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thesis entitled

SOCIAL EQUITY AND THE SPATIAL VARIATION OF THE
PROPERTY TAX STRUCTURE OF LANSING, MICHIGAN

presented by

Julie Louise Colby

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M.A. degree in Geography
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SOCIAL EQUITY AND THE SPATIAL VARIATION OF THE PROPERTY TAX
STRUCTURE OF LANSING, MICHIGAN

By

Julie Louise Colby

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF ARTS

Department of Geography

1999

ABSTRACT

SOCIAL EQUITY AND THE SPATIAL VARIATION OF THE PROPERTY TAX
STRUCTURE OF LANSING, MICHIGAN

By

Julie Louise Colby

lnequities in the property tax structure are tested for and examined using
various econometric and GIS methodologies. The inequities under investigation
include, vertical, spatial and those based on demographic, social-economic, and
housing characteristics. Five years (1994 — 1998) of property sale transaction
data from Lansing, Michigan are utilized. First, econometric models are used to
test for vertical equity. In all cases, regressive vertical inequity is concluded.
Next, the data are partitioned by sale year and subjected to semivariance
analysis, Moran’s [analysis and kriging to create surface maps. Results show
that both positive and negative spatial autocorrelation exist at a variety of lag
distances. Patterns are revealed that show there are spatial disparities and
geographic inequities in the property tax structure. Finally, other biases of the
assessment ratio are examined and identified in multiple regression analysis at
three geographic scales. The property tax structure of Lansing, Michigan has
significant inequity based on sale amount, race, income, housing stock and
market dynamics. These inequities may be caused by dynamics in the real
estate market and the assessment process itself. The assessment process must

change so that inequities are alleviated.

ACKNOWLEDGMENTS

I would like to acknowledge and thank the following people; without them I
could not have produced this document. | wish to thank Jim Myers at the
Lansing City Assessor’s Office who provided digital access to the parcel sale
transaction data used in this thesis and spent time helping me understand the
assessor‘s role in the property tax structure. I am also extremely grateful to the
Michigan State University Geography Department and my supportive
classmates. Mark Bowersox acted as my GIS consultant and put up with
repetitive brain-picking as well as occasional freak-outs, and Colleen Garrity
generously allowed me the use of her computer and home office. I owe special
thanks to family members and Tim Priest for editing suggestions and general
support. Finally, I would like to express appreciation for Bruce Pigozzi, my
advisor, who is an excellent role model and provided me the inspiration and

unending encouragement I needed to finish my Master's thesis work.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................... vii
LIST OF FIGURES .............................................................................................. iix
LIST OF ABBREVIATIONS .................................................................................. x
INTRODUCTION .................................................................................................. 1
CHAPTER 1
TAXATION AND ASSESSMENT ......................................................................... 3
The Importance of Equity ................................................................ 3
Regressive Taxation ....................................................................... 5
Assessment Process ...................................................................... 7
Ideal Property Tax Structure ......................................................... 10
Purpose ......................................................................................... 12
CHAPTER 2
DATA AND PREPARATION ............................................................................... 14
Data from the Lansing City Assessor’s Office ............................... 14
Census Data ................................................................................. 14
Data Preparation ........................................................................... 14
Three Scales of Analysis ............................................................... 19
CHAPTER 3
VERTICAL EQUITY TESTING ........................................................................... 24
Paglin and Fogarty Model ............................................................. 27
Cheng Model ................................................................................. 30
The International Association of Assessing Officers Model ........... 34
Clapp Two-Stage-Least-Squares Simultaneous Equations Model 36
Controlling for Year ....................................................................... 38
Conclusion .................................................................................... 40
CHAPTER 4
SPATIAL AUTOCORRELATION AND SURFACES OF INEQUITY ................... 42
Research Question 1: Is the Assessment Ratio Spatially
Autocorrelated? ............................................................................. 44
Semivariance Analysis ....................................................... 46
Semivariance Results ......................................................... 48
Moran’s I Tests ................................................................... 49
Morans’ I Results ................................................................ 53
Research Question 2: What do the Spatial Patterns Look Like?... 54
Semivariograms ................................................................. 54
Kriged Surfaces .................................................................. 63

Expected Map Patterns ...................................................... 64

Description of Map Patterns ............................................... 71
Research Question 3: Do the Patterns Vary From Year to Year?. 72
Map Interpretation and Results .......................................... 72
Research Question 4: Is an Assessment Correction Process
Present? ........................................................................................ 76
Correlation Results ............................................................. 77
Conclusion .................................................................................... 78
CHAPTER 5
EXPLANATION OF THE ASSESSMENT RATIO VARIATION ........................... 80
Framing the Hypotheses ............................................................... 8O
Correlation .................................................................................... 85
Ordinary Least Squares Modeling ................................................. 87
Tract Scales ....................................................................... 89
Block Group Scale .............................................................. 97
Parcel Scale ..................................................................... 100
Conclusion .................................................................................. 104
Sources of lnequity ..................................................................... 105
CHAPTER 6
CONCLUSION ................................................................................................. 108
Findings ...................................................................................... 108
Causes of Bias ............................................................................ 109
Future Research ......................................................................... 111
Implementing Geographic Process and Methodology ................ 111
APPENDIX A
SEMIVARIANCE VALUES FOR 1994 DATA ................................................... 118
APPENDIX B
SEMIVARIANCE VALUES FOR 1995 DATA ................................................... 120
APPENDIX C
SEMIVARIANCE VALUES FOR 1996 DATA .................................................. 122
APPENDIX D
SEMIVARIANCE VALUES FOR 1997 DATA ................................................... 124
APPENDIX E
SEMIVARIANCE VALUES FOR 1998 DATA ................................................... 126

APPENDIX F

1994 MORAN’S I .............................................................................................. 128
APPENDIX G

1995 MORAN’S I .............................................................................................. 132
APPENDIX H

1996 MORAN’S | .............................................................................................. 136
APPENDIX I

1997 MORAN’S l .............................................................................................. 140
APPENDIX J

1998 MORAN’S | .............................................................................................. 144
APPENDIX K

CENSUS TRACT SCALE CORRELATIONS .................................................... 148
APPENDIX L

BLOCK GROUP SCALE CORRELATIONS ..................................................... 150
APPENDIX M

PARCEL SCALE CORRELATIONS ................................................................. 152
BIBLIOGRAPHY .............................................................................................. 154

vi

LIST OF TABLES

Table 1 - Estimated Paglin and Fogarty Model ................................................... 29
Table 2 — Estimated Cheng Model ..................................................................... 33
Table 3 — Estimated I.A.A.O. Model ................................................................... 36
Table 4 - Estimated Clapp ZSLS Simultaneous Equation Model ....................... 38
Table 5 — Year Specific Cheng Model Results ................................................... 40
Table 6 - Chapter 4 Research Questions .......................................................... 44
Table 7 — Null Hypothesis Rejections by Year .................................................... 53
Table 8 - Semivariogram Results ...................................................................... 63
Table 9 — Correlation Statistics ........................................................................... 78
Table 10 — Hypothesized Correlation with AR .................................................... 86
Table 11 - Bivariate Pearson’s Correlation Results at Three Scales ................. 86
Table 12 — Simple Regressions, Dependent Variable = AR, df = 40 .................. 90
Table 13 — Tract Multiple Regression 1 .............................................................. 93
Table 14 — Tract Multiple Regression 2 .............................................................. 94
Table 15 - Simple Regressions, Dependent Variable = AR, df = 116 ................ 97
Table 16 - Block Group Multiple Regression 1 .................................................. 98
Table 17 — Block Group Multiple Regression 2 .................................................. 98
Table 18 — Simple Regressions, Dependent Variable = AR, df = 8055 ............ 100
Table 19 - Parcel Multiple Regression 1 .......................................................... 102

vii

LIST OF FIGURES

Figure 1 - Parcel Sale transactions Observations .............................................. 19
Figure 2 — Census Tract Boundaries in Lansing, MI ........................................... 21
Figure 3 — Census Block Group Boundaries in Lansing, MI ............................... 22
Figure 4 - Parcel Scale Transaction Observations on Block Groups ................. 23
Figure 5 — Paglin and Fogarty Model (under the null) ......................................... 28
Figure 6 - Estimated Paglin and Fogarty Model ................................................. 30
Figure 7 - The Cheng Model (under the null) ..................................................... 32
Figure 8 — Estimated Cheng Model .................................................................... 34
Figure 9 - The I.A.A.O Model (under the null) .................................................... 35
Figure 10 — Estimated I.A.A.O. Model ................................................................ 36
Figure 11 - 1994 Semivariogram with Maximum Active Lag Distance ............... 55
Figure 12 - 1995 Semivariogram with Maximum Active Lag Distance ............... 56
Figure 13 - 1996 Semivariogram with Maximum Active Lag Distance ............... 56
Figure 14 — 1997 Semivariogram with Maximum Active Lag Distance ............... 57
Figure 15 — 1998 Semivariogram with Maximum Active Lag Distance ............... 57
Figure 16 — ldealized Semivariogram with Parameters ...................................... 59
Figure 17 — 1994 Truncated Semivariogram with Exponential Model ................ 60
Figure 18 — 1995 Truncated Semivariogram with Exponential Model ................ 60
Figure 19 — 1996 Truncated Semivariogram with Exponential Model ................ 61
Figure 20 — 1997 Truncated Semivariogram with Exponential Model ................ 61
Figure 21 — 1998 Truncated Semivariogram with Spherical Model .................... 62

Figure 22 - Grid of Kriged Surface of AR from 1994 Sale transaction Data ....... 66

viii

Figure 23 - Grid of Kriged Surface of AR from 1995 Sale transaction Data ........ 67
Figure 24 - Grid of Kriged Surface of AR from 1996 Sale transaction Data ....... 68
Figure 25 — Grid of Kriged Surface of AR from 1997 Sale transaction Data ....... 69

Figure 26 — Grid of Kriged Surface of AR from 1998 Sale transaction Data ....... 70

Figure 27 — 1995 Surface Subtracted by 1994 Surface ...................................... 73
Figure 28 - 1996 Surface Subtracted by 1995 Surface ...................................... 74
Figure 29 — 1997 Surface Subtracted by 1996 Surface ...................................... 74
Figure 30 — 1998 Surface Subtracted by 1997 Surface ...................................... 75
Figure 31 — 1998 Surface Subtracted by 1994 Surface ...................................... 75
Figure 32 — 1997 Surface Subtracted by 1994 Surface ...................................... 76
Figure 33 — Census Tract Residuals .................................................................. 96
Figure 34 — Block Group Residuals .................................................................... 99
Figure 35 - Parcel Residuals ........................................................................... 103
Figure 36 - Three Dimensional Residual Surface ............................................ 104

LIST OF ABBREVIATIONS

AR ............................................................................................. Assessment Ratio
AV .................................................................................................. Assessed Value
BLACK .............................................................. percent of population that is black
HISPANIC ................................................... percent of population that is Hispanic
INC ............................................................................... median household income
LNAV ............................................................................................ natural log of AV
LNS ............................................................................................... natural log of S
MV ..................................................................................................... Market Value
PUBLIC ............................................. percent of households on public assistance
RENTOCC .................................. percent of housing units that are renter occupied
S ............................................................... dollar amount of parcel sale transaction
SEASON .................................................... binary variable: summer = 1,winter = 0
VACANT ................................................... percent of housing units that are vacant
WHITE ............................................................... percent of population that is white
YEAR ......................................................................... median age of housing stock

INTRODUCTION

The problem under investigation in this thesis is to understand the nature
and extent of inequity in the property tax structure of Lansing, Michigan. The
objective of this research is to synthesize previous research efforts as well as to
incorporate some new GIS technologies in a methodologically comprehensive
examination of the social and geographic equity or inequity in property taxation.

This work is not intended to be a case study that exemplifies all property
tax structures in the United States. However, in using several years worth of
data from one midsize, midwestern capital city, it is hoped that this study will
identify any problems and dynamics in the assessment process that are affecting
the level of social equity realized by tax payers.

The assessor’s offices in Lansing, Michigan and many other cities of
similar size and nature are still forced to operate in pre-GIS systems due to lack
of appropriated funding and expertise. In fact, in Lansing, the assessor’s office
does not use geographic methods or evaluate assessments spatially. While the
lack of GIS technology would never by the underlying cause of any inequity, this
particular technologic and geographic deficiency in local government is especially
vexing because the property tax assessment process is inherently geographic
and could be greatly aided by the implementation of a geographic information
system. Therefore, an additional goal of this thesis is to demonstrate how a
precise spatial evaluation of property tax equity could be important for local
assessors offices. However, it should be understood, that although GIS is a

powerful tool and could be used to better the assessment process, when used

incorrectly, it could just as easily not improve the system. Just because a
municipality uses GIS to assess properties, that municipality is not necessarily
eliminating inequity from its property tax structure.

The research is empirical by nature and highly data intensive. Many
different software packages are used in this thesis. They include ArcView,
ARC/INFO, Access, Excel, 68+, SPSS, SYSTA Tand Transcad. In addition,
many procedures and methods are applied to the data. For this reason, the
organization of this thesis will be atypical. Rather than one traditional
“methodology" chapter, this thesis will detail methodologies as they are
discussed in the body of the paper. Aside from the first two chapters, which are
“T axation and Assessment” and “Data Preparation” respectively, there are three
distinct research avenues in this thesis: Chapter 3, “Vertical Equity Testing”,
Chapter 4, “Spatial Autocorrelation and Surfaces of lnequity”, and Chapter 5,
“Explanation of the Assessment Ratio Variation”. Each of the three has separate
research questions, hypotheses, literature grounding, methodologies, results and
conclusions. The final chapter, Chapter 6, will be dedicated to interpreting the
meaning and use of this thesis to assessors’ offices as well as to recommend

future research foci.

CHAPTER 1

TAXATION AND ASSESSMENT

This chapter is intended to provide the reader background on the thesis
topic, the property assessment process and the legal obligations of taxing bodies
to provide taxpayers with a fair and equitable property tax. The term equitable,
throughout this thesis, will refer to anything “having or exhibiting equity" of which
equity is “freedom from bias or favoritism” (Mish 1998, p. 392). These terms
should not be confused with equality or any others coming from the root equal.
Things equal are “of the same measure, quantity, amount, or number.” That is,
to be equal is to be “identical in mathematical value” (Mish 1998, p.391 ). If
property taxes were equal, all property owners would pay the same dollar
amount in taxes and that would not be equitable.

The Importance of Equity

There have been two main theories of taxation, the benefit theory and the
ability to pay theory, from which modern day property taxation was born. Central
to both the benefit theory of taxation and the ability to pay theory of taxation are
their principles of equity (McCluskey et. al, 1998). Without equity, taxation of any
kind by local government is vulnerable to the acceptance and approval of
taxpayers. In fact, a pillar of American democracy is jeopardized if equity is not
present. Because of this, equity in the property tax structure is of vital
importance and has been the topic of many research endeavors and papers. It

shall also be the topic of this thesis, in which property tax inequities will be

evaluated, spatially examined and statistically explained. The objective of this
research is to synthesize previous research efforts in a methodologically
comprehensive paper aimed at examining the social and geographic equity of the
property tax structure in Lansing, Michigan.

It was the primary purpose of an international research project to examine
the efficacy of market value as a basis of assessment for an equitable property
taxation system (McCluskey et al, 1998). While the research effort has not yet
concluded, initial findings identify a need for criteria from which the collection of
revenues by local government can be measured. One such criterion the group
asserts is the status of equity, a term used interchangeablywith the phrase, “how
well the property tax is administered and assessed.”

Typology of Equity

Generally, there are two kinds of equity recognized in the literature
surrounding property tax equity, horizontal and vertical. Horizontal equity exists
when properties of the same value are, on average, taxed the same. Vertical
equity exists when properties of different value are, on average, taxed at the
same proportion of market value. These “principles of both horizontal and
vertical equity are vital to the perceived equity of any landed property-based tax,
and, therefore, its social acceptability" (McCluskey et. al 1998).

Horizontal and vertical equity are very important. They are much
discussed in taxation literature and this paper will deal with them. However,
spatial equity, which incorporates the principles of both horizontal and vertical

inequities, will be the primary target of investigation in this paper. For the

purpose of this paper, spatial equity will refer to the situation that exists if all
geographic areas of the taxing jurisdiction are taxed at the same rate. This
means that if spatial equity exists, properties that are over- or under-taxed are
spatially independent or not spatially autocorrelated. Indeed, if spatial equity
exists there will be no discernable spatial pattern to the property tax structure.

It is sensible to study spatial equity in addition to horizontal equity and
vertical equity, which are generally studied separately and aspatially, because if
horizontal or vertical inequity exists, the inequity is likely to be spatial as well.
Both vertical and horizontal inequities tend to represent themselves spatially due
to the nature of human settlement to organize itself in homogenous
neighborhoods within a larger segregated urban area. Furthermore, if equity,
regardless of type, is studied spatially, more information is revealed about the tax
structure. Spatial equity not only assures both horizontal and vertical equity, it
generally assures that a property tax is not regressive, which, as is described in
the next section, is most inequitable in a property tax structure.

Regressive Taxation

Vertical inequity may be regressive or progressive. Generally, regressive
vertical inequity is defined as a situation in which lower valued properties are
taxed at a higher rate or larger proportion of market value than higher valued
properties. Conversely, a progressive vertical inequity exists when higher valued
properties are taxed at a higher rate or larger proportion of market value than
lower valued properties. Because lower valued properties tend to cluster

together in a taxing jurisdiction and higher valued properties tend to cluster

together, vertical inequity can most often be visualized spatially, revealing a
spatial pattern to the over and under assessed areas.

The same spatial distribution might also be true of horizontal inequity.
Consider the example of horizontal inequity detected by Beveridge in his
research for the New York Times (Schemo, 1994). Beveridge found that in forty-
four percent of the sixty-one major United States cities and suburbs that he
studied, properties of the same value were taxed differentially depending on the
race of the property owner. Property owners who are black were taxed
significantly more than property owners who are white for similarly valued
properties in the same taxing jurisdictions. This is an example of a horizontal
inequity, but it also has a spatial pattern due to the strong degree of racial
segregation in his study areas. For the purposes of this paper, the inequity
described above will also be considered regressive. That is, the definition of a
regressive tax will be expanded to include not only regressive vertical inequities
but also horizontal inequities like this one discovered by Beveridge.

Thus, the definition of a regressive property tax will be expanded to one in
which properties that are of low value, owned by a poor person or person of a
marginalized group, or living in an area occupied by poor and or marginalized
populations are taxed at a higher proportion of market value than properties of
high value, owned by a wealthy person or someone living in an area occupied by
wealthy and or politically powerful mainstream populations. In other words, a

regressive tax is one that benefits the politically powerful and puts at a

disadvantage the poor and marginalized. Conversely, a progressive tax benefits
the opposite populations.

Researchers such as Massey recognize this kind of regressive spatial
inequity. In his discussion of property tax inequity as one of the problems
inherent to segregation by race and/or class, he warns of a “rising inequality and
its geographic expression." He sites data that show increasing segregation of
both poor and minority populations, as well as affluent populations. Because of
this increased geographic segregation, Massey concludes that property tax
inequity may become increasingly unequal as well (Massey, 1996).

A regressive tax structure is politically unpopular and exists in contrast to
the ability to pay theory, one of the historical underpinnings on which the property
tax is based. It is an extremely undesirable form of inequity to society
(McCluskey et. al, 1998). Therefore, the purpose of this paper is to test for and
examine any inequity with an emphasis on any regressive inequity that may exist
in the property tax structure.

Assessment Process

Throughout the United States and much of the world, property tax is
determined by some assessment of the property’s market worth. It follows, then,
that the status of equity is dependent on the quality of the assessment process.
This paper will utilize empirical assessment data from Lansing, Michigan.
Lansing will serve as an interesting case study that in some ways typifies the

American property tax structure and, in other ways, does not.

The taxing process employed in Lansing, Michigan is representative of
that present elsewhere in the United States. Each parcel of land is taxed
according to its assessed value multiplied by the municipality’s millage rate,
which is constant for each property type. A millage rate is defined as the amount
of taxation in mills (one tenth of a cent) per dollar of valuation and the assessed
value of the property is the property tax assessors best guess of the market
value. The market value can be defined as:

“the most probable price expressed in terms of money that a
property would bring if exposed for sale in the open market, in an

arm’s length transaction between a willing seller and a willing

buyer, both of whom are knowledgeable concerning the uses of the

property"(Miles and Wurtzbach, 1987, p. 751 ).

Clearly, market value is impossible to measure if a property isn’t “exposed
for open sale” and one could never even say with certainty that a particular
transaction was an “arm’s length transaction.” That is why assessors can only
estimate or guess the market value and the result of this estimation is their
assessment. Equation 1.1 shows how the property tax for parcel i is calculated.
Equation 1.1: Property tax. = AV. * millage rate

There are also some ways in which the taxing process in Lansing is
different from other taxing bodies. For example, in most tax jurisdictions, the
assessor estimates market value and sets their assessment equal to the
estimate. In other words the assessment has a theoretical one — to — one
relationship with market value. This is not the case in Michigan. Although

assessed value is still directly proportional to market value it is not a one — to —

one relationship. Instead, state statute dictates that property in Michigan is

assessed at half of market value. Therefore, by state law, assessors in Michigan
are required to estimate market value of each property and divide their estimate
in half to arrive at the assessed value of each property. This does not mean that
property taxes are lower in Michigan than in most other places. Michigan simply
has higher millage rates to make up for the lower assessments. One could
question why this is done. Perhaps the assessment system has something to
gain from the smaller assessments. If assessments are half of market value they
make the magnitude of the property tax base seem less unpalatable for
taxpayers who are than less likely to challenge or dispute the assessments. This
could potentially save the assessors office from having to defend their
assessments against questioning taxpayers.

In evaluating the quality of assessments, it is often helpful to express
assessed value (AV) as a proportion of market value (MV), or some measure of,
and is called the assessment ratio (AR).

Equation 1.2: AVJMV. = AR.

Due to the legal reasons stated above, assessment ratios should be 0.50 for all
properties in Lansing. Assessment Ratios reveal the accuracy of the
assessments. If they are more than 0.50, the assessment is too high. If they are
less than 0.50 then the assessment is too low. Because MV is an immeasurable
variable (see above definitions of market value), ARs are often constructed with a
surrogate for MV — usually the sale amount (S) of the property transaction.
Equation 1.3: AWS. = AR.

Equation 1.3 can be solved for AV to get Equation 1.4.

