter Plot for 1969. As is well known, the four quadrants of the Moran Scatter Plot reflect different types of spatial association between a state’s income (referred to as “own” in what follows) and that of its neighbors. Quadrant I (north east or NE) contains states that had above average incomes and were surrounded by states that, on average, also had high incomes. Found in quadrant II (north west or NW) are states that had below average incomes but whose neighbors had above average incomes. Quadrant III (south west or SW) consists of relatively poor states surrounded by other relatively poor states, while quadrant IV (south east or SE) is home to states that are above average in incomes but surrounded by relatively poor states.
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1
In the example that follows, W is defined using simple rook contiguity.
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All computations are carried out using the library PySAL (Rey and Anselin, 2007).
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[Fig. 1 about here.]
Our interest here is not so much in the cross-sectional view of this Scatter Plot but rather the contrast between it and the one for 2008, the latter being shown in Figure 1(b). At first glance, the overall impression is one of a NE shift, as the distribution of points in quadrant III is now more concentrated and closer to the center in 2008 relative to 1969. At the same time, the pattern in quadrant I appears to be more dispersed in 2008 relative to that found in the same quadrant in 1969. This comparative static view of the Moran Scatter Plot provides an overall impression of the spatial dynamics yet it may mask, or even misidentify, individual movements of economies and their neighbors in the income distribution. A more integrated view of these movements can be seen in the Directional Moran Scatter Plot shown in Figure 2(a). Here the transition for each economy is represented as a movement vector with the arrowhead pointed at its location in the 2008 period. Examination of this visualization reveals that while there are indeed seven observations in quadrant I in both periods, those observations do not represent the same seven economies as there is movement both in and out of the quadrant over the two periods. Moreover, the directionality for the five economies that are found in quadrant I in both periods is rather heterogeneous. At the same time, similar heterogeneities in the movement vectors are found in the other three quadrants. As we return to below, the characteristics of these movement vectors reflect the degree of spatial integration in the distribution dynamics.
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In order to focus more closely on the direction of the movements, we can normalize the vectors to develop the standardized directional Moran Scatter Plot as is done in Figure 2(b). Here the vectors are origin-standardized, that is, reflecting movements from their position in the 1969 Moran Scatter Plot. The four quadrants now take on a different meaning from that in the original static Moran Scatter Plot. Moves to the NE in the standardized plot reflect gains, or positive co-movement, of a state and its spatial lag in the income distribution. In contrast, movements to the SW reflect a worsening of the relative positions of a state and its neighbors in the distribution, or negative co-movement. These two types of movements could be considered evidence of spatially integrated dynamics along the lines suggested by Rey (2001). With these standardized vectors in hand, a number of characteristics of the movements can be examined. Drawing on circular statistics we can also develop a circular histogram which provides insight as to the frequency of moves across different directions. The rose diagram associated with the directional Moran Scatter Plot is shown in Figure 3(a). Here the directional vectors are placed into one of P = 4 classes, or circular segments, based on the angular motion. The predominant direction involves upward moves of both a state and its neighbor in the relative income distribution. Following this are downward moves in the distribution, that is a state and its neighbor lose ground in the income distribution over this period. Much less common are movements involving opposite trajectories for a state and its neighbors. Taking these together, there is striking visual evidence of strong spatial integration in evolution of
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the state income distributions. The visual dominance of the upward moves suggests that the convergence pattern has been an asymmetric one, with the reduction in the variance of the income distribution being driven by relatively faster growth in incomes for the poorer states compared to the states at the top of the relative income distribution.
We can manipulate the rose diagram to provide a more detailed view of the income dynamics by increasing the number of circular sectors. This is done in Figure 3(b) where P = 8 which splits each of the four circular sectors from 3(a) into two different sectors. Doing so reveals several interesting patterns. For both types of spatially integrated moves, the moves where the state’s position improves (worsens) more than the neighbors are more frequently encountered than the reverse situation. For example, of the 32 positive co-movements, 19 involve larger relative moves for the state than its neighbors, while for 13 states the neighbors gain relatively more. In the 9 negative co-movements, 7 involve the state loosing more ground than its neighbors, while 2 states lose less ground than their neighbors.
