Abstract
This chapter gives an exploratory description of the spatial distribution of relative local tax bases and private wealth as well as the growth rate in these variables across Swedish municipalities during the period 1992–2013. The main aim is to test the hypothesis that municipalities with relatively high tax bases and high private wealth, such as relative capital incomes or private property values and changes in these variables, are more spatially clustered than could be caused by pure chance. The chapter is purely descriptive where we make use of two frequently used statistical tests for spatial correlation, the global Moran’s I and the local G * i (d)-statistic, as well as maps to identify what we refer to as regional ‘hot spots’. That is, clusters of municipalities with high local tax bases and private wealth in combination with high growth rates in these variables. This chapter also serves as a guide to how the global Moran’s I and the local G * i (d)-statistic could be used with application to the spatial distribution of local tax bases and private wealth across Swedish municipalities. Even though this paper focuses on local tax bases and private wealth, the method applied could of course be used to identify other types of clusters such as industrial clusters, clusters of individuals and/or industries with specific human capital and knowledge, different types of crimes, etc.
The author would like to thank Sofia Lundberg and Krister Sandberg for comments on previous versions of this paper.
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Notes
- 1.
A more in-depth discussion and description of the spatial weights matrices used in this paper is given in the section Definition of Neighbors below.
- 2.
\( SD(I)=\sqrt{E{\left[I\right]}^2-E{\left[I\right]}^2} \) where \( E{\left[I\right]}^2=\frac{A-B}{C} \); \( A=n\left[\left({n}^2-3n+3\right){S}_1-n{S}_2+3{S}_0^2\right] \); \( B=D\left[\left({n}^2-n\right){S}_1-2n{S}_2+6{S}_0^2\right] \); \( C=\left(n-1\right)\left(n-2\right)\left(n-3\right){S}_0^2 \); \( D=\frac{{\displaystyle {\sum}_{i=1}^n}{z}_i^4}{{\left({\displaystyle {\sum}_{i=1}^n}{z}_i^2\right)}^2} \); \( {S}_0={\displaystyle \sum_{i=1}^n}{\displaystyle \sum_{j=1}^n}{w}_{ij} \); \( {S}_1=\frac{1}{2}{\displaystyle \sum_{i=1}^n}{\displaystyle \sum_{j=1}^n}{\left({w}_{ij}+{w}_{ji}\right)}^2 \); and \( {S}_2={\displaystyle \sum_{i=1}^n}{\left({\displaystyle \sum_{j=1}^n}{w}_{ij}+{\displaystyle \sum_{i=1}^n}{w}_{ji}\right)}^2 \).
- 3.
Most commonly, the 95 % level of significance is used to evaluate the significance of econometric test statistics. However, as we will later use the 99 % level of significance when calculating the G * i (d) -statistics in order to single out the most significant regions, we use the same level of significance when evaluating the Moran’s I.
- 4.
Here \( SD\left[{G}_i^{*}(d)\right]=\sqrt{V\left[{G}_i^{*}(d)\right]}=\sqrt{\frac{W_i^{*}\left(n-{W}_i^{*}\right){Y}_{i2}^{*}}{n^2\left(n-1\right){\left({Y}_{i1}^{*}\right)}^2}} \) where \( {Y}_{i1}^{*}=\frac{{\displaystyle {\sum}_{j=1}^n}{y}_j}{n} \) and \( {Y}_{i2}^{*}=\frac{{\displaystyle {\sum}_{j=1}^n}{\displaystyle {\sum}_{i=1}^n}{\left({y}_i{y}_j\right)}^2}{n}-{\left({Y}_{i1}^{*}\right)}^2 \).
- 5.
\( E\left[{G}_i^{*}(d)\right]=\frac{{\displaystyle {\sum}_{i=1}^n}{\displaystyle {\sum}_{j=1}^n}{w}_{ij}}{n\left(n-1\right)} \); \( V\left[{G}_i^{*}(d)\right]=E\left[{\left({G}_i^{*}(d)\right)}^2\right]-E{\left[{G}_i^{*}(d)\right]}^2 \).
- 6.
The local tax base is in principle the sum of all inhabitants’ income from work.
- 7.
The federal capital tax income comes from a 30 % tax on personal incomes from interest rates, dividends, and net profits from property sales.
- 8.
The private property tax is a federal tax on an estimated market price on private housing and private apartments.
- 9.
See Lundberg (2014) for a discussion regarding the definition of W when testing for yardstick competition.
- 10.
See Qu and Lee (2015) for estimation of spatial autoregressive models with an endogenous spatial weights matrix.
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Lundberg, J. (2015). The Spatial Distribution of Wealth: A Search for Hot Spots. In: Ishikawa, T. (eds) Firms’ Location Selections and Regional Policy in the Global Economy. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55366-3_11
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