Equation 1.4: AV. = AR. * S.
in equation 1.1 to arrive at equation 1.5 as follows:
Equation 1.5: Property tax. = AR. * S. * millage rate

Equation 1.5 shows the dependency of the property tax on the
assessment ratio. If the assessment ratio is too high, the property tax will be
more than it should and if the assessment ratio is too low, the property tax will be
less than it should. For example, if the millage rate for a jurisdiction were
imposed at 0.075, a properly assessed $100,000 property would be taxed
(0.50)*$100,000*(0.075) or $3,750.00. However, the same $100,000 property
that is incorrectly under-assessed at 0.45 of market value would only be taxed,
(0.45)*$100,000*(0.075) or $3,375.00. The under-assessed property would pay
$375.00 less than the correctly assessed property. On the other hand, the same
$100,000 property that is incorrectly over-assessed at 0.55 of market value
would be taxed, (O.55)*$100,000*(0.075) rate or $4,125.00. The over-assessed
property pays $375.00 more than it would if the property were assessed
correctly. Worse still, the over-assessed property is charged $750 more than if it
were under-assessed at 0.45 of market value.
Ideal Property Tax Structure

In a perfect property tax system, all properties in the system would be
taxed at the same rate according to their market values. That is, the assessment
ratio, AR (commonly calculated by dividing an assessed value, AV, by its
measured market price, S) would be uniform across all properties. While

deviations from uniformity in the assessment ratio are inevitable and to be

10

expected, these deviations would occur randomly if the system were equitable.
Assuming equity, there would be no spatial pattern to the assessment ratio
distribution. However, in most cities in the United States, spatial uniformity does
not exist in property assessment ratios. Indeed, there is often a geographic
distribution, or pattern, in the under and over assessment of residential
properties. These patterns that are pervasive in the property tax structure are
indicative of inequity.

Taxing administrations do generally evaluate their assessments for equity
themselves. The Lansing City Assessor’s Office does this by dividing the
jurisdiction into neighborhoods. Every year before issuing new assessments,
each neighborhood’s average assessment ratio, based on sale transaction data,
is compared to the mean for the city. If the average neighborhood assessment is
too low, each assessment in the neighborhood is adjusted up by some multiplied
amount. In the same way, if the average assessment is too high, each
assessment in the neighborhood is adjusted down by some multiplied amount.
While this method may indeed correct some inequities, individual assessments
may become even more skewed. For example, some parcels that are over
assessed in a neighborhood that on average is assessed too low will become
even more over assessed. However, individual parcel assessment adjustments
are too costly for most assessors’ offices and can be biased if judged by a non-
market sale transaction.

One way some taxing administrations assure greater compliance to

standards of equity is by managing and evaluating their assessments using a

11

geographic information system (GIS). A GIS is a database management system
that can deal effectively with large amounts of spatially referenced data. The
property tax assessment process is an ideal use of GIS because it is inherently
geographic. Because of this, several taxing bodies and even entire states have
implemented geographic information systems to aid in the administration of the
property tax (Castly 1993, Hensley 1993, Rodda 1992 and Wills 1998). Here lies
another difference between the Lansing City Assessor’s Office and other places
in the United States. The city of Lansing has not appropriated the funds
necessary to implement a GIS. Therefore, one intent of this thesis is to
demonstrate how a detailed computerized spatial evaluation of property tax
equity would be important for local assessor’s offices like Lansing’s.
Purpose

The purpose of this research is to test for, describe and explain patterns in
the property tax structure by implementing a variety of methodologies and using
assessment and sales data for the city of Lansing, Michigan. Several
hypotheses will be tested with the end goal of addressing the following research
question: What is the extent, nature and future of property tax inequity? The
many hypotheses to be tested in this paper fall into three distinct categories.
Thus, the thesis research can be divided into three main parts. First, in Chapter
3, the status of equity in the property tax structure will be ascertained and
measured by employing several tests for vertical inequity. Second, in Chapter 4,
the assessment ratio pattern will be graphically described and evaluated. Finally,

in Chapter 5, the geographic patterns in the assessment ratio will be

12

mathematically accounted for or explained. The research questions and
hypotheses specific to each chapter’s topic will be detailed in their respective
chapters. Prior to this, however, Chapter 2 will introduce the data used
throughout the thesis and will explain the methods used to prepare it for

answering the research questions.

13

Chapter 2

DATA AND PREPARATION

The data for this research came from two sources: the Lansing City
Assessor’s Office and the US. Bureau of the Census.
Data from the Lansing City Assessor’s Office:

Parcel sale transaction data were obtained from the city assessor for all
8,660 transactions occurring from January 1994 to August 1998. The data, in
spreadsheet format, include assessed value at the time of sale (AV), sale amount
(S), parcel address, month and year of sale.
Census Data

1990 Census Bureau Tiger files containing the street index of Lansing,
Michigan were obtained from Environmental Research Institute (ESRI) on its
Internet web site (www.esri.com). In addition, census tract boundaries and
census tract block group boundaries were obtained from the same web site in
shape file format. Finally, several census tract and block group variables were
extracted from the 1990 Census of Population and Housing for Lansing, MI. The
census variables include measures of race, income and age of housing. They
will be used in conjunction with the property sale transaction data later in the
papen
Data Preparation

The 1990 Tiger street index of Lansing, MI was imported by Transcad as

were the parcel sale transaction data. The parcel sale transaction data were

14

then geo-coded, by the address matching function in Transcad, to the 1990 Tiger
Street Index. This was done to obtain decimal degree latitude and longitude
geographic coordinates for each sale transaction observation. The address
matching was ninety-five percent successful, or about ninety-five percent of the
transactions were geocoded. The process leaves 8,227 geocoded sales
observations in the sample. The five percent that could not be matched were
probably not successful due to errors in the Tiger street index or errors in the
street names, zip codes or street numbers of the parcel sale transaction. They
have been discarded and will not be used in the following analyses. An
assumption is made at this point, for the purposes of this research, that these
unmatched addresses are randomly distributed throughout the city of Lansing.
Although this assumption is not proven, it is known that the unmatched five-
percent of sale transaction observations represented all Lansing zip codes.

Next, assessment ratios (ARs) were calculated for each observation. This
was accomplished by dividing each assessed value by its corresponding sale
amount. As noted above, Michigan state statute dictates that each assessed
value should be half of market value - not sale amount. However, market value
is an immeasurable variable so sale amount is measured and will be used in its
place in this study as it has in many others (DeCesare, 1998; I.A.A.O, 1990).
Market Value may be immeasurable but it is assumed to be a nonstochastic fixed
variable while sale amount, S, is viewed as a random variant around the fixed

market value (Clapp, 1990).

15

Unfortunately, sale amount, S, is not always a good indicator of market
value. Sometimes S is not representative of an “arm’s length market transaction.”
Occasionally, properties sell for a small nominal value simply to transfer official
ownership. For example, an elderly landowner may decide to “sell” a property to
a younger relative for a dollar. In this case, the name on the deed may be
changed to represent a change in ownership but not a transfer of the property’s
market worth. This would create an extremely high AR regardless of the quality
of the assessment. Another example of a sale that doesn’t represent market
value, which would result in a false high AR, is that of a land contract sale.
Sometimes a parcel is sold by contract at the market value of the day. However,
the sale is not finalized and made official until the contract is paid off, in most
cases, years later. The official sale amount used to calculate the AR then is the
amount on the contract. Inflation and market value increases render the official
sale amount very low and the calculated assessment ratio much too high.

There are also instances when an especially low AR is not indicative of a
bad assessment but rather an unusual circumstance as well. Perhaps the
assessment is done correctly but the property buyer is someone unfamiliar with
the local market conditions. For example, someone from California, used to the
high cost of housing and living there, suddenly gets transferred to Lansing,
Michigan and must buy a house in a short amount of time, might purchase the
house at an inflated amount, mistaking it for a bargain. This would result in an
especially low assessment ratio not representative of an ann’s length market

transaction. According to the International Association of Assessing Officers

16

(I.A.A.O.), sale transaction observations are considered “outliers” if they have
very high or low “sales ratios” which are calculated identically to the ARs in this
study. The I.A.A.O. believes the outliers may result from 1) “poor or outdated
appraisals,” 2) “non-arm’s-Iength” sales, or 3) a mismatch between the property
sold and the property appraised (I.A.A.O 1990, p. 137).

So it seems, the sale amount (S) is not always a very good surrogate for
market value. Because of this, the data are trimmed, using the variable AR, to
get rid of data observations that are not representative of market transactions.
Such trimming is done in most assessment ratio studies that utilize empirical sale
transaction data (Denne 1993).

One such assessment ratio study was done by Gaston, who trimmed any
transactions with “market error" which he defined as “discrepancy between sale
price and actual market value”(Gaston 1984, p. 182). Although Gaston does not
explicitly detail what percent of transactions he eliminated, he does list causes of
market error that led him to eliminate as he did. He eliminated transactions with
possible departures from market value caused by “inflation/deflation relative to
lien date, associated personal property, title transfer instruments, interest rates,
and other relevant and estimable transaction factors”(Gaston 1984, p. 182).

The I.A.A.O recommends the following trimming procedures: 1) select
“cut-off” points and trim above and below them, or 2) trim the observations that
fall more than two standard deviations from the mean ratio, which is usually
about five percent of the observations (I.A.A.O. 1990, p. 137). However, the

organization suggests that “most sales are usable,” and the goal of the

17

researcher should be “not to find reasons to exclude sales.” In fact, “in large
samples, the accidental inclusion of a few invalid sales will have little effect on
ratio studies” (I.A.A.O. 1990, p.138)

For this study the observations were trimmed to get rid of extreme
assessment ratios. Each tail of the assessment ratio distribution was trimmed by
1.25 percent. The total trim resulted in the removal of 2.5% of the geocoded sale
transaction observations. The trim resulted in a sample population of 8069
observations with a mean of 0.46 and range of (0.236 - 0.846). The following
map, Figure 1, shows the locations of the sample observations.

The sample is now prepared for the analysis of Chapter 3 and Chapter 4.
However, some of the research in Chapter 5, involves analysis at three
geographic scales: census tract scale, census block group scale and parcel
scale. The property sale transaction data were prepared in three different ways

accordingly. These are described in the next section.

18

 

 

E Lansing Blocks

I ' SalesTranaadionsObsenrations

 

 

 

 

 

Figure 1 - Parcel Sale transaction Observations
Three Scales of Analysis

To prepare the data for analysis at the census tract scale, the 8069 geo-
coded observations in the trimmed sample were each assigned to a census tract
using ARC/INFO. This was accomplished by first making the geo-coded
observations into a point file and importing their corresponding data into
ARC/INFO and creating a data coverage. Next, ARC/INFO coverages were
made of the census tract boundary shape file. Then the points were assigned to
tracts using the Identity function of ARC/INFO. Finally, the census tract variables
from the census of population and housing were imported by ARC/INFO and
joined to the geographic census tract boundaries. The resulting database
contained the 8069 parcels with their associated assessed values, assessment

ratios, sale amount and their census tract information. The observations were

19

then aggregated to the census tract level. Aggregation was done in an Excel
pivot table by averaging the parcel data variables.

The boundary of the city of Lansing contains and cuts through a total of
forty-four census tracts. The sale transaction observations are fairly well
distributed throughout the city (See Figure 1). However, there are very few
observations in the central business district (CBD) located in the middle of the
city. This is due to the large amount of government and corporately owned
properties that do not turn over very quickly. In addition, the Grand River cuts
through this downtown section, and there is a large public park on both sides of
the river containing very few privately owned properties. So while forty-one of
the tracts in Lansing each have at least one-hundred observations which is
convenient statistically, three tracts have only two, three and six observations
each. These are the tracts that cover the CBD area of Lansing. They were
therefore eliminated from the analysis, as their sample sizes would not lead to
unbiased means in statistical analyses. Thus, after the aggregation process,
forty-one observations are left in the sample at the census tract scale. Figure 2
shows the Census tract observational unit boundaries and the Census tracts that

will not be included in the analysis for lack of data.

20

 

 

 

 

 

1.

 

 

 

 

 

 

 

 

M - no clata

Figure 2 - Census Tract Boundaries in Lansing, MI

 

 

 

 

Preparation of the data for analysis at the census block group scale was
done in much the same way. The observations in the trimmed sample of parcel
sale transactions were each assigned a block group and then all of the data were
aggregated to that scale. All of the block groups within the three tracks that were
eliminated due to small sample size at the tract scale were also eliminated at this
scale. In addition, some of the remaining block groups had less than twenty
parcel sale observations. If so, they were combined to create a group of block
groups with at least twenty observations. This was done for statistical purposes.
Otherwise the means could be biased. In all, there are 117 block group
observations or group of block groups observations in the sample at this scale of

analysis.

21

 

 

 

 

 

#57
L

 

 

 

 

_._I

 

 

 

 

 

- no data

Figure 3 - Census Block Group Boundaries in Lansing, MI

 

No aggregation was necessary to prepare the data for analysis at the
parcel scale. However, since census variables could not be collected at the
parcel scale, each parcel took on the characteristics of its block group, so, all that
had to be done was to assign each parcel to a block group. At this scale all
parcels in the same block group will take on the same values for the variables
collected at the block group scale. For instance, all of the parcels in block group
one will have the same percentage black population and the same median
household income. However, the month and year of the transaction will vary
within the observations of the block group. The observations that were
eliminated for statistical purposes at the other scales will also not be included in
the analysis done at the parcel scale. Hence, the sample size decreased from

8069 to 8056.

22

 

 

[1 Block Groups

 

 

 

 

Sales Transaction Observations.
Figure 4 - Parcel Sale Transaction Observations on Block Groups

 

Although the data are further manipulated and subjected to various and
many methods, these will be noted and discussed in the bodies of the next three

chapters where appropriate.

23

Chapter 3

VERTICAL EQUITY TESTING

According to the International Association of Assessing Officers (I.A.A.O.),
“Vertical inequities are differences in appraisal levels for groups of properties
defined by value.” (I.A.A.O 1990, p.516)

As was stated in Chapter 1, vertical inequity exists if properties are taxed
differentially depending on their values. Recall that a regressive (progressive)
vertical inequity exists if the assessment ratio decreases (increases) with an
increase in property value or increases (decreases) with a decrease in property
value. There are many different models in the tax and business literature that
serve to test for vertical equity in property taxes. The purpose of this chapter is
to employ some of these models to test for vertical inequity in the property tax
structure using the assessment data for Lansing, Michigan. There are two
general research questions that are addressed in this chapter. First, is the
property tax structure of Lansing, Michigan vertically inequitable? The null
hypothesis of this question is that no vertical inequity exists. The primary test
hypothesis is that the property tax structure has regressive vertical inequity. The
other alternative hypothesis is that progressive vertical inequity exists. This
question will be addressed by employing four vertical equity tests.

The second research question asks, does the status of equity vary from
year to year? The null hypothesis is that there is no variation in the status of

equity from year to year. A rejection of this null will lead to the conclusion that in

24

some years the assessment process leads to a more equitable situation than in
others. This question will be addressed by modifying the models to control for
year of sale and assessment. This issue involving dynamics in the property tax
structure will also be addressed in Chapter 4.

In 1972, Paglin and Fogarty developed the first econometric model to test
for vertical property tax inequity. It consists of a simple linear regression of
assessed value on sale amount (Paglin and Fogarty 1972). Other researchers
such as Edelstein (1979) tested this model with empirical data for various study
areas. In these studies a regressive property tax structure was concluded most
often. Soon after its development, a variation of the vertical equity test was
developed by Chang in 1974. Cheng assumed a different functional form. He
thought the relationship between Assessed Value and Sale amount should be
exponential, or log-linear. Other researchers added their own slight changes to
the vertical equity test model, but they left assessed value as the dependent
variable and sale amount as the independent variable.

The literature changed direction when the hypothesized relationship
between assessed value and sale amount was reversed by Kochin and Parks in
1972. They believed that sale amount should be the dependent variable and
assessed value the independent variable. Kochin and Parks assert that if market
values were fixed and nonstochastic and assessed values were perfectly
proportional to market values (their were no misassessments), then assessment
ratios would appear regressive if sale amounts were distributed randomly around

market value. So, they argued, if a property sold above market value the

25

assessment ratio would appear lower and if a property sold below market value,
the assessment ratio would appear higher, incorrectly detecting regressivity. For
example, assume that all market values are $100,000 and assessed values are
$50,000. If one parcel sells for $110,000 its calculated AR would be 0.45. If
another parcel sells for $90,000 its calculated AR is 0.55. Any of the traditional
tests for vertical inequity would label this example one of regressive vertical
inequity because the parcel that sold for more has a lower AR and the parcel that
sold for less, has a higher AR. Kochin and Parks would argue that in this
example, there is no inequity because only the sale prices have errors, not the
assessments. Indeed, empirical results of their test largely revealed progressive
property tax structures and they declared previous results from the old models
were biased toward finding regressivity.

However, the assumptions made by Kochin and Parks have been
challenged. After the Kochin and Parks paper was published, the literature was
inundated with arguments for and against their model. Several papers were
written that showed statistical evidence against the Kochin and Parks model and
successfully defended the traditional models. These included work by Kennedy
(1984), Gaston (1984), and Clapp (1990). It is now generally accepted that
market value is a fixed non-stochastic variable and both assessed value and sale
amount are subject to error.

Clapp not only showed that Kochin and Parks model was plagued by bias,

he developed his own model that would incorporate the benefits of Kochin and

26

Parks and those of the traditional model. In doing so he removed the biases of
the former models (Clapp 1990).

In this chapter, four models, the Pagin and Fogarty, the Cheng, the
I.A.A.O, and the Clapp model will be employed to show whether there are
patterns of vertical inequity in the property tax structure in Lansing. In all four
models, the null hypothesis is that the property tax structure is vertically
equitable. The test hypothesis is that the property tax structure is regressive and
vertical inequity is present. An alternative situation, that the property tax
structure is progressive, might also be concluded from the test results.

Paglin and Fogarty Model

The first model, by Paglin and Fogarty was developed in 1972. It is a
simple linear regression of sale amount on assessed value. The model is
estimated by the Ordinary Least Squares method. A t—test will be used to
determine the significance of the difference between hypothesized and actual
coefficients. The model is formally stated in the following equation:
AV=Bo+B1S+e.

lithe property tax structure behaved equitably, as it theoretically should,
the mean assessment ratio would be 0.5 and accordingly the slope of the line
should also be 0.5 with an intercept of zero. Therefore under the null hypothesis,
that no vertical inequity exists, the slope of the line is 0.5 and the intercept is
zero:

Ho: [30 = 0 and B1 = 0.5.

27

However, if the line goes through the origin, there is no vertical inequity
regardless of slope. If this is the situation, it doesn’t matter what the sale price is
because the corresponding estimated assessed value will be half of sale price
and all parcels will be taxed at the same rate. Hence, the test for regressivity
hinges on the intercept. If the intercept is zero, equity exists. It the intercept is
more than zero, regressive vertical inequity exists and if the intercept is less than
zero, progressive vertical inequity exists. Figure 5 graphically shows the Paglin

and Fogarty regression under the null hypotheses of equity. It is the line:

 

 

 

 

 

AV = 0 + 0.5 5.
200000 /
//
/
,1
5 1000001 //

o v
0 o o o o
O O o O
8 8 8 8
O C O o
v- N (‘0 v

s

 

 

 

Figure 5 - Paglin and Fogarty Model (under the null)

There are two test hypotheses. First, that the property tax structure is
regressive. In this case the intercept will be more than zero. In addition, since
the mean of all assessment ratios in the sample is known to be 0.46, the slope is
expected to be less than 0.5 under conditions of regressivity.

H1: Bo > 0 and B1 < 0.5.

28

The second hypothesis is that the property tax structure is progressive. In
this case the intercept will less than zero and the slope is expected to be more
than 0.5.

H1: 8.. < 0 and B. > 0.5.
Results

The estimated Paglin and Fogarty equation is displayed in Table 1. The
estimated equation is also shown graphically in Figure 6a. The t-test on the
intercept coefficient shows it is significantly greater than zero and the t-test on
the slope shows the estimated 6. is significantly less than 0.5. Therefore, the null

hypothesis is rejected at the 99% confidence level. The result of the Paglin and
Fogarty test for vertical inequity is consistent with the first test hypothesis of a
regressive property tax structure. For comparison, the estimated or actual
relationship between AV and S is shown in Figure 6b on the same graph as the
theoretical line under equity (slope, 0.5, and intercept, 0,) and another
hypothetical linear relationship between AV and S that is also indicative of equity.
This line has a slope equal to the sample mean of 0.46 and intercept of zero.

Table 1 - Estimated Paglin and Fogarty Equation

 

 

AV = I30 + 31*3
Estimated = 2217.79 + 0.412 S

AV

s.e. = 1 18.272 0.002

t-stat for = 18.752 -46.809

hypothesis

test
R Squared = 0.926

 

 

29

 

 

 

 

 

 

 

100000
° /
,/
//
120000 n j
3 800004
40000.
O 1 r
-100000 100000 300000
0 200000 400000
5

 

 

Figure 6 - Estimated Paglin and Fogarty Model

As can be seen by the estimated Paglin and Fogarty model and Figure 6,
lower valued properties are taxed at a higher rate than higher valued properties.
For example, given the results of the test, a property that sold for $75,000 has an
estimated assessed value of $33,117.79 and therefore an estimated assessment
ratio of 0.442. A higher valued property, one that sold for $175,000, has an
estimated assessed value of $74,317.79 and therefore an estimated assessed
value of 0.425. This example shows that the higher valued property pays a lower
estimated rate of taxes and the lower valued property pays a higher estimated
rate of taxes.
Cheng Model

The Cheng model was developed in 1974 as an improved alternative to
the Paglin and Fogarty model. The Cheng model allows a non-linear relationship
to exist between assessed value, the dependent variable, and sale amount, the
independent variable:

AV = Bo SB1 * 8.

30

It is log transformed to a linear form for ease of estimation:

InAV = In 30 + B1InS + Inc

The log - log form is then estimated by ordinary least squares. Again, a t—test
will be employed to test the estimated coefficients for significance. Like the
Paglin and Fogarty test, the null hypothesis is no inequity. Under the null, the
untransformed slope is 0.5 or Ho: I30 = 0.5. However, when the equation is log
transformed the log is taken of I30 and the null hypothesis is modified so that the
transformed intercept is the log of 0.5 or

H..: InBo = ln0.5 = -0.693

Also under the null, the power of S, or the coefficient of ms, is one:

Ho: B1 = 1

Unlike the Paglin and Fogarty model, regressivity in this test hinges on the
elasticity, B1, or the power of S in the untransformed exponential model. If (3.
proves to be more than one, progressive inequity exists, or higher valued
properties are taxed higher rates. If (3. proves to be less than one, then
regressive inequity exists and lower valued properties are taxed at higher rates.

Figure 7 shows the Cheng equation graphically under the null of equity.

31

 

 

200000

2 100000 « ./

 

 

 

0 200000 400000

S

 

 

 

Figure 7 - The Cheng Model (under the null)

The test hypotheses are verbally the same here as they were for the
Paglin and Fogarty model. The first test hypothesis is that the property tax
structure is regressive. In this case, taxes are regressive if the power of S, 8., is
less than one. In the transformed model, the power becomes the coefficient of

the log of S and if the taxes are regressive it will be less than one. H1: B1< 1.

The other alternative is progressivity. l-l.: 81> 1. There is no test hypothesis
about the intercept term and it will not be interpreted because to interpret the
intercept is to interpret beyond the data set.
Results

The estimated Cheng equation is displayed in Table 2 and it is shown
graphically in Figure 8. As can be seen from Figure 8, the data fit the estimated
Cheng model better than they do the estimated Paglin and Fogarty model shown
in Figure 6a. The t — test shows the estimated (31 is significantly less than one
and the null hypothesis is rejected at the 99% confidence level. The results are

consistent with the first test hypothesis of a regressive property tax structure.