This pattern of the own-moves being larger than the neighbor-moves, in part, reflects the underlying definition of the neighbor moves. The movement along the Y-axis in the directional Moran Scatter Plot is dampened by the spatial weights. More specifically, if the weight matrix W is row standardized, the variance of the vector W Y2000 − W Y1969 will be less than or equal to the variance of Y2000 − Y1969 , resulting in an elongation of the directional
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Moran Scatter Plot along the x-axis. This will have important implications for inference about these movements, as we examine in the next section. [Fig. 2 about here.] [Fig. 3 about here.]
3 Inference
While the rose diagrams provide visual indications of spatial integration in the income dynamics, it is important to examine if these patterns are different from what would be expected if incomes and growth were randomly distributed in space. Here these new forms of visualization are complemented by inferential methods to test a number of hypotheses regarding the direction, length, and direction-length movement patterns. In the initial implementation of the movement vector approach Murray et al. (2010) suggested a bi-variate test examining the association between direction and length of movement:
χ2 =
P X (Op − Ep )2 p
Ep
(2)
where p = 1, 2, . . . , P is an index of a sector of a circular histogram under the null, Op is the observed count of movement vector endpoints in sector p, and Ep is the expected count of movement vector end points in sector p. Two issues prevent the straightforward application of this approach to regional income dynamics and the Moran Scatter Plot movement vectors. The first pertains to the specification of the Ep . One possibility would be to assume
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movement vectors had random directionality, in which case Ep = n/P ∀ p = 1, . . . , P . However, in the case of regional income dynamics this assumption is likely to be violated given the lower variance for the changes in the lag of state incomes relative to the variance of the changes in state incomes. The second, and more problematic issue, is the lack of independence of the movement vectors. The dependence is due to the use of the lag operator to define the vertical coordinates in the directional Moran Scatter Plots. To address these issues, we rely on conditional randomization of the income values in space as a basis for inference. For each state we repeat M permutations of its neighbors, and for each permutation the spatial lag is recalculated, and our directional vectors obtained. We collect all the directional vectors and compare them to the original rose diagram. Here our null hypothesis is that the movements of a state and the movements of its neighbors in the income distribution are independent. Our test statistic is implemented as follows. For each of the circular sectors in the original rose diagram we compare the height of the sector to the distribution of heights for the same sector drawn from the realizations. We develop two forms pseudo p-values for the original sector. The first uses a normal approximation: zi =
¯∗ hi − h i sh∗i
(3)
where hi is the height of circular sector i = 1, 2, . . . , P in the original rose ¯ ∗ is the average height of the same sector over the random spatial diagram, h i permutations and sh∗i the standard deviation of the empirical height distribu-
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tion for sector i. The pseudo p-value is then taken as the cumulative probability from a standard normal distribution. The second p-value is non-parameteric being expressed as: p − rand(hi ) =
ψi + 1 M +1
(4)
where ψi is the number of height values for sector i from the randomized distribution that were as extreme as the observed value. The number of randomized permutations is M = 999. Applying these to the US incomes generates the results summarized in Table 1 and Figure 4. Focusing on circular segment 2, under the null, the expected number of vectors in this segment was 9.14 with a standard deviation of 1.94. Given the actual count of 13, this corresponds to a z-value of 1.99 and a pseudo p-value of 0.023 based on the normal approximation, and 0.041 on the non-parametric approach. The importance of adjusting for the variance heterogeneity between the states and the lag can be seen by examining the first segment in in Figure 3(b) and Table 1. Clearly this segment contained the largest number of movement vectors, yet when compared to its expectation under the conditional randomization, the observed count is not significantly large. By the same token, sector 8 with an observed count of 1 movement vector is significantly below the expected number of vectors under the null. All together, five of the eight sector counts are found to be significant. In Figure 5, circular sectors that are not significantly different from what would be expected under the null are white, while those circular sectors that were significantly larger than
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what was expected are colored red, and those that were significantly smaller than expected are colored blue.3 Three of these (segments 2, 5, and 6) have larger counts than expected while segments 4 and 8 are lower than expected, pointing to strong evidence of integrated spatial dynamics. [Fig. 4 about here.] [Fig. 5 about here.] [Table 1 about here.]