32

The estimated model can be transformed to solve for AV by taking the antilog of
both sides of the equation:
Estimated AV = 1.586 3“”

As an example, to see how the estimated Cheng model shows a
regressive vertically inequitable property tax structure, let 8 equal $50,000 and
$100,000 respectively. When S equals $50,000, AV is expected to be
$22,604.38 and therefore the expected AR is 0.452. When S equals $100,000,
AV is expected to be $41716.05 and therefore the expected AR is 0.417. As can
be seen, as S increases, AR decreases. Clearly, the estimated Cheng model
reveals a regressive vertical inequity. Table 2 and Figure 8 show the estimated
Cheng model.

Table 2 - Estimated Cheng Model

 

 

 

InAV = II'IBQ + B1|nS
Estimated = 0.461 + 0.884InS

InAV

s.e. = 0.048 0.004
t-stat for = 24.042 -26.304
hypothesis

test
R Squared = 0.833

 

33

 

14

 

 

 

 

 

e 10 1'2 14

 

 

LNS

 

Figure 8 - Estimated Cheng Model
The International Association of Assessing Officers Model

The third model is endorsed by the International Association of Assessing
Officers (I.A.A.O). Its dependent variable is the assessment ratio instead of the
assessed value. It too is estimated by the Ordinary Least Squares Method. The
equation is as follows:
AR = I30 + B. S + 5
Under the null of no regressivity, the assessment ratio (AR) should be 0.5 for all
properties. Thus, under the null, the slope of the line, (the coefficient of S,) is
zero. Ho: 61:0. In addition, if the property taxes are assessed as they should be,
the intercept is 0.50 H..: 60:0.5. Figure 9 shows the I.A.A.O model graphically

under the null of equity.

34

 

1.0

 

 

AR

 

 

0.0

 

 

 

 

Figure 9 - The I.A.A.O. model (under the null)

Again, in this model the test hypotheses are, first, that the property tax
structure is regressive: H1: (3. < 0, and second, that the property tax structure is
progressive, H1: 8. > 0.

Resufls

The estimated I.A.A.O. equation is shown in Table 3. It is graphically
depicted in Figure 10. The t-test shows the intercept is significantly more than
0.5 and the slope is significantly different from zero. This means that the lowest
valued properties are being assessed above the legally mandated proportion of
market value and as value increase assessment ratio decreases. However, as
can be seen from Figure 10, the intercept is outside the data range and therefore
not consequential. Regardless, the null hypothesis of equity is rejected at the
99% confidence level. The results are consistent with the first test hypothesis of
regressivity. Note that even though the null hypothesis is rejected, the low R

squared and Figure 10 show that the model does not fit the data well.

35

Table 3 - Estimated l.A.A.O Model

 

 

lXF‘ :: I30 4- DRE;
estimated AR = 0.511 - -8.75E-07 S
s.e. = 0.002 3.71 E-08
t-stat for = 5.5 -23.607
hypothesis test
R Squared = 0.254

 

 

 

 

10

AR

 

 

 

 

0 200000 400000

S

 

 

 

Figure 10 - Estimated I.A.A.O model
Clapp Two - Stage Least Squares Simultaneous Equations Model

The first three models used so far, have been criticized, most notably by
Kochin and Parks, to be biased toward finding regressivity. The fourth model
was created with the goal of eliminating the bias of the previous models. It is the
Clapp Two-Stage Least Squares Simultaneous Equations Model. lt
acknowledges that sale amount is not a perfect predictor of market value and
assumes that assessed value and market value are interdependent. Included in
the model is an instrumental variable, 2, that is highly correlated to both market

value and assessed value. 2, a binary variable, is equal to one if the observation

36

is in both the top third of all sale values and in the top third of all assessed
values. 2 is set equal to negative one if the observation is in both the bottom
third of all sales values and in the bottom third of all assessed values. 2 is set
equal to zero otherwise. Unlike an ordinary regression model that may be
estimated using ordinary least squares, this model does not assume a one-way
causal relationship. In this model, assessed value is both an independent and a
dependent variable. Because AV is mutually dependent, not independent, it is
an endogenous variable.

The model is a set of two simultaneous equations. It must be estimated
using the Two-Stage-Least-Squares method. In this method, the parameters of
one of the equations are estimated taking into account the information given in
both equations. Ordinary Least Squares (OLS) can not be used in this case
because not all of the information known about AV would be taken into account.
For this reason, the coefficients, if estimated using OLS, would be inconsistent or
biased even in large samples. OLS coefficients would not converge to their true
population values and would be inefficient resulting in very large variances
(Gujarati 1995, part 4). This model was developed in 1990.

LnS =Bo+B1InAV+81
LnAV=0ro+or1Z+82

Under the null hypothesis of no inequity, B. is statistically equal to one,
Ho: 6. = 1. The test hypotheses are as follows: 1)The property tax structure is
regressive, H1: 6. > 1, and 2) The property tax structure is progressive,

”12 B1 <1.

37

Results

The estimated coefficients of the Clapp equation are in Table 4. T-test
results lead to a rejection of the null hypothesis at the 99% confidence level.
Results are consistent with the first test hypothesis, a regressive property tax
structure.

Table 4 - Estimated Clapp 2SLS Simultaneous Equation Model

 

 

 

 

 

LnS = Bo + B.lnAV
Estimated = 0.556039 + 1.023555 InAV
InS
s.e. = 0.062991 0.006252
t-stat for = 8.827 3.77
hypothesis
test

 

 

 

 

 

 

R Squared = 0.76892
Controlling for Year

The above models were estimated with the entire sample of property
transactions that occurred over five different years. Because properties are
given a new assessment every year, it seems logical to question whether the
amount of inequity varies from year to year. To test this question, the year of the
sale and assessment must be controlled. To this end, dummy variables
representing years, are inserted in the Cheng Model. The following dummy
variables were created from the data:
D1994 equals one if the observation was collected in 1994, zero otherwise.
D1995 equals one if the observation was collected in 1995, zero otherwise.
D1996 equals one if the observation was collected in 1996, zero othenrvise.
D1997 is set equal to one if the observation was collected in 1997, zero

othenrvise.

38

Note that there is not a dummy variable for every year as that would cause
perfect colinearity making Ordinary Least Squares estimation impossible. As can
be seen, there is a dummy variable for 1994, 1995, 1996 and 1997 but there is
no D1998 for 1998. Nevertheless, observations from 1998 are accounted for. If
an observation occurred in 1998, all other dummies are set to zero and it can be
assumed that the remaining model coefficients are valid for 1998. The dummy
variables are entered into the Cheng model as intercept dummies. They are also
used in multiplicative dummies or slope dummies by multiplying each by S.

The Cheng model with the intercept and slope dummies is stated as,

InAV = 8.. + 8. ms + 62D1997 + 83 lnSD1997 + B4D1996 + 65 InSD1996 +
[3.01995 + 87 lnSD1995 + 3091994 + 69 lnSD1994 + e.
The estimated Cheng model with dummies is as follows:

InAV = .113 + .913 ms + .289 D1997 + .028 lnSD1997 + .370 D1996 -.031
InSD1996 + .239 D1995 - .018 lnSD1995 + .287 D1994 - .017 InSD1994
Assuming all coefficients are significantly different from zero, the year specific
relationship between InAV and ms can be calculated by adding the coefficient on

ms, 6., to the specific year’s slope dummy. To calculate the year specific
intercept, the intercept, Bo, is added to the specific year's intercept dummy. For

example, in 1997, all variables except the intercept, Ins and the 1997 dummies
are set equal to zero and the estimated relationship between InAV and ms is

InAV = (I30 + 82D1997) + (6. ms + I33 lnSD1997). The year specific Cheng model

results are in Table 5.

39

Table 5 — Year Specific Cheng Model Results

 

 

 

 

 

 

 

 

 

 

 

 

LRAV1994 = 0.397 + 0.896 “151994
LRAV1995 = 0.352 + 0.895 “131995
LnAV1996 = 0.483 + 0.882 lnS1996
LnAV1997 = 0.402 + 0.885 InS1997
LDAV1993 = 0.113 + 0.913 IDS1993
Resuns

Results show that equity does vary from year to year. 1996 has the
smallest slope indicating the most regressive inequity. 1998 has the slope closet
to one and the least inequity.

Conclusion

All four models, Paglin and Fogarty, Cheng, I.A.A.O., and Clapp, show
vertical inequity that is statistically significant. The results in each case lead to a
rejection of the null of equity. All results are consistent with a regressive property
tax structure. Each test assumes a different functional relationship between AV
and S. Some of these functional forms fit the data better than others. As can be
seen by Figures 2.4 through 2.6, the estimated Cheng model (Figure 2.5) fits the
data quite well. The l.A.A.O model (Figure 2.6) on the other hand, clearly doesn’t
fit the data well at all, and the residuals have heteroskedasticity problems.

When dummy variables are included in the Cheng model to control for
year variations in equity, it is clear that although the level of equity changes
slightly from year to year, inequity is pervasive throughout. The yearly variations
in assessment equity will be further examined in the next chapter. This chapter's
results have great significance, demonstrating inequitable and undesirable non-

uniform assessments across properties. In Lansing, MI, higher priced properties

40

pay property taxes at a lower rate than lower priced properties. The finding that
property taxes are regressive with respect to sale values justifies the following
examination of the nature and spatial distribution of the assessment ratio. This
examination could reveal more information about the groups of people that are

most negatively affected by the regressive nature of the property tax structure.

41

Chapter 4

SPATIAL AUTOCORRELATION AND SURFACES OF INEQUITY

It has now been asserted that regressive vertical inequity is present in the
property tax structure of Lansing, Michigan. The test results of the previous
chapter show that higher valued properties are taxed at a smaller proportion of
market value than lower valued properties and than is legally permitted.
Because cities are made up of neighborhoods of like characteristics of which
housing value is one, it seems likely that the vertical inequity previously detected
only econometrically, manifests itself spatially as well. In addition, as was
discussed in Chapter 1, the assessor’s office divides Lansing into neighborhood
units to deal with assessments and assessment adjustments. If the average
assessment ratio of an assessor-designated neighborhood is too low or two high
the whole neighborhood is adjusted up or down accordingly. Because of this, it
is likely that residual or boundary effects of this process can be seen in the
spatial patterns of assessment. Are some areas of the city taxed more than
other areas? The subsequent chapters deal with this question of spatial inequity.

There has not been very much literature on the description of spatial
patterns in property taxation. Perhaps this is due to the absence of geographers
participating in the body of literature surrounding social equity and the property
tax. Most researchers in this field have come from real estate disciplines and
econometrics. Many of the geographers who have studied spatial inequity have

done so statistically and econometrically using cross sectional data with little

42

spatial referencing. If maps were created, they were usually based on data that
had been aggregated to relatively large areas. They did not tend to examine
spatial patterns visually in attempt to create hypotheses with sophisticated
surface maps. This is likely because when most of the geography literature in
this subject was created, computer GIS was just being developed and was in an
inaccessible, time-consuming stage. There is one notable exception. Thrall, a
geographer who has worked on this topic, has looked at map patterns in two
studies. The first was in done in 1978 (Thrall 1978). He had to manually locate
the addresses of the sale transactions on a map and then write code to create a
crude trend surface map of the assessment ratio. In 1993, Thrall did a similar
study to see if “geographic equity” was present in the tax structure of St. Lucie
County (Thrall 1993). This time he had use of the best GIS equipment made for
personal computers. However, Thrall did not create surface maps or even
analyze spatial autocorrelation. Instead, he simply mapped the locations of the
over assessed properties (properties with assessment ratios at least one
standard deviation over the mean) and under assessed properties (properties
with assessment ratios at least one standard deviation under the mean) and then
compared their patterns visually (Thrall 1993).

The purpose of this chapter is to display and describe any spatial patterns
that may exist in the property tax structure of Lansing, Michigan. To accomplish
this, the data will be subjected to several procedures. First the property tax
structure will be formally modeled and tested for spatial inequity by using

semivariance analysis and Moran’s Itests respectively. Next, spatial patterns of

43

tax inequity will be displayed in kriged surface maps. The process of kriging is
explained in detail later in this chapter. Finally, those maps will be graphically
compared and tested for correlation to identify any dynamic patterns that occur in
the data. The contents of the chapter will be organized around the four research
questions in Table 6 and each will be addressed in a section of this chapter. The
methodologies used to test each of the research questions will be explained in
the beginnings of these chapter sections.

Table 6 - Chapter 4 Research Questions

 

 

 

 

 

 

 

 

 

Research Questions Null Hypotheses Test
Hypotheses
1 Is the assessment ratio (AR) There is no spatial AR is spatially
syatially autocorrelated? autocorrelation in AR. autocorrelated.
2 What do the spatial patterns in the There are concentric There are other
data look like? ringpatterns. patterns.
3 Do the patterns vary or shift from The patterns do not The patterns
year to year? change. are dynamic.
4 Is an assessment correction No assessment A correction
process apparent in the data? correction process is process is
apparent. apparent.

 

Research Question 1: Is the Assessment Ratio Spatially Autocorrelated?
This is the first research question partly because, if the null hypothesis is
accepted, there is really no need to discuss spatial inequity and the rest of the
chapter, and partly because the methodology used to answer this question is
also needed to answer the subsequent research questions. There will be two
methods used to address this question; semivariance analysis and Moran’s I.
However, it is first important to establish just what the first research question is

asking.

44

 

As was defined in previous chapters, the assessment ratio, AR, for a
particular parcel, is the proportion of market value at which that parcel is
assessed. All assessment ratios are supposed to be 0.50. Therefore, there
shouldn’t be any pattern to the assessment ratio surface. However, since vertical
regressive inequity was detected in Chapter 3, it is already known that the
assessment ratio does not behave as it theoretically should and it is likely that
the vertical regressive inequity manifests itself spatially as well. If the vertical
regressive inequity does have spatial implications, then the variable AR will have
a spatial pattern and its high and low values will be spatially autocorrelated.
Spatial autocorrelation is the dependence of a value of a variable at any point on
the values of the variable at near by points. Spatial autocorrelation is present,
“when similar values cluster together on a map” (Odland 1988, p. 7). Spatial
autocorrelation will result in some areas in the city averaging lower ARs and
some areas averaging higher ARs. Spatial autocorrelation may be positive or
negative. If positive spatial autocorrelation exists between two points, then their
values will be similar. If negative spatial autocorrelation exists between two
points, their values will be dissimilar. Both kinds of spatial autocorrelation will
result in distinct spatial patterns. If there is no spatial inequity, then AR will not
be spatially autocorrelated and there will be no spatial pattern to the assessment
ratio. The first step taken in testing for spatial patterns is a graphical approach
that models any spatial autocorrelation that exists in the variable AR. The
method employed in this paper to model the spatial autocorrelation of AR is

semivariance analysis.

45

Semivariance Analysis
One way to examine spatial patterns and spatial autocorrelation is by
semivariance analysis. Semivariance is mathematically defined as:
Y I") = [N(h)/2] * InZIZI - 21441]2
Where:
7 (h) is the average semivariance of a pair separated by a lag distance of h
h = the lag distance
N(h) = the number of pairs separated by h
z. = the value of the variable (in this study 2 is AR) at location i.
2....1 = the value of the variable at the location a lag away from i.
Essentially, semivariance is the average difference between the values of
a pair of points that are separated by a given distance or lag. So, in this study,

y(h) is the difference in the values of AR for a pair of locations that are separated

by a lag of specified distance.

If positive spatial autocorrelation exists, the difference between values of
pairs that are closer together will be smaller than the difference between values
of pairs that are farther apart. Put differently, under classic spatial
autocorrelation circumstances, the semivariance will be less at smaller lag values
and increase until there is no spatial autocorrelation (dependence) between pairs
of points or until negative spatial autocorrelation is present.

The software GS-r- will calculate the average semivariance for all pairs (of
points) in a lag class for all lags up to a specified active lag distance. The active

lag distance is specified by the researcher and is the longest distance by which a

46

pair will be separated in the analysis. Its greatest value may be as much as the
distance between the pair of points that are the farthest away from each other in
the study area, or it may be any distance shorter. Often, a researcher will not be
interested in the difference between values of pairs that are extremely far apart
(spatially) because it is hypothesized that they will have little effect on each other
(pairs are not spatially autocorrelated after a certain distance).

In addition, GS+ lets the researcher specify the lag distances. The lag
distances may be at equal intervals or uneven intervals. That is, the active lag
distance may be divided evenly into lags, or the lags may vary in distance.
Specifying the appropriate lag interval — or the number of lags, is something of an
art form because spatial structure may become apparent at different lags for
different kinds of data. Usually the researcher undergoes a trial and error
process in choosing a lag that best represents the spatial dependence of the
data. At the beginning of the process it is beneficial to study the dependence of
the variable in question at the maximum active lag distance and with very short
lag intervals. Therefore, in this thesis, semivariance analysis will first be
conducted at the maximum active lag distance which is the length of the
distances between the pairs of points that are the farthest apart. The maximum
active lag distance for the Lansing study area is approximately 13,600 map
units,‘which is about 13.6 kilometers, or 8.5 miles. The actual maximum active
lag distance varies slightly between years of data because the locations of

sample points vary from year to year. In addition, the semivariance analysis will

47

first be done with very small lag classes. The active lag distance will be broken
into approximately 136 lags creating lag classes of 100 map units or 110 yards.
This will show information at the neighborhood scale.

A semivariance analysis is done for each year of data separately and the
results are found in tabular form in Appendices A through E. The data are
examined year by year because of the dynamic nature of the research questions
listed in Table 6.

The results in the appendices are shown in four columns. The first column
identifies the lag class. As was stated above, the lag classes are 100 map units,
and since the maximum lag distance is about 13,600 map units, there are
approximately 136 lags in the tables. The next column is “average distance,”
which is the average length of separation of points in the lag. For example, in the
first lag not all of the pairs are separated by exactly 100 map units. Instead, each
pair is separated by some distance up to 100 map units. The “average distance”
is the average of the distances between pairs in the first lag class. For example,
in Appendix A (1994 data), the average distance in the first lag is 56.47 map
units (approximately 62 yards). The third column in the tables is the “average
semivariance,” or the difference between the assessment ratios in each pair, and
the last column indicates the number of pairs in each lag class.

Semivariance Results
Recall from Table 6, that the null hypothesis of the first research question

is no spatial autocorrelation, and the test or alternative hypothesis is that spatial

 

' The map units are in UTM, Universal Transverse Mercator meters, units. The latitude and
longitude units derived from geocoding in Transcad were reprojected to UTM coordinates in a

48

autocorrelation does exist. The results of this first semivariance analysis shown
in the tables in Appendices A through E reveal a pattern of autocorrelation, which
is consistent with the test hypothesis. The first few lag classes have smaller
average semivariances indicating that values closest together are more similar.
The semivariances increase with lags until they begin to decrease again at about
50 lags. Actually, the values in the tables reveal that pairs of points that are
about 1000 map units apart are the most autocorrelated. Their semivariances
are even smaller than the pairs that are the closest together in the first couple of
lags. The significance of these patterns will be further examined and tested in
the next section, Moran’s lTests. Semivariance analysis will also be revisited
later in this chapter in a section called Semivariograms.

The results of this first semivariance analysis indicate that within small
areas or neighborhoods the assessment ratios are similar. In addition, these
results suggest that neighborhoods separated by great distances also have
similar assessment ratios. This analysis gives reason to reject the null
hypothesis of the first research question and conclude that the semivariance
analysis is so far consistent with spatial autocorrelation in the assessment ratio.
However, a more formal test for spatial autocorrelation, Moran’s I, has also been
performed and will be described in the next section to confirm the hypothesis test
for the first research question.

Moran ’s lTests
Moran’s I is an autocorrelation statistic that is constructed to test for

spatial autocorrelation. It is a “product moment” statistic that is similar in nature

 

coordinate conversion program called Corpscon For Windows.

49

and purpose to the Pearson’s Correlation statistic. Moran’s lvaries from
negative one to positive one. Positive values of Moran’s I represent the
presence of positive spatial autocorrelation. A variable is said to have positive
spatial autocorrelation when similar values are clumped together. If Moran’s l is
zero, no spatial autocorrelation exists and the variable exihibits randomness. It
follows logically then, that negative values of Moran’s I represent the presence of
negative spatial autocorrelation. Negative spatial autocorrelation exists when
disimilar values are consistently located in close proximity to one another.

The process of calculating Moran’s [involves applying a spatial weighting
function to the map of values of the variable in question. According to Odland, a
spatial weighting function is a set of rules for assigning values to pairs of places
in a way that represents their arrangement in space (Odland 1988, p. 9). When
the function is applied to the map, a set of weights in matrix form can be
calculated. The weights are the relative locations of the places on the map.
Generally, if places are closer together they have more effect on each other’s
variable values than places that are further apart. Weights can be simply binary
and are in this thesis. For instance, if an adjacency rule were applied, the weight
for a pair of locations that are adjacent to each other would be one while the
weight for a pair of locations that are not adjacent to each other would be zero.
In this paper, the weights will be binary based on distance lags. So if a pair is
separated by the lag for which the statistic is being calculated, the weight is one.
Otherwise, the weight of the pair will be zero.

The Moran’s I test statistic is mathematically defined as follows:

50

r = anEw.. * [22w.,(z. — z)(z. - z) / ."2(z.-z)2] (Odland 1988, p. 10)

Z. = the value of the variable in question (AR for the purposes of this study) at
location i. The double summation indicates summation over all pairs and lag
distances.

The null hypothesis of a Moran’s [test is always that no spatial
autocorrelation exists and that the variable in question is distributed either
normally or randomly. In this thesis, it is assumed that the assessment ratio
under the null is distributed normally. This is because it is known that the
assessment ratio should have a mean of 0.50 and any deviations from the mean
should not result in spatial patterns. Therefore, it is assumed that under the null,
there is no spatial autocorrelation in the variable assessment ratio at any lag, and
it is distributed throughout the study area in a normal way.

The value of Moran’s I under the null approaches zero for large samples
and is asymptotically normal. Spatial autocorrelation can affect pairs of locations
at different lags or distances apart. Because of this, a separate Moran’s!
statistic is calculated for every lag that is tested for spatial autocorrelation. For
example, if it is desirable to test for spatial autocorrelation at three lags: between
locations that are less than one mile apart, less than two miles apart and less
than three miles apart, then Moran’s I must be calculated and tested three times.
The program, 68+, will be used to calculate the Moran’s Istatistic. GS+
calculates the statistic for every lag of specified length in the active lag distance
described above in the section on semivariance analysis. The lags used in this

test will be 100 map units and the active lag distance will be set to the maximum

51

possible for the study area, approximately 13,600 map units. The calculated
Moran’s I statistics are listed in Appendices F through J.

The null hypotheses of no spatial autocorrelation are tested using a 2 test
of significance. A separate significance test is done for each lag. The sample
sizes are relatively large in this analysis and are equal to the number of pairs in
each lag class. Because of this and the asymptotically normal characteristic of
Moran’s I, significance will be decided by comparing a 2 statistic (1.645) to z
scores. Many steps are taken to calculate the z scores. First, the expected
Moran’s Iunder the null is calculated using the following formula:

E(I) = -1I(n-1)

Where: E(l) is the expected Moran’s I under the null and approaches zero when
the sample size gets very large. Then, the expected Moran ’s lstandard
deviation is calculated. It approaches one when the sample size gets very large.
The standard deviation is calculated as follows:

so = soar ((n2-n+9)l(n2-1))

Next, the standard error for the calculated Moran’s l is estimated as follows:
s.e. = SDISQRT(n)

Finally, 2 scores are calculated for each lag as follows:

2 score = (I - E(l))ls.e.

The null hypotheses were rejected at the 95% confidence level if the absolute

values of the z scores were greater than 1.645.

52

 

Moran’s I Results

The hypothesis test results, standard error, 2 score and the expected
Moran’s I under the null for each lag of each year are listed in the Appendices F
through J along side the calculated Moran’s I for the lag class and the number of
pairs in the lag class.

Table 7- Null Hypothesis Rejections by Year

 

Year Number of Lags Lags with Lags with

with Spatial Positive Spatial Negative Spatial
Autocorrelation Autocorrelation Autocorrelation

(Rejections of (number of lags (number of lags
null hypothesis) with significant with significant

 

 

 

 

 

 

 

 

 

negative ls) positive Is)
1994 220f135 10 12
1995 14 of 135 9 5
1996 51 of 134 29 22
1997 3001135 13 17
1998 15 of 130 7 8

 

It is interesting to note that at the smallest lags there are many instances
of positive spatial autocorrelation. This indicates that within small areas or within
neighborhoods, the assessment ratios are similar. This is expected, because it is
known that assessors adjust the ratios of entire neighborhoods up or down at
once. It is also interesting to note that there are almost as many negative
spatially autocorrelated lags as there are positive. This too, is not unexpected.
The fact that both kinds of spatial autocorrelation exist simply means that some
areas in the city are alike and some are different. If all of the lags had positive
spatial autocorrelation, the assessment ratio wouldn’t differ from neighborhood to

neighborhood or from area to area.

53

The results of the Moran’s I analysis leads to a formal rejection of the null
hypothesis of the first research question. The results are consistent with the
existence of spatial autocorrelation in the assessment ratio.

Research Question 2: What do the spatial patterns in the data look like?

It is apparent from rejecting the first research question hypothesis that the
data are spatially autocorrelated and that there are spatial patterns in the data.
To address research question two, the spatial patterns shall be viewed in surface
maps. The process used in this thesis to create the surfaces is kriging, which is
described in detail later on in this chapter. In order to create the kriged surfaces
however, the spatial autocorrelation must be modeled and described with
parameters. The semivariance analysis that has been done will be used to
create semivariograms that are in turn used in the kriging process to create
surface maps.

Semivariograms

A semivariogram is a graph of the autocorrelation present in the data
structure; that is, a semivariogram shows the semivariance of the data. The
software package, 68+, is used as a research tool in this paper to construct
semivariograms. A semivariogram plots the difference between values (in this
case the difference between AR in a pair of points) versus the distance between
points (pairs of values.) Typically the difference between values increases with
distance. This is as expected according to distance decay.

However, the semivariance analysis described above reveals that the

difference between values of AR does increase with distance until they begin to

54

decrease again. Semivariograms have been constructed from the data in

Appendices A through E and they appear in Figures 11-15.

 

Isotropic Varlogram

0.009827 I 1:1

  

 

 

 

0.007370 ‘-

o

.3 h I. D

._ 0 004913« .3

5 Q]

to q [E
0.002457 1

. El
0.000000 .L : 42 4 A. : : : : : ‘r :
0.00 3391 .85 6783.70 1 01 75 .54 1 3567.39

Separation Distance

 

 

 

Figure 11 -1994 Semivariogram with Maximum Active Lag Distance

55

 

 

Isotropic Variogram

   

 

 

 

 

0.013333 Cl
El
0.010000 E
° El
i ' ' :9
E 0.006667 1 ‘ " U
0.003333: D
0.000000 1 1 : : + f 3 e : : : s
0.00 3375.63 6751.66 10127.49 13503.32

Separation Distance

 

 

Figure 12 -1995 Semivariogram with Maximum Active Lag Distance

 

Isotropic Variogram

0.009809

0.004905

Semivariance

0.002452

 

J

0.000000 - 4 fi‘ 4‘ 1 : r : . : 4- L 4
0.00 3394 .09 6788.18 101 82.28 1 3578.37

Separation Distance

 

 

 

Figure 13 -1996 Semivariogram with Maximum Active Lag Distance

56

 

Isotropic Variogram

0.009809

0.007357

Semivariance

0.002452

 

L J A
r

1 01 82 28 1 3578 .37

8788.1 8

Separation Distance

0.000000 - t t t t
0 .00 3394.09

 

 

 

Figure 14 —1997 Semivariogram with Maximum Active Lag Distance

 

 

 

Isotropic Variogram
0.01 3221
0.00991 6
0
8
a
3 0.006611
5
to
0.003305
El
0000000 . t 1 Jr 1 1 *‘fi 1 1 1 I i 1
0.00 3269 .70 6539 .40 9809.1 1 1 3078.81
Separation Distance

 

 

 

Figure 15 —1998 Semivariogram with Maximum Active Lag Distance

57

Figure 11 is a graph of the data in Appendix A. The semivariogram clearly
shows that pairs closest together, in the very first lags have more similar values
than those do in the middle lags, but that those pairs farthest apart are the most
alike in terms of AR value.

After the semivariogram is plotted, it is modeled by a known distribution
such as a gausian, spherical, linear or exponential. Obviously, none of these
known distributions resemble any of the maximum active lag distance
semivariograms seen plotted in Figures 11 through 15. Because of this, a
smaller active lag distance is chosen. For example, if the semivariogram only
displays semivariances for pairs separated by up to one quarter of the total
extent of the map unit distance, or about 43,000 map units, the known
distributions will fit the pattern better.

Therefore, the active lag distance is truncated before the semivariances
decrease, and new semivariograms showing just a portion of the maximum
active lag distance are created. The known distributions are then fit to the
semivariogram plots, and the errors are calculated between the actual plot and
the known distributions. The distribution with the lowest mean square error is
then chosen to model the semivariogram plot. Figures 17 through 21, are the
truncated semivariograms with the best-fit distributions modeling the
semivariances.

Three parameters, the nugget, sill, and range are used to summarize the
modeled semivariogram plot. The nugget is the y-intercept of the model. The sill

is the y-value or semivariance value at which the model levels off and the range

58

is the distance over which spatial dependence is apparent in the modeled
semivariogram. Figure 16 is an idealized semivariogram with nugget, sill and

range shown.

 

Range

C!

i

 

   
  

 

Nugget (y - intercept)

 

 

Figure 16 - ldealized Semivariogram with Parameters

Figures 17 through 21 are the truncated semivariograms with active lag
distance of approximately 4,300 map units or about one third of the maximum
active lag distance. The most appropriate lag intervals for these semivariograms
are found by iteratively examining how well semivariograms fit the established
models. The following semivariograms have larger lags, like 500 map units
instead of only 100 map units. This allows the exponential distribution, the best-
fit model for these semivariograms, to better model the spatial dependence of the

assessment ratio.

59

 

 

 

Isotropic Variogram
0.006951 ["1 r1 r1 ['1 D n
H U L] H L]
0.00521 4
. 0.003476
6
(0
0.001 738
0000000 . : : : .L : : 4 4. : 4 : :
0 .00 1 000.00 2000.00 3000.00 4000.00
Separation Distance

 

 

 

Figure 17 —1994 Truncated Semivariogram with Exponential Model

 

 

  
 

 

 

isotropic Variogram
0.008369 0 n E. .... n n El
0 '-' u u ._. _
0.006277 4’
O 0
8
.9
.g 0.004184
g .
m 4
0.002092 -
-r
0000000 1 3 i 2 t i i i i i 4 i
0.00 1050.00 21 00.00 3150.00 4200.00
Separation Distance

 

 

 

Figure 18 -1995 Truncated Semivariogram with Exponential Model

60

 

Semivariance

0.0081 24 '-
0.008093 "

0.004082 -

0.002031 ‘-

 

0.000000
0 .00

Isotropic Variogram

A A
7

550.00

1100.00

DUDE]

A A
1'

1 850.00

DUE!

2200.00

Separation Distance

 

 

 

Figure 19 - 1996 Truncated Semivariogram with Exponential Model

 

 

 

Isotropic Variogram
0.007728 [3 El 1:] n
D U U U "'
0.005796
0
3
a
.g 0.003664
6
(0
0.001932
0.000000 . : : 4 4 c 4. : : : % : :
0.00 11 25.00 2250.00 3375.00 4500.00
Separation Distance

 

 

 

Figure 20 -1997 Truncated Semivariogram with Exponential Model

61

 

Isotropic Variograrn

0.00871 7

0.008537

0.004358

Sern ivariance

0 .0021 79

 

L 4L 1 L

I
1*

0.000000 . i i i
0.00 1 075.00

‘1

21 50 .00 3225.00 4300.00

Separation Distance

 

 

 

Figure 21 —1998 Truncated Semivariogram with Spherical Model

Now, the known models fit the data much better, and parameters (nugget,
sill, and range) describing the model will more accurately describe the data. The
nugget is how much difference between AR is observed at zero distance
between points. The sill is how much difference between AFt is observed when
the semivariogram levels off and the range is the distance between pairs of
values when the difference between the values is at its highest - or when the
semivariogram levels off. At distances less than the range, there exists positive
spatial autocorrelation between points, and at distances greater than the range,
there is assumed no dependence between the values of points. Autocorrelation
is assumed to be the same between pairs of the same lag regardless of where

the pairs are located. These semivariograms are a necessary step in the

62

 

creation of the kriged surfaces that will be used to examine and describe the

spatial patterns in the assessment ratio.

The model parameters and fit of the semivariograms that will be entered

into the kriging process are listed in Table 8.

Table 8 - Semivariogram Results

 

 

 

 

 

 

 

 

 

 

 

 

 

Semivariogram Model Type Nugget Sill Range R RSS
Squared
1994 Exponential 0.0022 0.0068 700 map 0.92 1.01E-07
units
1995 Exponential 0.0026 0.0082 441 map 0.73 1.26E-07
units
1996 Exponential 0.0024 0.0077 111 map 0.41 1.78E-06
units
1997 Exponential 0.0023 0.0074 606 map 0.7 3.09E-03
units
1998 Spherical 0.0019 0.0081 326 map 0.49 2.01E-06
units
Kriged Surfaces

Kriged surfaces are constructed to describe the spatial assessment ratio

distribution. Kriged surfaces are superior to other surfaces such as trend

surfaces because they can measure the error associated with each estimated

value on the surface. The most likely value at each point is known as well as

how likely that value is to occur there. Kriging is a distance weighting method

that calculates estimated values of the variable - in this case the assessment

ratio - at all points in the study area, sampled and unsampled. The parameters

for the method are determined by interpreting the semivariograms shown in

Figures 17 through 21.

63

 

The parameters, nugget, sill, and range, from the semivariograms derived
in the previous section and listed in Table 8, are entered into the kriging model to
estimate the weights that should be applied to each point to estimate the value of
the assessment ratio at any unknown location. So kriging calculates the
weighted average of neighboring known values and assigns it as the value at any
given point in the study area. GS+ will once again be employed, this time to
create the kriged surfaces. The kriged surfaces are then brought into ARC/INFO
and converted to raster grids.

The raster grids are helpful for two reasons. First, in raster format, the
patterns in the maps are clear and discernable, and second, the patterns can be
compared more readily from year to year. These raster grids of the kriged
surfaces are shown in Figures 22 through 26. In all of the maps, darker colors
represent higher assessment ratios, and lighter colors represent lower
assessment ratios.

Expected Map Patterns

Recall from Table 7 that the null hypothesis of the second research
question, “What do the spatial patterns in the data look like?” is that concentric
patterns will appear radially out from the center of the city. This is an expected
pattern that comes from the literature. It is widely acknowledged that the oldest
parts of cities tend to be over assessed due to the depreciation of the housing
stock and the amount of rented property. It is also accepted that lower
assessment ratios tend to congregate in the newer parts of the city near the

edges of the city or closer to the suburbs. One reason is because assessments

64

usually err toward the mean, and newly developed properties that tend to be
worth more are therefore underassessed (Paglin and Fogarty 1972). The actual
patterns in the data will be described in the next section. Figures 22 through 26

show the raster grids of the kriged surfaces for each year from 1994 to 1998.

65

.03

. a...

 

Data

Ion

Figure 22 - Grid of Kriged Surface of AR from 1994 Sale transact

66

 

Figure 23 - Grid of Kriged AR surface from 1995 Sale transaction Data

67

n5;

a . fiem?

 

Figure 24 - Grid of Kriged AR surface from 1996 Sale transaction Data

68

.63“

. a

«Ema»

..
aw.

a

 

Figure 25 - Grid of Kriged AR surface from 1997 Sale transaction data

69

 

Figure 26 - Grid of Kriged AR surface from 1998 data

70

Description of Map Patterns

ln the1994 map, a pattern of concentricity does appear with darker colors
representing over assessed areas in the center of the city and lighter colors
representing under assessed areas near the periphery of the city. However, the
other maps for 1995, 1996, 1997 and 1998 do not show a pattern of concentricity
clearly. The null hypothesis in this case, that patterns of concentricity are
present, may be rejected for 1995, 1996, 1997 and 1998. Even though the 1994
map is consistent with the null hypothesis, it is clear that for all of the maps, other
patterns of over and under assessment are dominant. Chapter five will attempt
to statistically explain the patterns in figures 22 through 26. Why are some areas
over assessed and some underassessed? What characteristics determine
assesment ratio? These questions will be addressed, but more important is the
question of causation. With out identifying possible causes, it is impossible to
remedy inequity. One possible cause lies in the assessment correction process.

Recall that assessors do a yearly evaluation of their work by averaging
assessment ratios in each of their designated neighborhoods. If a
neighborhood’s average AR is too high it is adjusted down by applying a constant
multiplier to each assessment ratio. If an average AR is too low, it is adjusted up
by applying a constant multiplier to each assessment ratio. Therefore, each
parcel within a neighborhood is treated the same way as all the others in the
neighborhood regardless of its actual individual value. Thus, if the average AR in
a neighborhood happens to be too low, even the assessments that are too high

in that neighborhood are adjusted still higher and if the average AR in a

71

neighborhood is too high, even the assessments that are too low are adjusted
still lower. It can be shown that this process would lead to a boundary effect, that
is, properties on the edges of neighborhoods would be most likely to have
extreme values of assessment ratios. This process may actually cause more
spatial inequity and indeed the patterns of over and underassessment seen in
the maps.

Research Question 3: Do the patterns vary or shift from year to year?

This next research question will be addressed by graphically comparing
each year’s raster grid to the previous year’s grid. The comparison is done in
ARC/INFO by subtracting the value of each cell in the later year by the
corresponding cell in the earlier year. The resulting grid maps are shown in
Figures 27 through 32. The null hypothesis for this research question is that the
patterns do not change. Under the null then, areas that are over and under
assessed in 1994 will remain so in ensuing years. The test hypothesis is that the
map patterns from Figures 27- 32 are dynamic — areas that are under and over
assessed do change from year to year.

Map Interpretation and Results

In these maps, the darkest colors are areas where the later year’s
assessment ratio was much higher than the earlier year, so darker areas
represent areas where the assessment ratio has increased. The lightest colors
are areas where the later year’s assessment ratio was much lower than the

earlier year, so the lighter areas represent places where the assessment ratio

72

has decreased. The medium gray colors represent places where the
assessment ratio has changed little from the earlier year to the later year.

So under the null hypothesis of no change in the patterns of over and
under assessed areas, the map patterns will be boring, mostly gray maps with
out areas of great change. Conversely, if the null hypothesis is rejected, the
map patterns will be distinct with areas of great change represented by dark and
light colors.

This research question will not be tested by means of statistics, only by a
subjective viewing of the maps. Because the maps do show distinct areas of
dark and light colors, the null hypothesis — that the patterns do not change — is

rejected, and it will be concluded that the assessment ratio is dynamic.

 

Figure 27 - 1995 Surface Subtracted by 1994 Surface

73

 

Figure 29 - 1997 Surface Subtracted by 1996 Surface

74

 

Figure 31 — 1998 Surface subtracted by 1994 Surface

75

 

Figure 32 — 1997 Surface subtracted by 1994 Surface
Research Question 4: Is an Assessment Correction Process Present?

The results of the previous section indicate that the patterns of over and
under assessment are not static but do change from year to year. This research
question asks whether or not these dynamics in the property tax structure are the
products of correction processes or other forces. There are two kinds of
correction processes that could be present. The first comes from the assessor’s
office. The assessment process does attempt to target neighborhoods that are
under or over assessed and adjust them with the goal of achieving more equity.
The other kind of correction could come from the market. In this case, the
assessments are assumed to be correct, and it is the sale prices that are
assumed to be too high or too low. If there were a market correction at work, the

sale prices would be adjusting back to true market values, thereby correcting

76

assessment ratios. This question will test to see if the dynamics result from an
assessment correction or a market correction.

To answer this question, a correlation statistic will be generated by
correlating the grids in Figures 22 through 26 in ARC/INFO. A separate
correlation statistic will be calculated for each year and its subsequent year. In
addition a correlation statistic will be calculated for 1994, the earliest year of the
data, and 1998, the latest year of the data. The null hypothesis is that no
assessment correction process is apparent. Under the null, spatial inequities are
getting worse or staying the same. Thus, under the null, the property tax
structure is not becoming more equitable. Under the null, the correlation statistic
r will be zero or positive. If the correlation statistic is zero there is no correlation
between the values of cells in the grids from year to year. If the correlation
statistic is positive, then cells with high values correspond to high values and
cells with low values correspond to low values indicating the spatial inequities are
worsening. The test hypothesis is that the property tax structure is becoming
more equitable. If this is the case, then the correlation statistics will be negative.
If the correlation statistic is negative, than cells with high values correspond to
low values and vice versa.

Correlation Results

The correlation statistics are listed in Table 9. Because the statistic is
calculated by means of a cell to cell comparison, the sample size of each
correlation is 3283, the number of cells the grids have in common. With such a

large sample size, the standard error of the statistic will be very low and the

77

statistics are likely to be significant. However, this test is biased because the
correlations (like the grids) are based on interpolated data, and there are only a
few thousand actual measured observations in each year’s data. Regardless,
five of the six correlations are positive and thus consistent with the null
hypothesis. However, the correlation between 1994 and 1998, the first and last
years of the analysis, is negative. Although this correlation is the smallest in
magnitude, it does indicate that perhaps inequities correct themselves over a lag
of a few years. Nonetheless, it is clear that over all, the results are consistent
with the null hypothesis — that there is no apparent corrective process in the data
and in fact, patterns of inequity are getting worse.

Table 9 - Correlation Statistics

 

 

 

 

 

 

 

 

 

 

 

 

 

1994 1995 1996 1997 1998
1994 0.24648 0.05492 -0.00383
1995 0.14745
1996 0.15218
1997 0.09061
Conclusion

The goal of this chapter was to see if the vertical regressive inequities
detected in Chapter 3 manifest themselves spatially, and if so, do patterns of
under and over assessment change from year to year and is this change the
result of an assessment correction? The results of the hypothesis testing are as
follows; the assessment ratio is spatially autocorrelated because over and under
assessments are clustered together. The spatial patterns are dynamic, changing
from year to year. However, there is apparently no correction of the inequity.

Unfortunately, it seems that inequity is either becoming more concentrated or at

78

best, it is not getting any less concentrated. The next chapter, Chapter 5, will
attempt to explain the spatial inequities and patterns in the assessment process
that have been shown in this chapter as well as identify other biases that may

exist in the property tax structure.

79

Chapter 5

EXPLANATION OF THE ASSESSMENT RATIO VARIATION

The purpose of this chapter is to attempt to explain the property tax
inequities identified in Chapters 3 and 4, as well as to identify further inequities
and possible biases in the property tax structure. Already known is that the
property tax structure is regressive with respect to sale amounts (sale amount, S,
is inversely linearly related to AR). Are there any other regressive biases of the
property tax structure? Why are some geographic areas over assessed and
some under assessed? Are there different forces influencing the assessment
process at different scales? These questions will be addressed in this chapter.
The hypotheses for this chapter, discussed in the next section, are addressed by
testing the significance of cross sectional, geographic variables in explaining the
assessment ratio. These hypothetical explanatory variables are derived from the
literature on this topic, which is reviewed in the following pages. First the
hypotheses will be tested with bivariate Pearson’s Correlation tests. However,
the primary method used to test hypotheses for explanatory power in this thesis
is multiple regression analysis. Throughout Chapter 5, the hypothetical variables
will be referred to by their short code names. A description of each can be found
in the codebook under “List of Abbreviations” on page xi.

Framing the Hypotheses
Property tax literature has long recognized a geographic discrepancy in

property tax assessment. For example, an article appearing in the National Tax

80

Journal in 1965 concludes, “the tax burden on real estate and personal property
in Iowa varies substantially depending on the location of the property being
taxed.” (Meyer 1965) Although Meyer was studying the variation in the effective
tax rate between counties and did not test determinants of assessment ratio
variation, he does speculate that income and the extent that a population uses
public services can affect the rate of taxation. Essentially, Meyer implies that the
tax structure is regressive with respect to economic group and biased against
those on public income. Meyer’s speculations will be represented and tested in
this chapter by the variables INC and PUBLIC.

In 1972, Black published an article that aimed to identify and test the
determinants of the variation in the assessment ratio (Black 1972). He used
regression analysis on data aggregated to census tracts. His dependent variable
was equivalent to the AR in this thesis, and his study area was the city of Boston.
The results of Black’s study reveal that systematic variation in the assessment
ratio is positively correlated to the percentage of properties with multiple housing
units, inversely related to the median family income of a census tract, and
positively related to percent nonwhite. That is, areas with more multiple unit
housing, lower median incomes and higher percentages of nonwhite populations
were taxed higher. Conversely, areas with more single family housing units,
higher median incomes, and higher percent whites were taxed lower. In
addition, areas with built environments of poorer quality were also found to be
taxed at a higher rate. Black does not offer explanation as to why the above

determinants explain the assessment ratio, but he suggests that increased

81

federal or centralized control over the assessment process would alleviate some
of the inequity. Some of Black’s results will be retested here at three scales with
the variables INC, WHITE and RENTOCC.

Black’s study prompted similar studies including a comprehensive
appraisal of the nature of regressive inequities in the property tax by Edelstein.
Although the main purpose of his paper is measurement of property tax
regressivity, Edelstein also empirically finds “regressivitiy of the property tax is
principally caused by its poor and de facto discriminatory administration”
(Edelstein 1979). Several possible explanations for the observed regressivity
are discussed in the article. First, Edelstein postulates that miss-assessment
could be the result of the user-benefits principle which theorizes that households
consuming more public goods and/or services should be taxed more to cover
their share of the local public expenditures.

Another explanation or cause of regressivity laid out in the Edelstein
article is pure discrimination. That is, the assessment process may discriminate
against marginalized households in order to “avoid confrontations” caused by
advantaged home-owners who are more likely to “protest assessment changes.”
Both of these possible causes assume that individual assessors are biased
against poor, minority and politically unpowerful groups. This explanation is
highly unlikely because individual assessors rarely readjust a single parcel’s
assessment and it would be difficult for them to ascertain what kind of person

owns any given parcel.