4 Conclusion
The new visualization devices introduced in this paper provide unique insights on the spatial dimensions of income distribution dynamics. While developed in the context of regional income convergence, we feel the approach could be applied to a wide set of spatial dynamics. To that end we are working on four main extensions of this work: [1] multi-period moves (time paths); [2] alternative approaches towards inference; [3] tests for association between direction and length of movements; [4] integration of the rose and dynamic Moran Scatter Plot into interactive geovisualization software.
References
Anselin, L. (1995). Local Indicators of Spatial Association-LISA. Geographical Analysis, 27(2):93–115. 3
In Figure 5 we give each sector equal height to focus on relative significance as reflected
by the hue of the sector.
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Basile, R. and Gress, B. (2005). Semi-parametric spatial auto-covariance models of regional growth in Europe. R´egion et D´eveloppement, n ◦ 21-2005:93– 118. Carrington, A. (2006).
Regional convergence in the European Union: A
stochastic dominance approach.
International Regional Science Review,
29(1):64. Fischer, M. M. and Stumpner, P. (2008).
Income distribution dynamics
and cross-region convergence in Europe. Journal of Geographical Systems, 10:109–139. L´ opez-Bazo, E., Vaya, E., Mora, A. J., and Suri˜ nach, J. (1998). Regional economic dynamics and convergence in the European Union. Annals of Regional Science, 36:1–28. Magrini, S. (1999). The evolution of income disparities among the regions of the European Union. Regional Science and Urban Economics, 29:257–281. Murray, A. T., Liu, Y., Rey, S. J., and Anselin, L. (2010). Exploring movement object patterns. Working paper GeoDa Center for Geospatial Analysis and Computation. Rey, S. J. (2001). Spatial empirics for economic growth and convergence. Geographical Analysis, 33(3):195–214. Rey, S. J. and Anselin, L. (2007). PySAL: A Python library of spatial analytical methods. The Review of Regional Studies, 37(1):5–27. Rey, S. J. and Le Gallo, J. (2009). Spatial analysis of economic convergence. In Mills, T. and Patterson, K., editors, Palgrave Handbook of Econometrics
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Volume II: Applied Econometrics, pages 1251–1290. Palgrave McMillan. Trede, M. (1998). Making mobility visible: a graphical device. Economics Letters, 59(1):77–82.
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List of Figures 1 2 3 4 5
Moran Scatter Plots US Relative Incomes . . . . . . . . . . . Directional Moran Scatter Plots . . . . . . . . . . . . . . . . . Rose Diagrams: Moran Movement Vectors . . . . . . . . . . . Distribution of Counts in Second Segment under H0 . . . . . Probability Rose Diagram for Spatial Income Mobility (P=8)
. . . . .
15 16 17 18 19
FIGURES
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(a) 1969
(b) 2008 Fig. 1 Moran Scatter Plots US Relative Incomes
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FIGURES
(a) Unstandardized
(b) Standardized Fig. 2 Directional Moran Scatter Plots
FIGURES
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(a) P=4
(b) P=8 Fig. 3 Rose Diagrams: Moran Movement Vectors
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Fig. 4 Distribution of Counts in Second Segment under H0
FIGURES
FIGURES
Fig. 5 Probability Rose Diagram for Spatial Income Mobility (P=8)
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FIGURES
List of Tables 1
Conditional randomization tests of directionality . . . . . . . .
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TABLES
21 Segment 1 2 3 4 5 6 7 8
Count 19 13 3 2 7 2 1 1
Expected 18.157 9.141 4.587 6.947 1.924 0.543 1.223 5.478
s 2.356 1.940 1.412 1.720 1.467 0.720 1.019 2.060
z 0.358 1.989 -1.124 -2.876 3.460 2.024 -0.219 -2.174
Table 1 Conditional randomization tests of directionality
p-norm 0.360 0.023 0.131 0.002 0.000 0.021 0.413 0.015
p-rand 0.432 0.041 0.233 0.010 0.005 0.092 0.638 0.013