82

Finally, Edelstein describes a market phenomena explanation of tax
inequity. This theory is based on the differential market behavior of geographic
sub-areas: that there is an inertial quality to assessed values which cause them
to have a delayed reaction to market values. Thus, geographic sub-areas with
increasing market values usually have smaller assessment ratios, and
geographic subareas with decreasing market values usually have larger
assessment ratios (Edelstein 1979). This “market phenomena” will be directly
tested in this chapter with the variable SEASON.

Another article, written by Thrall and also published in 1979, dealt with
spatial inequities in tax assessment in Hamilton, Ontario. Thrall’s findings were
consistent with the prior research described above, but his methodological
design varied slightly. Instead of using assessment ratios as the dependent
variable, he borrows from Paglin and Fogarty’s vertical inequity test and
postulates that, “the quality of assessment depends upon the relative behavior of
AV with respect to MV” (Thrall 1979). He asserts that the ideal assessment
system should be characterized by2
AV = a + bMV.

Where a = 0.00 and b = 1.00 (although for Michigan this coefficient is 0.50).
Thrall then states that,
“T he manner in which the actual assessment system

deviates from the ideal may be identified by establishing how

the constant and the slope in the equation behave over a set

of characteristics: by determining what social, economic,

geographic, institutional, and other characteristics lead to
variation in the intercept a and slope b (Thrall 1979).”

 

2 Thrall uses this linear functional form but stipulates that any two—parameter (intercept and
slope) functional forms would be appropriate (Thrall 1979).

83

However, Thrall did not measure such “characteristics” directly. Instead,
he partitioned his study area of Hamilton, Ontario into 16 subareas. He uses
dummy variables to represent the 16 area partitions and inserts them into his
regression design. If any of the dummy variables have significant coefficients
that diverge from the ideal intercept and slope coefficients he concludes that the
assessment process is geographically biased. So, Thrall does not test any
demographic variables like racial or income variables, directly. However, he
does imply that he is testing social, economic, geographic, institutional and other
characteristics indirectly because he claims he divided his study area into
partitions with similar characteristics. The manner in which he did this is unclear
but he asserts that each area is relatively homogenous. He also partitioned the
sale prices into three categories: low, medium and high, and created dummy
variables to represent the categories inorder to test for vertical regressive
inequity.

In this thesis, geographic inequity will not be studied like Thrall does by
dividing the study area into dummy variables. Instead, all of the variables are
geographically referenced and in the event that they are significant, they
represent geographic bias with respect to the characteristic they stand for. In
doing so, the geographical bias discussed by Thrall is being implicitly studied.

Thrall found that indeed the assessment process in Hamilton, Ontario was
geographically biased. Some of the 16 areas were over taxed and some were

under taxed. He concluded that areas near the central business district and

industrial areas were over taxed and areas with new subdivisions were under
taxed. Thrall’s results were also consistent with regressive vertical inequity.
Properties with higher sale prices were found to be under taxed, and properties
with lower sale prices were found to be over taxed.
Correlation

Recall that the data was prepared for analysis at three scales: census
tract, block-group and parcel from largest to smallest scales respectively. This
was done so that hypotheses can be tested at three distinct scales. lf
relationships between scales are found to be the same regardless of scale,
perhaps the assessment process, which is scale specific, isn’t the source of the
bias, but rather market interactions are causing assessment discrepancies. In
this section, many of the hypotheses and determinants found in the literature will
be tested at all three scales with simple Pearson’s Correlations. The nature and
direction of the determinant relationships are hypothesized in Table 10. Please
refer to the “List of Abbreviations” on page xi where descriptions of each variable
used in Table 10 and throughout this thesis are located. All of the verbal
hypotheses found in Table 10 postulate that areas dominated by marginalized

populations are correlated with the highest assessment ratios.

85

Table 10 - Hypothesized Correlation with AR

 

 

 

 

 

 

 

 

 

 

 

 

 

Explanatory Variable Hypothesized Verbal Hypothesis
Direction

WHITE - As white percent increases, AR
decreases

BLACK + As black percent increases, AR
increases

HISPANIC + As Hispanic percent increases,
AR increases

INC - As median household income
increases, AR decreases

RENTOCC + As renter occupancy percent
increases, AR increases

PUBLIC + As public assistance percent
increases, AR increases

YEAR - As median year built increases,
AR decreases

SEASON (binary - AR decreases in the summer

variable: summer = 1,

winter = 0)

VACANT + As vacant percent increases,
AR increases

 

Correlation Results

Table 11 - Bivariate Pearson’s Correlation Results at Three Scales

 

 

 

 

 

 

 

 

 

 

 

Explanatory Census Tract Census Parcel
Variable Block-
Group

WHITE r = 0.382“ r = 0.341 ** r = .063“
BLACK r = -0.273* r = -0.252* r = -0.045**
HISPANIC r = 0.480" r = 0.380" R = 0.077"
INC r = 0.531“ r = -0.42** R = 0.087"
RENTOCC r = 0334* r = 0.292“ R = 0.057“
PUBLIC r = 0.555“ r = 0.421 ** R = 0.087"
YEAR r = -0.027 r = -0.033 r = 0.003
SEASON (binary not applicable not r = -0.033**
variable: summer applicable
= 1, winter = 0)
VACANT r = 0.425“ r = 0.400“ R = 0.067“

 

 

 

 

 

** r is significant at .01 level
* r is significant at .05 level

86

 

As can be seen in Table 11, the results are consistent with all of the
hypotheses except that with YEAR. There is no significant correlation between
the age of housing and the assessment ratio. All other correlations are
statistically significant at least at the 95 percent confidence level. The meaning
of the results is this; the regressive nature of the property tax structure in
Lansing, Michigan goes much farther than just regressive vertical inequity alone.
Higher AR values are correlated to areas with larger minority populations. The
tax structure is regressive and biased against black and Hispanic populations.
AR values are also positively correlated to areas that have larger poor
populations so the tax structure is regressive and biased against poor
populations. AR values are also positively correlated to politically unpowerful
groups like those renting property instead of owning property and those groups
on public assistance. Thus, the tax structure is regressive and biased against
renters and people on government assistance. The regressive nature of the
property tax appears to go much further than being biased against lower valued
properties alone.

OLS Modeling

Next, the significant variables, identified in the correlation table above, will
be used in linear regression models to explain variation in the assessment ratio.
In addition, a form of S, sale amount, will be used, because as the I .A.A.O. model
in Chapter 3 demonstrated, S is not only correlated to, it also helps to explain the

variation in AR. For that reason, the natural log of S will be used as an

87

explanatory variable in some of the following models.3 Again, linear models will
be built at each of the three scales of analysis. However, because many of the
hypothesized determinants listed in Table 10 are correlated with each other, and
not independent, they can not all be used in the same linear regression models.
For this reason, two models are created and displayed for each scale of analysis.
Full tables of correlations, which show the linear bivariate relationships between
all pairs of variables are located in Appendices K, L and M.

This thesis is concerned with the variation from the expected 0.50 in the
assessment ratio. Recall that if the property tax structure were equitable, any
variation from the expected 0.50 in the assessment ratio would be randomly
distributed. Under equity, the random errors would be both spatially neutral and
uncorrelated to other variables with spatial reference. However, it is already
known from tests done in Chapter 3 and Chapter 4 that the assessment errors
are not completely neutral. In Chapter 3 vertical inequity testing showed that the
assessment ratio is biased with respect to sale amount and Chapter 4 showed
that instances of deviation from 0.5 in the assessment ratio are not spatially
random. Nonetheless, much of the error in the assessment process, which
causes AR to deviate from 0.5, is most likely randomly distributed. AR will
therefore be used as the dependent variable in the following regression analyses
to try to find out what explains the variation in the assessment ratio. Thus, R-

Squares are not expected to be very large, and thus, the focus of the

 

3 The natural log transformed version of S, LNS, has a better linear relationship with AR than S
and LNS will therefore be used in the following linear regression models.

88

investigation will be on the statistical significance of the hypothesized explanatory
variables.

The residuals from one model at each scale will be examined for spatial
autocorrelation and possible sources of additional AR explanation. if there
appears to be geographic patterns in the residuals, then some explanatory
variables related to assessment bias are probably still missing from the models.
If the residuals appear to be randomly distributed, (this could be confirmed by
formal tests for spatial autocorrelation), then the models can be considered
complete, and there are no more spatial variables causing assessment bias.

Alternatively, like Thrall’s regression models, assessed value, AV, or its
natural log, could also be used as a dependent variable when coupled with sale
amount, 8, as an independent variable. In this case, R-Squares should be high
and S should explain most of the variation in AV; any other variables found to be
significant would represent property tax bias. From a model of this nature, the
variation of AR is only implicitly examined. The Paglin and Fogarty model as well
as the Cheng model, which were discussed in Chapter 3, already demonstrated
how regressive vertical inequity could be explained by such a model.

Census Tract Scale

Each of the explanatory variables that were found to be significantly
correlated to AR are entered as an explanatory variable in a simple, single
independent variable, regression model. The results are displayed in Table 12.
For organizational purposes, the explanatory variables and their regression

models have been divided into three categories: those having to do with racial

89

characteristics of the population, those having to do with income and sale

amount of property and those having to do with housing stock.

Table 12 — Simple Regressions, Dependent Variable = AR, df = 40

 

 

 

 

 

 

 

 

 

 

 

 

 

Explanatory Estimated Estimated Beta t-stat R

Variable Intercept Slope Square

Race WHITE 0.487 -3.33E-04 -0.382 -2.584 0.146
Variables

BLACK 0.457 2.53E-04 0.273 1.771 0.074

HISPANIC 0.451 1.40E-03 0.480 3.416 0.230

Wealth LNS 0.757 -2.71E-02 -0.622 -4.957 0.371
Variables

INC 0.486 -8.54E-07 -0.531 -3.914 0.264

PUBLIC 0.450 2.39E-03 0.555 4.166 0.308

Housing VACANT 0.451 1.81E-03 0.425 2.934 0.181
Variables

RENTOCC 0.453 2.51E-04 0.334 2.212 0.111

 

 

 

 

 

Each of the simple regression models at the tract scale, displayed in Table
12, has significant explanatory power. All of the explanatory variables have
significant slope coefficients at the 95% one-tail confidence level. This shows
that neighborhood and geographic characteristics do play a role in determining
assessment ratios and therefore property taxes of individual taxpayers.

The slope coefficients represent the expected change in AR for a one-unit
increase in the explanatory variable. For instance, a one-unit (one percent)
increase in HISPANIC will bring about an expected 0.00140 increase in the
assessment ratio. In the same way, a one-unit (one percent) increase in
population on public assistance (PUBLIC) will bring about an expected 0.00239
increase in the assessment ratio.

The R-Squares in the models vary from 0.07 to 0.37. It is important to

recognize the meaning of the magnitude of the explanatory power of these

90

individual variables alone. When regression designs are used to create
predictive or forecasting models, it is important that they have large R-Squares
and explain a large amount of the variation in the dependent variable. However,
the regression models in this chapter are not intended for predictive purposes.
They are only used for diagnostic purposes, to test explanatory variables for
significance and thus test the property tax structure for geographic bias. In
addition, none of the hypothesized determinants should theoretically, under
equity, explain any of the variation in AR. For these reasons, low R-Squares are
expected and have been realized. Nonetheless, the R-Squares for the models in
Table 12 at the tract scale do show explanation of up to 37 percent. In fact,
each of the race variables alone explains between nearly 7.5 percent and 23
percent of the variation in AR. A single wealth variable explains between 26
percent and 37 percent of the variation in AR and individual housing variables
explain between 11 and 18 percent of the variation in the dependent variable.
To account for even more of the variation in AR, explanatory variables will
be used together in multiple regression models. However, they can not all be
used in the same model since they are not all independent from each other
thereby violating an assumption of linear regression. The variables within each
category (race, wealth and housing) are highly interdependent. When variables
within each category are used in the same multiple regression model there is
likely to be multicolinearity. For this reason, the use of more than one variable
from any category will be avoided. However, there is also dependency between

the categories making it difficult to include more than any two of the variables in

91

the regression designs especially at the tract scale that has fewest degrees of
freedom. Check the full correlation tables in Appendices K-M for information on
what the correlation is between variables. The following multiple regression
models are an attempt to combine relatively independent explanatory variables in
models that explain what is determining the variation in AR. The tolerance of the
explanatory variables will be included in the tables that display the multiple
regression models. The tolerance is the percent of the explanatory variable that
is not explained by the other variables. It ranges between zero and one. Low
tolerances represent the presence of multicolinearity, under the presence of
which can cause coefficient bias and unefficiency. The variables included in the
models were chosen by first entering a wealth variable, and then a race variable
and then a housing variable. If the variables were not significant, they were
removed from the design.

Another reason for creating multiple regression models is to test for bias
holding other variables constant. The bivarite correlations and simple
regressions already revealed the explanatory power of the individual independent
variables. The multiple regression models will show the effect of individual
variables holding other variables constant. For example, the racial variables
have shown to significantly explain AR. When combined with a wealth variable in
a multiple regression model, the impact of the racial variables will be known
holding income or sale amount constant. In addition, since it is known that the

property tax structure is biased against racial minority groups, the models will

92

able to distinguish between expected AR values for rich racial minority
populations and poor racial minority populations.

Table 13 - Tract Multiple Regression 1.

 

 

AR = b0 + B1INC + 82WHITE
Estimated AR = 0.494 -1.68E-04 INC -7.23E-07 WHITE
Beta 0450 -0.193
t-stat 53.62 -3.033 -1 .299

sig. 0.000 0.004 0.202
tolerance 0.822 0.822

R - Square = 0.313

F-stat = 8.64

df = 40

 

 

Each independent variable coefficient should be interpreted as the
marginal expected change in AR, the dependent variable, for a one-unit change
in the independent variable holding all other explanatory variables constant.
Therefore, as the amount of white population increases by one percent holding
median household income constant, the assessment ratio can be expected to
decrease by 0.000000723. Likewise, as median household income increases by
one dollar, the expected assessment ratio decreases by 0.000168 holding
WHITE constant. What impact does this inequity have on the average taxpayer?
For example, holding race constant, a $100,000.00 property that is subjected to a
$100.00 dollar decrease in median household income would increase its
estimated AR from 0.494 to 0.510. This could increase the effective property
taxes by more than $50 dollars if the millage rate is 0.033. The results of Tract
Multiple Regression 1 in Table 13 show a bias in the property tax structure that
financially disadvantages neighborhoods as INC and WHITE decrease. The

standardized beta coefficients of the independent variables can be compared to

93

show the relative effects of the independent variables on AR. In this case, the
betas show that INC has more than two times the effect on AR than does
WHITE. Tract multiple regression 1 shows that not only is the property tax
structure biased with respect to income and race, expected AR values are lower
for rich white populations than for poor white populations.

Table 14 - Tract Multiple Regression 2.

 

 

AB = b0 + b1LNS + szLACK
Estimated AR = 0.740 -2.580E-02 LNS 1.091 E-04 BLACK
Beta 0591 0.118

t-stat 1 1.857 -4.534 0.902

sig. 0.000 0.000 0.373
Tolerance 0.931 0.931

R — Square = 0.398

F-stat = 12.572

df = 40

 

 

The results of the second multiple regression model in Table 14 show
similar biases. Holding sale amounts constant, as the black population increases
by one-unit (percent), the assessment ratio will increase by 0.0001091. While
this discrepancy doesn’t seem large at first glance, a fifty percent increase in
population that is black, can mean more than $24 in effective property tax dollars
for a parcel that sold for 100,000 dollars. In addition, results also show that
holding other explanatory variables constant, a one- percent increase in sale
amount, decreases the assessment ratio by an expected 0.0258.4 This time the
betas show the wealth variable has five times more effect on AR than the race

variable.

 

‘ Notice that the interpretation for the coefficient on LNS is slightly different than for the linear
variables. In this situation, the coefficient gives the marginal expected change in the dependent
variable for a one percent change in the log linear variable, S, holding all other variables constant.

94

Residual Examination — Census Tract Scale

The multiple regression model with the largest R square, shown in Table
14, has been chosen to be tested for completeness by examining its residuals.
Residual analysis is very important in model building because the residuals show
what the model fails to explain. If the residuals appear spatially autocorrelated, it
is likely that there is another geographic factor that explains variation in the
assessment ratio. If the residuals appear randomly distributed, it can be
assumed that random forces cause the variation that remains unexplained by the
multiple regression design. The residuals, the difference between the expected
ARs given the model and the actual ARs, are standardized, or made into 2—
scores, and mapped in Figure 33. The darker colors represent tracts with large
positive residuals where the ARs are higher than the model predicts. The lighter
colors represent areas with large negative residuals where the ARs are lower

than the model predicts.

95

 

 

Census Tract Residuals

-2.158 - -2.15
' , -2.15 - -o.953

  
 

66:73.1 -0.406 - 0.006

Manx

 

 

 

 

Figure 33 - Census Tract Residuals

It appears that large positive residuals are located in the center city tracts
and in the southwestern tracts of Lansing, while large negative residuals are
located in the north and northwestern tracts of the city. However, the residuals
closest to zero, tracts the model explains well, seem to be distributed evenly
throughout Lansing. A formal test for spatial autocorrelation is needed to
conclude with statistical confidence that the residuals are either clustered or
independently distributed. However, it is beyond the scope of this thesis to
gather new data and form new explanatory hypotheses and variables. Future
research efforts should use residual analysis in an interactive iterative process to
examine map patterns, develop new hypotheses from residuals, collect variables
to represent the hypotheses, and finally, to create better models that explain

more of the variation in the assessment ratio.

96

Block Group Scale

Table 15 - Simple Regressions, Dependent Variable = AR, df = 116

 

 

 

 

 

 

 

 

 

 

 

 

 

Explanatory Estimated Estimated Beta t-stat R

Variable Intercept Slope Square

Race WHITE 0.497 -4.49E-04 -0.341 -3.894 0.117
Variables

BLACK 0.457 3.68E-04 0.252 2.798 0.064

HISPANIC 0.451 1.46E-03 0.380 4.402 0.137

Wealth LNS 0.821 -3.31E-02 -0.503 -6.237 0.253
Variables

INC 0.491 -9.92E-07 -0.420 -4.961 0.176

PUBLIC 0.450 2.44E-03 0.421 4.974 0.1 17

Housing VACANT 0.450 2.35E-03 0.400 4.682 0.160
Variables

RENTOCC 0.451 3.39E-04 0.292 3.273 0.085

 

 

 

 

The results of simple regression models at the block group scale are very
similar to those at the census tract scale. The t-statistics show that each model
has significant explanatory power and each of the explanatory variables’
coefficients are significantly different from zero at the 99 percent confidence
level. Again, this indicates that geographic, social and economic variables do
play a role in determining assessment ratios, and the assessment process is
biased and inequitable. The move to a smaller scale with more smaller
observational units appears not to affect the conclusion of bias in the assessment
process.

Differences in the regression models at this scale lie in the R squares.
The R squares at the block group level are lower than those at the census tract
level. At this scale, each variable only explains between six percent and 25
percent of the variation in the assessment ratio. Again, multiple regression

designs are formulated with a combination of the explanatory variables above.

97

Two of these are displayed at this scale in Tables 16 and 17. The variables were
picked arbitrarily, one from each category, and inserted in the design if they
proved significant.

Table 16 - Block Group Multiple Regression 1

 

 

AR = bo-l- b1lNC + b2WHITE
Estimated = 0.507 -8.010E-07 -2.780E-04
AR INC WHITE
Beta 0339 -0.21 2
t-stat 57.165 -3.771 -2.355
sig. 0.000 0.000 0.020
Tolerance 0.853 0.853

R - Square 0.214

F-stat = 15.564

Df = 116

 

 

The model shown in Table 16 is slightly different from its census tract
scale counter part in the relative proportions of the betas. In this model at the
block group scale lNC’s beta is only 1.6 times greater than WHITE’s beta.

Table 17 - Block Group Multiple Regression 2

 

 

AR = bo+ B1LNS + BzBLACK + 33VACANT
Estimated = 0.731 -2.55E-02 1 .70E-04 9.47E-04
AR LNS BLACK VACANT
Beta 0388 0.116 0.162
t-stat 10.533 -4.091 1 .404 1 .696
sig. 0.000 0.000 0.163 0.093
Tolerance 0.700 0.917 0.693

R — Square 0.289

F-stat = 15.308

df = 116

 

 

Table 17 shows a significant regression design that includes three of the
explanatory variables. The results at this scale are consistent with those at the

census tract scale, except the R-Square is lower. The three explanatory

98

variables together explain just less than thirty percent of the variation in AR. The
standardized residuals from this model will be mapped and the resulting map
patterns will be examined. This model was chosen because it explains more of
the variation in AR than the other block group model in Table 16.

Residual Examination — Block Group Scale

 

 

Block Group Residuals

-2.597 - —2.066
-2.066 - -0.972
-0.972 - -0.465
0465 - 0.103
0.103 - 0.691
.... 0.691 - 1.801
1.801 - 3.836
m no data

 

 

 

 

Figure 34 — Block Group Residuals

It seems as if there are more positive residuals in the southern half of
Lansing, tracts with larger calculated ARs than the model would predict, and
more negative residuals in the northern half of Lansing, tracts with smaller
calculated ARs than the model would predict. This means there is probably
another bias of the assessment ratio that is not represented in the regression

model. An example of an appropriate hypothesis for this missing element is as

99

follows; perhaps there is a characteristic of the real estate market, affecting
property supply and demand. This hypothetical variable would affect
differentially the northern and southern parts of Lansing. If, for example, the real-
estate characteristic is correlated with lower demand in the southern block
groups, (thus, decreasing sale prices there), and higher demand in the northern
block groups, (thus, raising sale prices there), then calculated assessment ratios
would be higher in the southern block groups than the model in Table 17 predicts
and lower in the northern block groups than the model predicts. This hypothesis
is consistent with the pattern seen in the residuals. Unfortunately, it is beyond
the scope of this thesis to measure a variable that could explain this
phenomenon, instead it is offered here as a suggestion for further research.
Parcel Scale

Table 18 - Simple Regressions, Dependent Variable = AR, df = 8055

 

 

 

 

 

 

 

 

 

 

 

 

 

Explanatory Estim Estimated Beta t-stat R
Variable ated Slope Square
Interc
ept
Race WHITE 0.488 -3.55E-04 -0.063 -5.633 0.004
Variables
BLACK 0.456 2.89E-04 0.045 4.082 0.002
HISPANIC 0.451 1.36E-03 0.077 6.934 0.006
Wealth LNS 1.133 -6.19E-02 -0.322-30.519 0.104
Variables
INC 0.486 -8.37E-07 -0.087 -7.803 0.008
PUBLIC 0.451 8.17E-04 0.087 7.795 0.007
Housing VACANT 0.452 1.73E-03 0.067 6.032 0.004
Variables
RENTOCC 0.452 2.83E-04 0.057 5.164 0.003
Season SEASON 0.464 -5.76E-03 -0.033 -2.934 0.001
Variable

 

 

 

 

 

Again, at the parcel scale, all of the simple regressions have significant

explanatory power, and all explanatory variables are statistically different from

100

zero at the 99 percent confidence level. The R-Squares are remarkably smaller
at this scale. This is because the dependent variable was collected at the parcel
scale and the independent variables were collected at the block group scale and
then disaggregated to the parcel scale. The models at this scale are called “fixed
x” models. Because there is more variation in the dependent variable than in the
independent variables, less can be explained by them. However, the concern in
this chapter is not how much of the variation in AR is explained, but which
variables do explain variation.

Notice at this scale, the explanatory variable SEASON, is included in the
analysis. This dummy variable, which denotes the season that the sale
transaction took place, can only be used at the parcel scale because dummy
variables collected at a smaller scale can not be aggregated to a larger scale.
The results of the simple regression show that when SEASON equals one, or in
the summer time, AR decreases. This is as expected because more parcels sell
in summer, which represents more demand for parcels. More demand drives
sale prices up and since sale price is the denominator of AR, the assessment
ratio will appear smaller in the summer. Thus, there is an inverse relationship
between SEASON and AR.

The following multiple regressions have several significant independent
variables. More explanatory variables can be used at the parcel scale because
the number of degrees of freedom is high and significance is more easily

achieved.

101

 

Table 19 - Parcel Multiple Regression 1

 

AR = bo+ b1WHlTE + b2HlSPANIC + balNC + b.5EASON
Estimated AR = 0.488 -1.602E-04 5.469E-04 -5.544E-07 -4.641E-03
WHITE HISPANIC INC SEASON
Beta = -0.028 0.031 -0.057 -0.026
t-stat = 64.742 -2.288 2.209 -4.198 -2.370
Sig. = 0.000 0.022 0.027 0.000 0.018
Tolerance = 0.805 0.621 0.659 0.995
R Square = 0.010
df = 8055

 

 

The results of this multiple regression model are consistent with those at
the other scales of analysis. Because of the large number of degrees of
freedom, this model contains two race variables. So the effect of one, holding
the other constant may be ascertained. It is apparent that as the percentage of
Hispanic people increase, AR increases, even holding the percent of white
people, median household income, and season of sale transaction constant.
Again, INC is the variable with the largest Beta and the most relative impact on
the assessment ratio. The standardized residuals from this model will be kriged

and mapped to examine the residual map patterns.

102

Residual Examination - Parcel Scale

 

Parcel Residuals

1.575
1.053
.531
.008
-.514
-1.038
-1.558

 

 

 

 

 

 

Figure 35 — Parcel Residuals

Figure 35 is a kriged surface map of the residuals from the model in Table
19. Recall from Chapter 4 that the process of kriging involves interpolation of the
surface based on the parameters of a semivariogram. The semivariance
analysis reveals that dependence in the residuals exists only within a very small
lag distance. In fact, Moran’s l statistics show the residuals are spatially
autocorrelated only at lag distances of less than 350 UTM meters (less than .35
kilometers). This means that the model from which the residuals came does not
treat large areas of Lansing differentially. In other words, there are not large
scale patterns in the residuals from which hypotheses could easily by developed.

Because the spatial autocorrelation only exists at very small scales it is very

103

difficult to conjecture what variables, if any, are missing from the model. Another
way to see that large and small residuals are fairly well distributed on the map, is
to look at a three dimensional representation of the residuals. Figure 36, shows
the same surface map in three dimensions. The map is shown as if Lansing is
being looked at from west to east. As is shown in this map, it is easy to see that
the large positive residuals, shown by dark colored peaks, are often
accompanied by large negative residuals, shown by light colored holes. This is
further indication that there aren’t easily measured variables missing from the
model. It is more likely that if there is an element missing, it is a subtle market
affect that is highly geographically variable.

Parcel Fles'duals

1.575
1.053
.531
.008
-.514
-1.038
-1 .558

   

Figure 36 - Three Dimensional Residual Surface
Conclusion

Under an equitable property tax structure, the variation in the assessment
ratio would be unexplainable and randomly distributed. The results of this
chapter find nine geographic variables that explain significant variation in the
assessment ratio. Eight of these are significant at all three scales. Each
significant variable represents a bias in the assessment process. Much of the
bias is of a regressive nature. Results of the research in this chapter show that

areas dominated by marginalized, low income, minority populations, rented

104

housing or vacant housing are also those areas that are explaining higher
assessment ratios. It was also determined that members of poor racial minority
groups are the most discriminated against in terms of expected assessment
ratios. In addition, a seasonal bias was detected at the parcel scale. Parcels
that sell in summer have significantly higher assessment ratios. In general,
geographic, economic and demographic variables all play a role in determining
assessment ratios.

Sources of lnequity

There are three possible sources of the inequity that has been identified
throughout this thesis. First, the literature notes that individual assessors are the
source of inequity (Edelstein 1979). Second, it could be the assessment process
or the fault of the assessment readjustment process. Third, the workings of the
real estate market dynamics could be the source of inequity.

The first of these causes can be ruled out due to the three-scaled
methodology, which gives some clue as to the cause of the bias. At the small,
parcel scale, the neighborhood characteristics have a very small influence on the
assessment ratio (as evidenced by the small R square). This indicates that
neighborhood characteristics do not play a big role in determining an individual
parcel’s assessment, giving evidence against the argument that individual
assessors are causing inequity. However, at the larger scales, (census tract and
block group scales), the parcel data is aggregated to a smaller number of
observational units and more variation in the assessment ratio is explained. This

indicates that much of the random assessment error tends to be averaged out,

105

and that at these scales neighborhood characteristics do play a big role in
determining average assessment ratios. These observations are evidence
against the argument that it is individual assessors causing inequity.

It is much more plausible that it is the assessment process and market
dynamics causing inequity. Because the biases are most apparent at the
neighborhood scale (tract and block group scales) they are most likely caused by
processes that are inherently based on neighborhood units and their
characteristics. The assessment process and the real-estate market are both
such processes.

As has already been discussed, the assessment process involves
reassessing entire neighborhoods up or down if the average AR is too low or to
high. This process produces a boundary effect that creates areas of extremely
high or low AR values. Since geographic areas tend to be demographically
homogenous at small scales, the boundary effect could cause certain
demographic groups to be differentially affected by the assessment process.

Another likely source of inequity is various real estate market dynamics.
The significance of the variable SEASON is already proof that the cyclic market
dynamic causes inequity. There are many other dynamics causing supply and
demand differentials that could also determine AR values. Part of the reason for
this is the extreme segregation of real estate markets. In the United States, real
estate markets are as diverse as there are niches of population. Each real estate
market caters to a different kind of household. Each market likely has its own

ups and downs and supply and demand patterns. It is out of the realm of this

106

thesis to try to correct segregation in the settlement patterns in the United States.
However, it can safely be postulated that correcting the market dynamics that
cause inequity would be a difficult job indeed. Nonetheless, it is legally
incumbent upon the assessment process to correct any inequity, regardless of
source. Therefore, bias and inequity in the property tax structure can and should
be the responsibility of the property tax assessment process. The next chapter
will present an outline for correcting the biases and inequities identified in this

chapter.

107

Chapter 6

CONCLUSION

This chapter will begin with a short review of important findings from the
research of this thesis. A discussion of the possible causes for the inequity and
bias will follow. Next, the limitations of this thesis and ideas for future research
are laid out. Ultimately, the purpose of this chapter is to outline related policy
implications as well as recommendations for the city of Lansing Assessor’s
Office.

Findings

The property tax structure of Lansing, Michigan was first found to be
inequitable with respect to sale amounts. Four models tested in Chapter 3;
Paglin and Fogarty, Cheng, I.A.A.O. and Clapp all show significant, regressive,
vertical inequity. This is a clear demonstration that higher priced parcels are
taxed at a lower rate than lower priced parcels.

lnequities with respect to geographic location or distinct neighborhoods in
Lansing were also found. This finding comes out of the analyses described in
Chapter 4 which show that positive spatial autocorrelation exists between parcels
separated by small distance lags and that both positive and negative spatial
autocorrelation exist between parcels separated by medium and large distance
lags. These results show that some neighborhoods are taxed higher than other

neighborhoods. This represents a geographic bias because it shows that

108

taxpayers, on average, pay taxes on different proportions of market value based
on where they live.

Results from the linear model testing in Chapter 5 show that the property
tax structure in Lansing, Michigan is also inequitable with respect to many
demographic, social-economic, and market variables. These biases exist even
when sale amounts are held constant. If the property structure were equitable,
no measurable variables would explain variation in the assessment ratio because
the variation would only consist of random error. Instead, the property tax
structure is inequitable. Three racial variables: percentages of black, white, and
Hispanic populations, all prove to be property tax biases as they significantly
explain variation in the assessment ratio. Also, income variables like median
household income and percentage of population on public income are significant
variables, revealing bias based on wealth. All of these results were found at
three geographic scales. Furthermore, a seasonal dummy variable also shows
significant bias of the property tax with respect to the real estate market’s
seasonal dynamics.

Causes of Bias

This thesis sought to identify and examine property tax inequities and
biases. However, the question this thesis does not answer directly is what
causes the property tax inequities that are persisting. No scientific hypothesis
testing will be done to answer that question. Instead, only logical speculation is
made here. It seems there are two possible sources or causes of the inequity.

First, the assessment process itself could be biased, causing inequity. Second,

109

the dynamics of the real estate market could be such that assessments are
unavoidably error prone. It is likely that it is some combination of the two sources
causing the identified inequities. Another possibility, one that should be ruled
out, is that malicious individuals, who are actively seeking to over assess some
parcels while underassessing others, cause the inequities. This supposition,
which is mentioned in the literature, is highly unlikely as the assessment process,
(see Conclusion section in Chapter 5), would seem to prevent the manifestation
of this kind of intent (Edelstein 1979).

The work in this thesis provides evidence for both of the other possible
sources of bias. Evidence that the assessment process is causing inequity lies in
the work done in Chapter 4. Because the assessors’ office has carved Lansing
into discrete neighborhoods and uses them as units to adjust up or down, it
seems likely that some of the spatial patterns identified in the maps in Figures 22
through 26 are boundary effects produced by this neighborhood assessment
technique.

There is also evidence obtained in this study that indicates inequity is
caused by real estate market dynamics. The best example of this is the variable
SEASON. SEASON equals one if the parcel sold in the summer and zero if the
parcel sold in the winter this dummy variable is inversely linearly correlated to
AR5. The result reflects a supply and demand effect that raises sale amounts in
the summer making the assessment ratio appear lower for those parcels.

Certainly, it is not the assessment process assessing parcels sold in the summer

 

5 This variable is measured at the parcel scale and therefore is only tested at that scale in the
linear models.

110

lower than those sold in the winter. This is the only variable in the study that is
necessarily independent of the assessment process so that it only describes a
market dynamic. Nonetheless, its’ significance indicates that market dynamics
do, to some degree, determine assessment ratios.
Future Research

Future research should concentrate on further isolating the root(s) of the
inequities. One of the limitations of this thesis is the short time frame over which
the observations were collected. A longer time frame could perhaps reveal
dynamic spatial patterns in the property tax structure that would indicate market
characteristics causing bias. In addition, a longer time study could reveal if the
assessment correction process already being used by Lansing assessors was
actually correcting inequity.
Implementing Geographic Process and Methodology

Regardless of what is at fault concerning the inequity of the property tax
structure, there are some clear policy implications that come from the research in
this thesis. The policy implications involve changing the assessment process
rather than the real-estate market, for two reasons. First, and obviously, the
workings of the real-estate market can not be changed, and ultimately, it is
incumbent on the assessment process to overcome market forces that cause
bias and inequity anyway. Second, the policy implications are aimed at
improving the assessment process so that it can correct its own errors and
alleviate its biases more efficiently. In general, the policy implication is this: to

incorporate geographic evaluation at all stages in the assessment process. The

111

best and most efficient way to deal with large amounts of geographic data and
analyses is to create and use a GIS database management system. Please note
that an introductory GIS textbook should be consulted for in depth discussion of
what a GIS is and how specific spatial operations could be performed. With that
said, GIS is a powerful tool that could probably eliminate inequity in the property
tax structure of Lansing, Michigan. The next paragraphs will outline a step-by-
step procedure that could be used by the city of Lansing to implement a GIS and
a more precise spatial evaluation of the property tax structure.

GIS should be used at the very beginning of the cyclic assessment
process in the data storage stage. GIS is a powerful tool for managing large
quantities of geographically referenced data and Assessment data are just that.
The assessor's office already stores its data in databases. But these non-spatial
databases do not allow assessors to perform spatial analyses, spatial queries, or
view assessments spatially. This is a huge limitation of the current assessment
process since anything related to real-estate markets is inherently spatial. This
limitation is even more severe, in light of the findings of this thesis that the
property tax structure is inequitable with respect to geographically distinct areas.
If the assessment data were transferred to a GIS database management system,
then the data could be used for assessment purposes more efficiently. There are
many different ways GIS should be used in the assessment process. The first,
and most elementary of these, is for organization and display data purposes.

A GIS platform could help the assessor organize the multitudes of data that

are collected for the purposes of assessment. Usually, the first step to

112

implementing a GIS for municipal and assessment purposes is to enter the city’s
plat maps in digital format. Next, aerial photos of the city should be taken and
made digital. Plat maps and digital air photos would together make up the base
map for the GIS. Together they would have the parcel boundaries, street
network, sidewalk delineation, hydrology, electrical and sewer lines, parks and
public lines etc. Next, each parcel will be geographically referenced, not only
with address and zip code as is done now, but also by latitude and longitude. As
was done for the purposes of this research and discussed in Chapter 2, each
parcel can then be assigned to other geographic scales to allow for multiple scale
analysis. For instance, an assessor could easily bring up all parcels located
within a two-block radius from the capital or any other point. The main
advantage of this is the elimination of the assessor neighborhood.

After the data has been stored, geo-reference and organized, GIS can be
used for display and queries. GIS would aid in the assessment of new parcels or
recently altered parcels through spatial analyses and queries. Existing
databases in the assessment process can do aspatial queries, but GIS is
intended for doing spatial queries. For example, if an assessor wanted
information on how a certain kind of property was assessed in the past, he/she
could query to find all similar parcels with respect to designated characteristics
and perhaps within certain geographic boundaries and then use statistics on the
query to guide the assessment. Other layers of data could also be
geographically referenced and assigned / attached to each parcel. So, for

example, parcels could be selected based on the census characteristics

113

surrounding them. Of course, GIS can also be used for easy mapping and
viewing of assessments, parcels and related data. There are many ways GIS
would make the actual assessing more efficient and accurate. However, as this
thesis has demonstrated, perhaps the most important role for GIS in the
assessment process comes at the evaluation stages of the assessment cycle.
GIS is most powerful when used as a mode for analysis rather than just data
collection and storage.

The assessors do evaluate the assessments on a yearly basis. However,
they do so completely aspatially. The techniques used in this thesis that
identified many inequities could be easily implemented by the assessor’s office
for evaluation. Once the evaluation was over, precise adjustments could be
made to the property tax structure that would alleviate inequity.

Currently, all assessment analyses and adjustments are done based on
static neighborhood designations. Each parcel was manually assigned to a
neighborhood and then, the neighborhood designations were entered into the
database. A GIS would give the analyses much more flexibility. Instead of using
discrete boundaries for readjustment, a moving window sometimes called a
“neighborhood smoothing” techniques, could be passed over Lansing and each
of an infinite number of windows (areas of a designated size) could be readjusted

(Chrisman 1997, p.179).6 This technique would rid the assessment structure of

 

6 A moving window, or a moving average, can be thought of as a box or circle that is layed over
each part of the study area, the average assessment ratio could be taken for each moving
window and adjusted. Moving windows do away with discrete boundaries, moving window
boundaries are also changing.

114

the boundary affects, seen to cause geographic inequity, caused by the assessor
neighborhood boundaries.

Another way GIS could be used to readjust assessments to get rid of
inequity is through surface analysis and functions. The assessment ratio
surfaces could be smoothed by some function that would eliminate all major
peaks and valleys. Then, the surface could be ungenerated to arrive at new
assessments for each parcel. This process is only theoretical and would have to
be adjusted for the particular software and resources of the implemented GIS.
But, in doing these more detailed adjustments and analyses, GIS would be used
to correct the inequities caused by the real estate market in addition to correct
any remaining inequities overlooked by the assessment process.

Naturally, the implementation of any municipal GIS takes a lot of capital in
the form of time, hardware, software and human resources. This would require
an overhaul of the current assessment office structure and a large financial
commitment from the city.

In addition, the implementation of a GIS and the methods used to assess
and correct assessments that are described above, would require a paradigm
shift on the part of the assessor's office, but more importantly, a paradigm shift
for tax payers. Currently, taxpayers compare their assessments to other
assessments on their block, or in their neighborhood. If the systems described
above were implemented, taxpayers could do some querying of their own and
compare their assessments to those of parcels with similar characteristics and

circumstance as well as to the sale amounts of similar parcels.

115

 

The city of Lansing, must decide if the costs and changes involved with the
implementation of a GIS are worth the increased precision and equity it would

bring to the property tax structure.

 

116

APPENDICE

117

APPENDIX A

SEMIVARIANCE VALUES FOR 1994 DATA

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lag Average Average Pairs
Distance Semi-
vafiance
1 56.47 0.005450 1 338
2 1 52.05 0.005294 3402
3 253.27 0.005716 4950
4 351 .83 0.005547 6279
5 450.51 0.005903 7426
6 550.81 0.006078 8343
7 650.21 0.006128 9044
8 750.15 0.006158 9800
9 850.06 0.006467 1 0302
10 951.16 0.006855 1 1435
11 1050.21 0.006719 12036
12 1150.01 0.006902 12317
13 1 250.60 0.006721 1 2759
14 1350.57 0.007017 13070
15 1450.59 0.006741 14159
1 6 1 550.93 0.006755 14668
17 1650.01 0.006756 15168
18 1750.46 0.006847 15732
1 9 1 849.82 0.006779 1 6405
20 1949.71 0.006705 17100
21 2050.45 0.006645 1 7582
22 21 50.45 0.006720 1 8084
23 2250.1 0 0.006573 1 8233
24 2349.83 0.006721 1 8922
25 2449.65 0.006825 1 9225
26 2550.31 0.006808 1 9827
27 2650.44 0.006659 20429
28 2750.07 0.006726 20649
29 2850.32 0.006767 21084
30 2950.03 0.006899 21719
31 3050.09 0.006931 22169
32 31 50.22 0.006831 22479
33 3249.84 0.007023 22653

 

 

118

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

34 3349.92 0.00701 9 22658
35 3449.90 0.006952 23058
36 3549.86 0.006828 2351 0
37 3649.97 0.006729 23381
38 3750.26 0.006755 23545
39 3850.68 0.006629 23927
40 3949.70 0.00681 6 23522
41 4049.68 0.006762 2381 2
42 4150.37 0.006651 24122
43 4250.08 0.006669 2381 5
44 4349.81 0.006570 23462
45 4449.26 0.006376 23604
46 4550.05 0.006404 22558
47 4649.78 0.006480 22489
48 4749.68 0.006364 22140
49 4850.37 0.006141 21806
50 4949.98 0.006231 21408
51 5049.82 0.006407 20840
52 51 49.93 0.006398 20367
53 5249.45 0.006463 1 9980
54 5349.83 0.006468 1 961 9
55 5449.31 0.006434 1 9309
56 5549.85 0.006236 1 9060
57 5650.04 0.006328 1 9295
58 5750.02 0.006431 1 8795
59 5849.47 0.006386 1 8791
60 5950.35 0.006151 18149
61 6050.31 0.006220 1 7709
62 61 49.53 0.005960 17478
63 6249.52 0.006034 1 71 75
64 6349.89 0.006043 1 6854
65 6449.30 0.005967 1 6622
66 6549.89 0.005961 161 13
67 6650.1 1 0.005882 15833
68 6750.25 0.005940 1 5664
69 6849.60 0.006079 1 5201

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

70 6949.47 0.005920 14641
71 7049.99 0.005932 1 3754
72 7149.96 0.005904 13416
73 7249.39 0.005922 1 2770
74 7349.83 0.005702 12161
75 7449.15 0.005841 1 1691
76 7549.35 0.005895 1 1 185
77 7649.20 0.0061 58 1 0667
78 7748.91 0.006078 1 0380
79 7850.35 0.005989 9982
80 7949.46 0.006249 9903
81 8049.78 0.0061 31 9097
82 8149.97 0.006023 8754
83 8250.01 0.005945 8585
84 8349.09 0.005756 81 51
85 8449.60 0.005879 7868
86 8549.78 0.0061 69 7739
87 8649.88 0.005875 741 8
88 8749.21 0.006000 7234
89 8849.79 0.005861 7049
90 8949.76 0.006032 6739
91 9049.60 0.005734 6335
92 91 49.38 0.005702 5699
93 9248.95 0.005798 5761
94 9349.63 0.005216 5121
95 9449.1 1 0.005492 4981
96 9550.1 8 0.005329 4796
97 9648.77 0.005414 4534
98 9749.55 0.005538 4320
99 9848.94 0.0051 1 0 4256
1 00 9950.26 0.0051 94 3996
101 10048.34 0.005116 3797
102 10150.61 0.005384 3422
1 03 1 0249.93 0.005288 31 64
1 04 1 0350.27 0.005324 3008
105 10448.91 0.005183 2704
1 06 10547.90 0.0051 73 2548
1 07 1 0649.06 0.005049 2223
108 1 0748.07 0.005459 2047
109 10850.27 0.005654 1810
1 10 10948.05 0.005651 1589
1 1 1 1 1049.32 0.005207 1464
112 11149.84 0.006329 1257
113 11250.55 0.005646 1112
1 14 1 1348.42 0.005636 1 020

 

 

119

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

115 11448.23 0.006215 820
1 16 1 1548.29 0.006859 686
117 11648.98 0.006615 621
118 11745.48 0.007018 492
119 11849.32 0.006991 445
120 1 1 947.85 0.004777 351
121 12045.70 0.003925 336
122 12147.93 0.004127 323
123 12245.27 0.006859 227
1 24 12346.91 0.007897 1 60
1 25 1 2446.56 0.007906 1 21
126 12553.91 0.008281 73
1 27 12647.90 0.008361 50
1 28 1 2741 .99 0.004452 36
129 12846.82 0.003445 32
1 30 1 2951 .89 0.004222 26
131 13050.27 0.003254 18
132 13155.17 0.005605 15
133 13242.74 0.006771 6
134 13337.97 0.001918 3
135 13442.00 0.009827 2

 

 

 

APPENDIX B

SEMIVARIANCE VALUES FOR 1995 DATA

 

E
10

Average
Distance

Average
Semi-
vafiance

Pairs

 

54.53

0.006720

1 448

 

152.82

0.007180

3937

 

253.29

0.007225

5352

 

352.09

0.008031

6969

 

451.38

0.007672

8558

 

551 .39

0.007734

9523

 

650.91

0.007810

1 0585

 

750.71

0.008037

11693

 

850.88

0.007709

1 2450

 

A
OCDQNODU'I-bOON-t

951.06

0.008318

1 3657

 

—L
—L

1049.98

0.008070

1 4307

 

—L
N

1150.35

0.008202

15163

 

.5
00

1250.37

0.008321

15671

 

_L
&

1350.54

0.008047

16215

 

—L
01

1450.27

0.008350

1 7687

 

.A
03

1550.28

0.007980

1 7875

 

.5
\1

1649.72

0.008020

19019

 

_L
m

1750.09

0.008088

1 8967

 

—L
(0

1849.87

0.007961

1 9747

 

N
0

1950.38

0.008127

20461

 

N
_L

2050.28

0.008213

20957

 

N
[0

2150.81

0.008104

21343

 

N
00

2250.16

0.007950

21944

 

N
.5

2349.97

0.008018

22695

 

N
01

2450.39

0.007996

24060

 

N
0)

2549.95

0.007945

24643

 

N
\1

2650.17

0.008173

25039

 

N
00

2750.58

0.008222

25815

 

N
(0

2849.83

0.008279

26215

 

00
0

2950.07

0.008285

26792

 

00
_L

3050.23

0.008308

26984

 

(D
10

3149.65

0.008084

27215

 

00
00

3250.02

0.008226

27386

 

8

3350.17

0.008295

27957

 

0)
01

3449.87

0.008208

28119

 

 

0.)
O)

 

3550.42

 

0.008289

 

27969

 

 

120

 

37

3650.14

0.008281

28251

 

38

3750.01

0.008361

28234

 

39

3850.04

0.008400

28335

 

40

3950.50

0.008513

28096

 

41

4049.81

0.008423

28208

 

42

4150.07

0.008277

27847

 

43

4249.94

0.008373

26700

 

44

4349.49

0.008049

26349

 

45

4449.57

0.008260

25695

 

46

4549.93

0.007974

25349

 

47

4649.49

0.008167

24849

 

48

4749.91

0.007834

24828

 

49

4850.22

0.007955

24262

 

50

4950.00

0.007994

23673

 

51

5049.80

0.007713

22775

 

52

5149.47

0.007813

22479

 

53

5249.60

0.007769

21925

 

54

5349.92

0.007981

21494

 

55

5449.54

0.007874

21325

 

56

5549.38

0.007551

21131

 

57

5649.75

0.007700

20987

 

58

5750.07

0.007660

20545

 

59

5850.05

0.007663

20628

 

60

5950.08

0.007686

1 9846

 

61

6049.87

0.007630

1 9285

 

62

6149.13

0.007317

1 9042

 

63

6249.25

0.007579

18513

 

64

6349.64

0.007542

1 8340

 

65

6449.73

0.007553

1 7526

 

66

6549.92

0.007368

1 6966

 

67

6650.17

0.007533

1 6300

 

68

6749.94

0.007377

1 5765

 

69

6849.47

0.007268

1 5454

 

70

6949.92

0.007408

14891

 

71

7049.20

0.007634

14501

 

72

7149.39

0.007554

1 3290

 

73

7249.85

0.007488

13313

 

74

7349.41

0.007378

1 2707

 

 

75

 

7450.28

 

0.007286

 

12100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

76 7549.77 0.007254 1 1420
77 7649.61 0.00751 1 1 0892
78 7749.70 0.007399 1 0681
79 7849.61 0.007130 10261
80 7950.47 0.006738 1 0088
81 8049.44 0.006989 9477
82 8149.58 0.006963 9119
83 8249.68 0.0071 44 8922
84 8349.58 0.006889 841 3
85 8449.93 0.006939 7906
86 8550.22 0.007101 7551
87 8649.81 0.006989 7341
88 8749.03 0.006678 7087
89 8848.1 9 0.006538 6512
90 8949.28 0.006659 61 06
91 9049.68 0.006359 5739
92 9149.11 0.006512 5180
93 9248.78 0.006486 4842
94 9349.78 0.006346 4448
95 9448.76 0.0061 44 4239
96 9549.31 0.005799 3907
97 9648.32 0.006253 3649
98 9749.75 0.005754 3377
99 9848.91 0.005462 31 95
1 00 9950.00 0.005631 2955
101 1 0049.33 0.006035 2615
102 10150.56 0.005838 2441
103 1 0248.82 0.005736 2320
104 1 0349.07 0.005905 2107
1 05 1 0449.22 0.006020 1855
106 10548.53 0.005188 1769
107 10649.00 0.005923 1421
108 1 0748.35 0.006256 1 176
109 10847.76 0.006379 916
1 10 10949.76 0.006747 842
111 11049.79 0.006156 745
112 11148.14 0.006394 611
1 13 1 1248.08 0.006537 546
1 14 1 1350.26 0.005803 523
115 11449.17 0.006147 548
116 11550.17 0.004476 445
117 11646.84 0.004377 421
1 18 1 1747.94 0.004723 406
1 19 1 1847.75 0.002825 323
120 1 1946.19 0.004965 229

 

 

 

 

 

121

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

121 12050.44 0.004517 201
122 12149.97 0.004981 170
123 1 2247.08 0.005347 122
124 1 2350.22 0.004692 144
125 1 2440.03 0.004298 97
126 1 2542.02 0.007232 58
1 27 1 2653.95 0.008262 24
128 12749.86 0.011914 24
129 12844.64 0.009285 22
130 12950.21 0.008062 1 1
131 13063.09 0.008229 1 1
132 13154.26 0.007866 11
133 13237.50 0.013333 1
134 13350.33 0.010594 9
1 35 1 3435.40 0.009782 2

 

 

 

SEMIVARIANCE VALUES FOR 1996 DATA

APPENDIX C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lag Average Average Pairs
Distance Semi-
vafiance
1 54.44 0.006535 1954
2 152.94 0.007132 4922
3 252.45 0.007050 7022
4 351.08 0.007437 8773
5 451.18 0.007633 10401
6 551.00 0.007407 11630
7 650.95 0.007244 13097
8 750.92 0.007459 13986
9 851.43 0.007352 14790
10 951.03 0.007521 15723
11 1050.07 0.007701 16573
12 1150.63 0.007740 17841
13 1250.54 0.007638 18732
14 1350.51 0.007881 19235
15 1450.45 0.007971 20334
16 1550.13 0.007992 21418
17 1650.17 0.008019 22408
18 1750.27 0.008120 23069
19 1850.15 0.007894 23738
20 1950.58 0.008001 24824
21 2050.37 0.008124 25087
22 2150.61 0.008011 26295
23 2250.50 0.008245 26639
24 2350.11 0.008261 27115
25 2449.91 0.008076 28065
26 2550.64 0.008018 28608
27 2650.31 0.007946 29526
28 2750.02 0.007982 30010
29 2850.12 0.008075 31189
30 2950.47 0.008366 31460
31 3049.96 0.008082 32673
32 3150.08 0.008191 32618
33 3250.20 0.008385 32900
34 3350.12 0.008166 33596
35 3450.08 0.008239 34274
36 3550.11 0.008177 33877

 

 

 

 

 

122

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

37 3649.85 0.0081 22 3371 3
38 3750.00 0.007990 33472
39 3849.77 0.007935 33785
40 3950.16 0.007780 33581
41 4050.03 0.007864 33972
42 4150.24 0.007921 33758
43 4250.25 0.007820 33762
44 4349.85 0.007868 33712
45 4449.65 0.007673 33266
46 4550.20 0.007792 33247
47 4649.86 0.007889 32836
48 4749.71 0.007918 31596
49 4849.80 0.007762 30982
50 4949.97 0.007945 30001
51 5049.40 0.007815 29319
52 5150.07 0.007731 28262
53 5250.09 0.007499 27984
54 5349.35 0.007564 27379
55 5449.55 0.007278 27094
56 5550.10 0.007420 27207
57 5649.98 0.007302 26516
58 5749.98 0.007434 25926
59 5850.15 0.007273 25443
60 5950.02 0.007281 251 16
61 6049.57 0.007217 24434
62 6149.61 0.007171 23872
63 6249.81 0.007188 23266
64 6349.88 0.007286 22762
65 6449.88 0.0071 66 21785
66 6549.87 0.0071 06 21 305
67 6650. 1 8 0.006885 21242
68 6750.23 0.007182 20960
69 6850.15 0.0071 15 20393
70 6949.91 0.00721 6 19760
71 7049.66 0.007364 19073
72 7149.78 0.007328 1 8365
73 7249.61 0.007287 17827
74 7349.94 0.007283 16776
75 7449.76 0.007330 1 6169

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

76 7549.88 0.007391 1 5634
77 7650.20 0.007237 14941
78 7749.76 0.007242 1 5090
79 7849.84 0.007294 1431 6
80 7949.79 0.007282 1 3940
81 8049.62 0.007257 1 3579
82 8150.20 0.007579 12874
83 8249.65 0.007401 1 2626
84 8349.75 0.007477 12043
85 8449.01 0.007547 1 2086
86 8549.57 0.007447 1 1621
87 8649.59 0.007389 1 0953
88 8749.30 0.007322 1 0627
89 8849.29 0.00721 3 9673
90 8949.58 0.007324 9295
91 9049.32 0.006980 8685
92 9149.97 0.006961 7858
93 9249.02 0.006897 751 7
94 9349.26 0.006789 7066
95 9449.67 0.006758 6407
96 9549.79 0.006454 61 86
97 9649.65 0.006053 5851
98 9749.31 0.005862 5492
99 9849.00 0.005982 5142
1 00 9949.65 0.005823 4996
1 01 1 0049.58 0.005699 4727
102 10149.65 0.005981 4479
103 10249.45 0.005573 4181
104 1 0349.46 0.005422 3917
105 10448.77 0.005553 3616

 

 

 

123

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 06 1 0549.49 0.005368 3212
107 1 0649.36 0.005480 2861
108 1 0748.95 0.005423 2559
109 10850.39 0.005581 2201
1 10 10950.16 0.005678 1862
11 1 1 1047.09 0.005246 1682
112 11149.61 0.004583 1600
113 11251.17 0.004306 1379
1 14 11348.99 0.004245 1239
115 11448.14 0.004342 1060
116 11547.46 0.004312 871
1 17 1 1647.29 0.004207 837
1 18 1 1747.94 0.003836 728
119 11848.96 0.004065 714
120 1 1949.03 0.003476 554
121 1 2050.33 0.004099 502
122 12149.52 0.003981 384
123 1 2247.75 0.004659 325
124 1 2351 .49 0.003746 280
125 1 2445.10 0.004936 226
126 12547.81 0.004397 139
127 12646.64 0.006301 99
128 12748.35 0.00451 1 65
129 12848.13 0.006767 29
130 12944.03 0.004301 21
131 13048.78 0.002397 22
132 13134.73 0.004838 5
133 13248.78 0.001558 11
1 34 1 3354.52 0.000542 5

 

 

 

 

SEMIVARIANCE VALUES FOR 1997 DATA

APPENDIX D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lag Average Average Pairs
Distance Semi-
vafiance
1 56.78 0.006328 1618
2 153.66 0.006311 4037
3 252.69 0.006400 5958
4 351.89 0.006377 7436
5 450.71 0.006604 8738
6 551.17 0.006858 10112
7 651.21 0.006717 11253
8 750.45 0.007156 12300
9 850.55 0.007403 13084
10 950.47 0.006912 14006
11 1050.23 0.007197 14839
12 1150.15 0.007418 15108
13 1250.66 0.007197 15745
14 1350.90 0.007062 16698
15 1450.64 0.007235 17343
16 1550.40 0.007240 18019
17 1650.73 0.007348 19048
18 1750.69 0.007252 19251
19 1850.23 0.007167 20168
20 1950.19 0.007102 20998
21 2050.54 0.007235 21168
22 2150.23 0.007184 21744
23 2250.20 0.007153 22338
24 2350.11 0.007261 23036
25 2449.94 0.007446 23782
26 2550.50 0.007418 24641
27 2650.16 0.007608 25623
28 2750.03 0.007646 26177
29 2850.01 0.007727 26566
30 2950.07 0.007838 27529
31 3050.41 0.007725 27380
32 3149.92 0.007765 27941
33 3250.33 0.007661 28701
34 3350.26 0.007784 28922
35 3449.99 0.007704 29258
36 3550.07 0.007675 29546

 

 

 

 

 

124

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

37 3649.85 0.007674 29495
38 3749.68 0.007618 29737
39 3849.83 0.007712 29441
40 3950.26 0.007459 29704
41 4050.1 1 0.007388 29713
42 4150.15 0.007658 29197
43 4249.67 0.007454 29513
44 4349.56 0.007441 28845
45 4449.96 0.007456 28186
46 4550.05 0.007376 27737
47 4649.71 0.007450 26796
48 4749.82 0.007359 26535
49 4849.86 0.007287 26091
50 4949.52 0.007128 25121
51 5049.87 0.007182 24064
52 5149.60 0.007200 23623
53 5249.36 0.007199 22939
54 5349.73 0.007061 22530
55 5449.86 0.007233 22223
56 5549.62 0.007040 21869
57 5649.97 0.006997 21363
58 5749.95 0.006898 20415
59 5850.07 0.006848 20379
60 5950.00 0.006881 20039
61 6049.81 0.006824 19500
62 6149.33 0.006718 19288
63 6249.44 0.006644 18949
64 6349.67 0.006790 18743
65 6449.75 0.006876 17837
66 6549.97 0.007079 17552
67 6650.36 0.0071 15 17286
68 6750.00 0.007021 16435
69 6849.52 0.007304 16257
70 6949.51 0.0071 72 1 5472
71 7049.88 0.0071 53 14933
72 71 49.89 0.007208 1 4247
73 7250.37 0.007074 13994
74 7349.67 0.007309 13340
75 7449.46 0.007334 12929

 

 

 

 

 

 

76

7550.02

0.007409

1 2353

 

77

7649.46

0.007248

11673

 

78

7749.41

0.007562

11333

 

79

7849.70

0.007557

1 0759

 

80

7949.40

0.007619

1 0228

 

81

8049.98

0.007603

1 0302

 

82

8149.97

0.007775

9649

 

83

8249.75

0.007606

9137

 

84

8348.72

0.007638

8655

 

85

8449.67

0.007535

8145

 

86

8549.54

0.007578

7883

 

87

8649.96

0.007228

7408

 

88

8750.27

0.007617

7099

 

89

8848.83

0.007502

6692

 

90

8949.23

0.007429

6569

 

91

9050.08

0.007520

6176

 

92

9149.84

0.007379

5831

 

93

9248.44

0.007278

5502

 

94

9349.64

0.007321

5384

 

95

9450.14

0.007018

4831

 

96

9549.08

0.007220

4570

 

97

9649.97

0.007074

4304

 

98

9748.78

0.006904

3926

 

99

9849.26

0.006979

3634

 

100

9949.57

0.006970

3513

 

101

1 0048.82

0.007417

3324

 

102

10150.38

0.007518

3009

 

103

1 0249.55

0.007625

2826

 

104

10348.36

0.007895

2562

 

 

105

 

10448.67

 

0.007867

 

2421

 

 

125

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

106 10549.97 0.007946 2114
107 1 0647.60 0.008052 1882
1 08 1 0748.00 0.008855 1641
109 10848.03 0.008174 1337
110 10949.46 0.009410 1150
1 11 1 1049.32 0.007976 1035
112 11146.86 0.007687 797
1 13 1 1247.30 0.009068 758
1 14 1 1350.77 0.007822 627
1 15 1 1450.52 0.007603 456
1 16 1 1549.25 0.005356 435
1 17 1 1647.69 0.004955 367
118 1 1747.92 0.004636 331
119 11849.54 0.004133 345
120 1 1949.63 0.006301 271
121 12045.86 0.004441 221
122 12150.83 0.006205 211
123 12251.67 0.006325 171
124 12345.10 0.004614 146
1 25 12444.26 0.005256 120
1 26 1 2545.93 0.007896 66
127 12652.70 0.008590 51
128 12739.81 0.008297 44
1 29 1 2855.38 0.008221 30
130 1 2943.35 0.007202 44
1 31 1 3056.74 0.009809 33
132 13161.80 0.008219 26
133 1 3224.42 0.008363 21
134 13325.71 0.005460 10
1 35 1 3463.19 0.003678 8

 

 

 

 

 

APPENDIX E

SEMIVARIANCE VALUES FOR 1998 DATA

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lag Average Average Pairs
Distance Semi-
vaflance
1 50.80 0.013221 43
2 152.47 0.006199 161
3 252.08 0.005455 172
4 349.56 0.007545 246
5 450.45 0.008892 280
6 551.71 0.008131 292
7 650.74 0.008526 319
8 747.79 0.008144 361
9 851.83 0.008473 365
10 949.46 0.007761 383
11 1051.24 0.008459 436
12 1149.71 0.006826 504
13 1250.10 0.007885 506
14 1352.80 0.008997 511
15 1451.31 0.009171 555
16 1551.82 0.008871 591
17 1649.62 0.008586 617
18 1749.74 0.008700 657
19 1849.30 0.008571 650
20 1949.94 0.007994 699
21 2050.44 0.007979 648
22 2149.52 0.008910 681
23 2250.20 0.007721 658
24 2347.38 0.008048 700
25 2450.57 0.008523 699
26 2548.89 0.008242 748
27 2651.64 0.007990 810
28 2749.69 0.007504 792
29 2850.32 0.007847 844
30 2949.90 0.008635 962
31 3049.95 0.008060 950
32 3149.84 0.008230 934
33 3249.09 0.008342 941
34 3349.99 0.007991 935
35 3447.07 0.008225 879
36 3548.23 0.007806 946

 

 

126

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

37 3649.86 0.007602 909
38 3749.54 0.008026 907
39 3850.00 0.007809 914
40 3949.91 0.007071 919
41 4050.43 0.007348 881
42 4148.16 0.007161 925
43 4249.61 0.0071 60 870
44 4348.86 0.008353 841
45 4449.26 0.006955 908
46 4549.1 6 0.006663 901
47 4650.07 0.00781 2 899
48 4750.60 0.007527 820
49 4850.08 0.007500 708
50 4951 .26 0.007326 737
51 5050.37 0.007310 700
52 5150.44 . 0.007563 697
53 5249.1 1 0.007275 690
54 5348.73 0.007434 646
55 5449.42 0.006877 629
56 5551 .31 0.006450 684
57 5649.73 0.0061 35 702
58 5749.31 0.007159 667
59 5849.05 0.007596 657
60 5951 .62 0.006549 61 6
61 6048.85 0.0061 25 572
62 6150.42 0.006905 587
63 6249.52 0.006740 588
64 6348.97 0.007625 581
65 6453.25 0.006649 51 7
66 6547.94 0.007600 574
67 6651 .56 0.007591 557
68 6751 .05 0.006448 525
69 6848.44 0.006835 484
70 6950.83 0.00631 0 479
71 7047.35 0.007371 434
72 7151.60 0.005800 411
73 7252.25 0.006845 425
74 7350.1 5 0.006383 351
75 7451 .43 0.006435 383

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

76 7550.21 0.007290 381
77 7648.32 0.006547 368
78 7749.53 0.006757 350
79 7848.86 0.006592 375
80 7949.50 0.006678 323
81 8050.90 0.006587 280
82 8148.48 0.006317 283
83 8247.92 0.00861 1 287
84 8347.80 0.006757 264
85 8445.55 0.006695 279
86 8546.27 0.00671 8 280
87 8648.73 0.006746 235
88 8749.81 0.006364 220
89 8846.47 0.006890 225
90 8949.99 0.005789 224
91 9051.14 0.006684 194
92 9149.06 0.007211 169
93 9244.44 0.006591 1 71
94 9344.1 7 0.005203 175
95 9452.39 0.004765 1 64
96 9551 .44 0.005680 164
97 9652.24 0.005407 146
98 9744.00 0.006899 126
99 9848.24 0.007435 97
1 00 9945.92 0.005724 1 08
1 01 1 0052.65 0.006407 99
102 10150.69 0.006143 112
1 03 1 0255.53 0.006449 87
104 1 0352.05 0.005087 92
105 1 0443.64 0.006996 78
106 10543.66 0.006019 52
107 10648.42 0.006013 46
1 08 1 0748.07 0.009880 43
1 09 1 0840.73 0.007471 44
1 10 10950.47 0.007570 35
111 11052.07 0.004531 32
112 11155.73 0.003609 24
113 11251.51 0.003947 14
114 11345.14 0.005832 22
115 11458.77 0.002577 15
116 11551.40 0.003613 19
1 17 1 1646.84 0.005005 29
118 11736.57 0.008245 15
1 19 1 1835.58 0.004289 22
120 1 1953.33 0.002439 18

 

 

 

 

 

127

 

121

1 2050.47

0.004813

14

 

122

12150.82

0.002949

 

123

12259.14

0.002537

 

124

12349.18

0.000990

 

125

 

126

1 2550.86

0.004926

 

127

12683.99

0.009524

 

128

12784.15

0.008910

 

129

12838.04

0.004494

 

 

130

 

1 2964.58

 

0.002484

 

—L
NMNAODOODQCO

 

 

APPENDIX F

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1994 MORAN’S l
Lag Moran’s l Pairs E(l) SD s.e. Z Hypothesis
(n) score Test
1 0.100755 1338 -7.5E-04 0.99963 0.027328186 3.7142 Reject Null
2 0.026976 3402 -2.9E-04 0.99985 0.017142304 1.5908 No Reject
3 0.041947 4950 -2.0E-04 0.99990 0.014211948 2.9657 Reject Null
4 0.050301 6279 -1.6E-04 0.99992 0.012618863 3.9988 Reject Null
5 0.005232 7426 -1.3E-04 0.99993 0.011603615 0.4625 No Reject
6 0.009738 8343 -1.2E-04 0.99994 0.010947448 0.9005 No Reject
7 0.012273 9044 -1.1E-04 0.99994 0.010514672 1.1777 No Reject
8 0.022854 9800 -1.0E-04 0.99995 0.010101011 2.2726 Reject Null
9 0.008704 10302 -9.7E-05 0.99995 0.009851859 0.8933 No Reject
10 -0.019313 11435 -8.7E-05 0.99996 0.009351105 -2.0560 Reject Null
11 0.002029 12036 -8.3E-05 0.99996 0.009114669 0.2317 No Reject
12 0013916 12317 -8.1E-05 0.99996 0.009010106 -1.5355 No Reject
13 0.000597 12759 -7.8E-05 0.99996 0.008852678 0.0763 No Reject
14 -0.038004 13070 -7.7E-05 0.99996 0.008746728 -4.3362 Reject Null
15 001178 14159 -7.1E-05 0.99996 0.008403658 -1.3934 No Reject
16 -0.015119 14668 -6.8E-05 0.99997 0.008256572 -1.8229 Reject Null
17 -0.018865 15168 -6.6E-05 0.99997 0.008119355 -2.3153 Reject Null
18 0.005242 15732 -6.4E-05 0.99997 0.007972495 0.6655 No Reject
19 0.001576 16405 -6.1E-05 0.99997 0.00780726 0.2097 No Reject
20 0.0063 17100 -5.8E-05 0.99997 0.007646968 0.8315 No Reject
21 0.009782 17582 -5.7E-05 0.99997 0.007541427 1.3046 No Reject
22 0.006472 18084 -5.5E-05 0.99997 0.007436023 0.8778 No Reject
23 0.014225 18233 -5.5E-05 0.99997 0.007405579 1.9283 Reject Null
24 0.006411 18922 -5.3E-05 0.99997 0.007269508 0.8892 No Reject
25 0011295 19225 -5.2E-05 0.99997 0.007211997 -1.5589 No Reject
26 0004429 19827 -5.0E-05 0.99997 0.007101671 06166 No Reject
27 0.007234 20429 -4.9E-05 0.99998 0.006996258 1.0410 No Reject
28 0.004306 20649 -4.8E-05 0.99998 0.00695889 0.6257 No Reject
29 -0.011919 21084 -4.7E-05 0.99998 0.006886732 -1.7238 Reject Null
30 0.001057 21719 -4.6E-05 0.99998 0.006785316 0.1626 No Reject
31 -0.001484 22169 -4.5E-05 0.99998 0.0067161 02142 No Reject
32 0009303 22479 -4.4E-05 0.99998 0.006669632 -1.3882 No Reject
33 0007449 22653 -4.4E-05 0.99998 0.006643968 -1.1145 No Reject
34 0009463 22658 -4.4E-05 0.99998 0.006643235 -1.4178 No Reject
35 0005407 23058 -4.3E-05 0.99998 0.006585364 -0.8145 No Reject
36 -0.003466 23510 -4.3E-05 0.99998 0.006521755 -0.5249 No Reject

 

 

 

 

 

 

 

 

 

128

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

37 -0.00793 23381 -4.3E-05 0.99998 0.00653972 -1.2060 No Reject
38 -0.00208 23545 -4.2E-05 0.99998 0.006516906 -0.3127 No Reject
39 0.003276 23927 -4.2E-05 0.99998 0.006464677 0.5132 No Reject
40 -0.014816 23522 -4.3E-05 0.99998 0.006520091 -2.2658 Reject Null
41 -0.001922 23812 -4.2E-05 0.99998 0.006480268 -0.2901 No Reject
42 -0.003594 24122 -4.1E-05 0.99998 0.006438495 -0.5518 No Reject
43 -0.003268 23815 -4.2E-05 0.99998 0.00647986 -0.4979 No Reject
44 0.006025 23462 -4.3E-05 0.99998 0.006528422 0.9294 NO Reject
45 0.000532 23604 -4.2E-05 0.99998 0.006508756 0.0882 No Reject
46 0.001182 22558 -4.4E-05 0.99998 0.006657943 0.1842 No Reject
47 -0.000504 22489 -4.4E-05 0.99998 0.006668149 -0.0689 No Reject
48 0.009513 22140 -4.5E-05 0.99998 0.006720497 1.4222 No Reject
49 0.017861 21806 -4.6E-05 0.99998 0.008771768 2.6443 Reject Null
50 0.01097 21408 -4.7E-05 0.99998 0.006834422 1.6119 No Reject
51 -0.000968 20840 -4.8E-05 0.99998 0.006926929 -0.1328 N0 Reject
52 0.002387 20367 -4.9E-05 0.99998 0.007006898 0.3477 No Reject
53 0.001478 19980 -5.0E-05 0.99997 0.007074429 0.2160 No Reject
54 -0.000254 19619 -5.1E-05 0.99997 0.007139216 -0.0284 No Reject
55 -0.010476 19309 -5.2E-05 0.99997 0.007196294 -1.4486 No Reject
56 -0.006111 19060 -5.2E-05 0.99997 0.007243145 -0.8365 No Reject
57 -0.009011 19295 -5.2E-05 0.99997 0.007198904 -1.2445 No Reject
58 -0.02092 18795 -5.3E-05 0.99997 0.007294026 -2.8608 Reject Null
59 -0.024516 18791 -5.3E-05 0.99997 0.007294802 -3.3535 Reject Null
60 0.002088 18149 -5.5E-05 0.99997 0.007422696 0.2887 No Reject
61 -0.006278 17709 -5.6E-05 0.99997 0.007514338 -0.8280 No Reject
62 0.002552 17478 -5.7E-05 0.99997 0.007563829 0.3450 N0 Reject
63 -0.000045 17175 -5.8E-05 0.99997 0.007630254 0.0017 No Reject
64 0.003284 16854 -5.9E-05 0.99997 0.00770257 0.4341 No Reject
65 0.00557 16622 -6.0E-05 0.99997 0.007756134 0.7259 N0 Reject
66 0.009362 16113 -6.2E-05 0.99997 0.00787768 1.1963 No Reject
67 0.028326 15833 -6.3E-05 0.99997 0.007947027 3.5723 Reject Null
68 0.014835 15664 -6.4E-05 0.99997 0.00798978 1.8647 Reject Null
69 -0.012028 15201 -6.6E-05 0.99997 0.008110538 -1.4749 No Reject
70 0.006397 14641 -6.8E-05 0.99997 0.008264181 0.7823 No Reject
71 0.002831 13754 -7.3E-05 0.99996 0.008526479 0.3406 No Reject
72 -0.006685 13416 -7.5E-05 0.99996 0.00863321 -0.7657 No Reject
73 -0.00431 12770 -7.8E-05 0.99996 0.008848865 -0.4782 No Reject
74 -0.003478 12161 -8.2E-05 0.99996 0.009067708 -0.3745 No Reject
75 -0.011205 11691 -8.6E-05 0.99996 0.009248166 -1.2023 No Reject
76 -0.008081 11185 -8.9E-05 0.99996 0.009455023 -0.8452 N0 Reject
77 -0.010528 10667 -9.4E-05 0.99995 0.009681854 -1.0777 No Reject
78 -0.001537 10380 -9.6E-05 0.99995 0.009814777‘ -0.1468 No Reject
79 0.001328 9982 -1.0E-04 0.99995 0.010008511 0.1427 No Reject
80 -0.013652 9903 -1.0E-04 0.99995 0.010048349 -1.3486 N0 Reject
81 0.003041 9097 -1.1E-04 0.99995 0.010484001 0.3005 No Reject

 

129

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

82 0.003357 8754 -1.1E-04 0.99994 0.010687397 0.3248 No Reject
83 -0.000399 8585 -1.2E-04 0.99994 0.010792066 -0.0262 No Reject
84 0.017771 8151 -1.2E-04 0.99994 0.011075617 1.6156 No Reject
85 0.00427 7868 -1.3E-04 0.99994 0.01127302 0.3901 No Reject
86 0.004513 7739 -1.3E-04 0.99994 0.011366573 0.4084 No Rfi'ect
87 0.003986 7418 -1.3E-04 0.99993 0.01160987 0.3549 No Reject
88 0.011322 7234 -1.4E-04 0.99993 0.011756574 0.9748 No Reject
89 0.014655 7049 -1.4E-04 0.99993 0.011909828 1.2424 No Reject
90 0.006564 6739 -1.5E-04 0.99993 0.01218064 0.5511 No Reject
91 0.005793 6335 -1.6E-04 0.99992 0.012562974 0.4737 No Reject
92 0.008206 5699 -1.8E-04 0.99991 0.013245325 0.6328 No Reject
93 -0.008069 5761 -1.7E-04 0.99991 0.013173872 -0.5993 N0 Reject
94 0.027661 5121 -2.0E-04 0.99990 0.013972698 1.9936 Reject Null
95 0.007953 4981 -2.0E-04 0.99990 0.014167663 0.5755 No Reject
96 0.007189 4796 -2.1E-04 0.99990 0.014438272 0.5124 No Reject
97 -0.020655 4534 -2.2E-04 0.99989 0.014849487 -1.3761 No Reject
98 -0.019239 4320 -2.3E-04 0.99988 0.015212759 -1.2494 No Reject
99 0.008908 4256 -2.4E-04 0.99988 0.015326687 0.5965 No Reject
100 0.015055 3996 -2.5E-04 0.99988 0.015817325 0.9676 No Reject
101 0.003635 3797 -2.6E-04 0.99987 0.016226418 0.2403 No Reject
102 -0.020801 3422 -2.9E-04 0.99985 0.017092151 -1.1999 NO Reject
103 -0.009434 3164 -3.2E-04 0.99984 0.017775153 -0.5130 No Reject
104 -0.022557 3008 -3.3E-04 0.99983 0.018230103 -1.2191 No Reject
105 -0.004237 2704 -3.7E-04 0.99982 0.019227226 -0.2011 No Reject
106 0.003536 2548 -3.9E-04 0.99980 0.019806849 0.1983 N0 Reject
107 0.018668 2223 -4.5E-04 0.99978 0.021204743 0.9016 No Reject
108 -0.007999 2047 -4.9E-04 0.99976 0.022097111 -0.3399 No Reject
109 -0.01 1169 1810 -5.5E-04 0.99973 0.023498567 -0.4518 No Reject
110 -0.012357 1589 -6.3E-04 0.99969 0.025078538 -0.4676 N0 Reject
111 0.00247 1464 -6.8E-04 0.99966 0.026126552 0.1207 N0 Reject
112 -0.064208 1257 -8.0E-04 0.99961 0.028194274 -2.2491 Reject Null
113 0.019601 1112 -9.0E-04 0.99955 0.029974642 0.6839 No Reject
114 0.032819 1020 -9.8E-04 0.99951 0.031296013 1.0800 No Reject
115 -0.012843 820 -1.2E-03 0.99940 0.034900474 -0.3330 N0 Reject
116 -0.074235 686 -1.5E-03 0.99928 0.038152745 -1.9075 Reject Null
117 -0.012073 621 -1.6E-03 0.99921 0.040096816 -0.2609 No Reject
118 -0.003 492 -2.0E-03 0.99900 0.045038574 -0.0214 No Reject
119 -0.142692 445 -2.3E-03 0.99890 0.047352451 -2.9658 Reject Null
120 -0.025001 351 -2.9E-03 0.99862 0.053302131 -0.4154 No Reject
121 0.104749 336 -3.0E-03 0.99856 0.054475649 1.9777 Reject Null
122 0.080309 323 -3.1E-03 0.99850 0.055557959 1.5014 No Reject
123 -0.183431 227 -4.4E-03 0.99789 0.066232427 -2.7027 Reject Null
124 -0.121673 160 -6.3E-03 0.99707 0.07882498 -1.4638 N0 Reject
125 0.07798 121 -8.3E-03 0.99620 0.0905638 0.9531 N0 Reject
126 -0.093424 73 -1.4E-02 0.99407 0.116347123 -0.6836 No Reject

 

 

 

 

 

 

 

 

 

130

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

127 0.122192 50 -2.0E-02 0.99196 0.140284967 1.0165 N0 Reject
128 -0.017457 36 -2.9E-02 0.98991 0.164985082 0.0674 No Reject
129 0.084539 32 -3.2E-02 0.98919 0.17486554 0.6679 No Reject
130 -0.115275 26 -4.0E-02 0.98808 0.193777856 -0.3885 No Reject
131 0.106116 18 -5.9E-02 0.98754 0.232765046 0.7086 N0 Reject
132 -0.000362 15 -7.1E-02 0.98878 0.255300943 0.2784 No Reject
133 -0.067886 6 -2.0E-01 1.05560 0.430945804 0.3066 No Reject
134 0.326682 3 -5.0E-01 1.36931 0.790569415 1.0457 No Reject
135 -0.277577 2 -1.0E+00 1.91485 1.354006401 0.5335 NO Reject

 

 

 

131

APPENDIX G

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1995 MORAN’S I
Lag Moran’s pairs E(l) SD s.e. Z score Hypothesis
I (nj Test

1 0.065807 1448 -6.9E-04 0.9997 0.02627 2.531 Reject Null

2 0.041966 3937 -2.5E-04 0.9999 0.015935 2.649 Reject Null

3 0.046781 5352 -1.9E-04 0.9999 0.013668 3.436 Reject Null
4 -0.00632 6969 -1.4E-04 0.9999 0.011978 -0.515 No Reject
5 0.003711 8558 -1.2E-04 0.9999 0.010809 0.354 No Reject
6 0.007416 9523 -1.1E-04 0.9999 0.010247 0.734 No Reject
7 000472 10585 -9.4E-05 1.0000 0.009719 -0.476 No Reject
8 -0.00342 11693 -8.6E-05 1.0000 0.009247 -0.361 No Reject
9 0.00226 12450 -8.0E-05 1.0000 0.008962 0.261 No Reject
10 -0.01156 13657 -7.3E-05 1.0000 0.008557 -1.343 No Reject
11 001187 14307 -7.0E-05 1.0000 0.00836 -1.411 No Reject
12 -0.0069 15163 -6.6E-05 1.0000 0.008121 -0.841 No Reject

13 001746 15671 -6.4E-05 1.0000 0.007988 -2.178 Reject Null
14 000505 16215 -6.2E-05 1.0000 0.007853 -0.635 No Reject
15 001051 17687 -5.7E-05 1.0000 0.007519 -1.390 No Reject
16 0.011078 17875 -5.6E-05 1.0000 0.007479 1.489 No Reject
17 0.010225 19019 -5.3E-05 1.0000 0.007251 1.417 No Reject
18 00031 18967 -5.3E-05 1.0000 0.007261 -0.420 No Reject
19 0.001367 19747 -5.1E-05 1.0000 0.007116 0.199 No Reject
20 0.011271 20461 -4.9E-05 1.0000 0.006991 1.619 No Reject
21 000211 20957 -4.8E-05 1.0000 0.006908 -0.298 No Reject
22 0.000956 21343 -4.7E-05 1.0000 0.006845 0.147 No Reject
23 000794 21944 -4.6E-05 1.0000 0.00675 -1.169 No Reject
24 -0.00213 22695 -4.4E-05 1.0000 0.006638 -0.315 No Reject
25 0.00421 24060 -4.2E-05 1.0000 0.006447 0.659 No Reject
26 000582 24643 -4.1E-05 1.0000 0.00637 0907 No Reject
27 000711 25039 -4.0E-05 1.0000 0.00632 -1.118 No Reject
28 00055 25815 -3.9E-05 1.0000 0.006224 -0.877 No Reject
29 00037 26215 -3.8E-05 1.0000 0.006176 -0.592 No Reject
30 0.002041 26792 -3.7E-05 1.0000 0.006109 0.340 No Reject
31 000551 26984 -3.7E-05 1.0000 0.006087 -0.899 No Reject
32 0.003483 27215 -3.7E-05 1.0000 0.006062 0.581 No Reject
33 0.00045 27386 -3.7E-05 1.0000 0.006043 0.081 No Reject
34 0.004205 27957 -3.6E-05 1.0000 0.005981 0.709 No Reject
35 -0.00434 28119 -3.6E-05 1.0000 0.005963 -0.722 No Reject

36 001186 27969 -3.6E-05 1.0000 0.005979 -1.977 Reject Null

 

132

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

37 0.00882 28251 -3.5E-05 1.0000 0.005949 1.488 No Reject
38 0.00093 28234 -3.5E-05 1.0000 0.005951 0.162 No Reject
39 0.009986 28335 -3.5E-05 1.0000 0.005941 1.687 Reject Null
40 0.005255 28096 -3.6E—05 1.0000 0.005966 0.887 No Reject
41 -0.00151 28208 -3.5E-05 1.0000 0.005954 -0.248 No Reject
42 0.005184 27847 -3.6E-05 1.0000 0.005992 0.871 No Reject
43 -0.0073 26700 -3.7E-05 1.0000 0.00612 -1.186 No Reject
44 -0.00508 26349 -3.8E-05 1.0000 0.00616 -0.818 No Reject
45 -0.00245 25695 -3.9E-05 1.0000 0.006238 -0.386 No Reject
46 0.004651 25349 -3.9E-05 1.0000 0.006281 0.747 No Reject
47 -0.01347 24849 -4.0E-05 1.0000 0.006344 -2.117 Reject Null
48 0.002747 24828 -4.0E-05 1.0000 0.006346 0.439 No Reject
49 0.00037 24262 -4.1E-05 1.0000 0.00642 0.064 No Reject
50 0.014049 23673 -4.2E-05 1.0000 0.006499 2.168 Reject Null
51 0.013965 22775 -4.4E-05 1.0000 0.006626 2.114 Reject Null
52 0.002025 22479 -4.4E-05 1.0000 0.00667 0.310 No Reject
53 0.005229 21925 -4.6E-05 1.0000 0.006753 0.781 No Reject
54 -0.00158 21494 -4.7E-05 1.0000 0.006821 -0.224 No Reject
55 -0.0033 21325 -4.7E-05 1.0000 0.006848 -0.474 No Reject
56 -0.00548 21131 -4.7E-05 1.0000 0.006879 -0.790 No Reject
57 -0.00044 20987 -4.8E-05 1.0000 0.006903 -0.056 No Reject
58 -0.01636 20545 -4.9E-05 1.0000 0.006976 -2.337 Reject Null
59 -0.00507 20628 -4.8E-05 1.0000 0.006962 -0.721 No Reject
60 -0.00949 19846 -5.0E-05 1.0000 0.007098 -1.330 No Reject
61 -0.00386 19285 -5.2E-05 1.0000 0.007201 -0.528 No Reject
62 0.003877 19042 -5.3E-05 1.0000 0.007247 0.542 No Reject
63 -0.00571 18513 -5.4E-05 1.0000 0.007349 -0.770 No Reject
64 0.00341 18340 -5.5E-05 1.0000 0.007384 0.469 No Reject
65 -0.00296 17526 -5.7E-05 1.0000 0.007553 -0.385 No Reject
66 0.000761 16966 -5.9E-05 1.0000 0.007677 0.107 No Reject
67 0.007216 16300 -6.1E-05 1.0000 0.007832 0.929 No Reject
68 0.01432 15765 -6.3E-05 1.0000 0.007964 1.806 Reject Null
69 0.008208 15454 -6.5E-05 1.0000 0.008044 1.028 No Reject
70 0.015479 14891 -6.7E-05 1.0000 0.008195 1.897 Reject Null
71 -0.00573 14501 -6.9E-05 1.0000 0.008304 -0.681 No Reject
72 -0.00925 13290 -7.5E-05 1.0000 0.008674 -1.058 No Reject
73 -0.00285 13313 -7.5E-05 1.0000 0.008667 -0.321 No Reject
74 -0.00961 12707 -7.9E-05 1.0000 0.008871 -1.075 No Reject
75 -0.00114 12100 -8.3E-05 1.0000 0.009091 -0.116 No Reject
76 -0.00283 11420 -8.8E-05 1.0000 0.009357 -0.293 No Reject
77 -0.00559 10892 -9.2E-05 1.0000 0.009581 -0.573 No Reject
78 -0.00225 10681 -9.4E-05 1.0000 0.009676 -0.223 No Reject
79 0.000754 10261 -9.7E-05 1.0000 0.009872 0.086 No Reject
80 0.008261 10088 -9.9E-05 1.0000 0.009956 0.840 No Reject
81 -0.00369 9477 -1.1E-04 0.9999 0.010272 -0.349 NO Reject

 

 

 

 

 

 

 

 

133

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

82 0.012732 9119 -1.1E-04 0.9999 0.010471 1.226 No Reject
83 -0.01212 8922 -1.1E-04 0.9999 0.010586 -1.135 No Reject
84 -0.00459 8413 -1.2E-04 0.9999 0.010902 -0.410 No Reject
85 0.002212 7906 -1.3E-04 0.9999 0.011246 0.208 No Reject
86 -0.00414 7551 -1.3E-04 0.9999 0.011507 -0.348 No Reject
87 -0.00142 7341 -1.4E-04 0.9999 0.011671 -0.110 No Reject
88 -0.01398 7087 -1.4E-04 0.9999 0.011878 -1.165 No Reject
89 -0.00172 6512 -1.5E-04 0.9999 0.012391 -0.126 No Reject
90 -0.0111 6106 -1.6E-04 0.9999 0.012796 -0.855 No Reject
91 -0.00147 5739 -1.7E-04 0.9999 0.013199 -0.098 No Reject
92 -0.01822 5180 -1.9E-04 0.9999 0.013893 -1.297 No Reject
93 -0.02537 4842 -2.1E-04 0.9999 0.01437 -1.751 Reject Null
94 -0.00518 4448 -2.2E-04 0.9999 0.014992 -0.330 No Reject
95 0.000583 4239 -2.4E-04 0.9999 0.015357 0.053 No Reject
96 0.006334 3907 -2.6E-04 0.9999 0.015996 0.412 No Reject
97 -0.01572 3649 -2.7E-04 0.9999 0.016552 -0.933 No Reject
98 0.005103 3377 -3.0E-04 0.9999 0.017206 0.314 No Reject
99 0.029529 3195 -3.1E-04 0.9998 0.017689 1.687 Reject Null
100 0.000247 2955 -3.4E-04 0.9998 0.018393 0.032 No Reject
101 -0.01231 2615 -3.8E-04 0.9998 0.019552 -0.610 No Reject
102 -0.00117 2441 -4.1E-04 0.9998 0.020236 -0.037 No Reject
103 -0.00188 2320 -4.3E-04 0.9998 0.020757 -0.070 No Reject
104 -0.00106 2107 -4.7E-04 0.9998 0.02178 -0.027 No Reject
105 -0.01923 1855 -5.4E-04 0.9997 0.023212 -0.805 No Reject
106 0.028018 1769 -5.7E-04 0.9997 0.023769 1.203 No Reject
107 -0.00173 1421 -7.0E-04 0.9997 0.026519 -0.039 No Reject
108 0.006468 1176 -8.5E-04 0.9996 0.029148 0.251 No Reject
109 -0.02159 916 -1.1E-03 0.9995 0.033023 -0.621 No Reject
110 0.021869 842 -1.2E-03 0.9994 0.034442 0.669 No Reject
111 0.00394 745 -1.3E-03 0.9993 0.036613 0.144 No Reject
112 -0.00765 611 -1.6E-03 0.9992 0.040423 -0.149 No Reject
113 -0.03677 546 -1.8E-03 0.9991 0.042758 -0.817 No Reject
114 -0.02401 523 -1.9E-03 0.9991 0.043686 -0.506 NO Reject
115 -0.06356 548 -1.8E-03 0.9991 0.04268 -1.446 No Reject
116 -0.00715 445 -2.3E-03 0.9989 0.047352 -0.104 No Reject
117 0.017908 421 -2.4E-03 0.9988 0.04868 0.417 No Reject
118 0.045792 406 -2.5E-03 0.9988 0.04957 0.974 No Reject
119 0.063189 323 -3.1E-03 0.9985 0.055558 1.193 No Reject
120 -0.00833 229 -4.4E-03 0.9979 0.065944 -0.060 No Reject
121 -0.00243 201 -5.0E-03 0.9976 0.070368 0.037 No Reject
122 0.027186 170 -5.9E-03 0.9972 0.076484 0.433 No Reject
123 0.0668 122 -8.3E-03 0.9962 0.090194 0.832 No Reject
124 0.053125 144 -7.0E-03 0.9968 0.083064 0.724 No Reject
125 0.031776 97 -1.0E-02 0.9954 0.101064 0.417 No Reject
126 -0.01406 58 -1.8E-02 0.9928 0.130366 0.027 No Reject

 

 

 

 

134

 

 

 

 

 

 

 

 

 

 

127 0.031879 -4.3E-02 0.9878 0.201624 0.374 No Reject
128 013606 -4.3E-02 0.9878 0.201624 -0.459 No Reject
129 0.243041 -4.8E-02 0.9875 0.210536 1.381 No Reject
130 0.072382 -1.0E-01 0.9958 0.300252 0.574 No Reject
131 0.176853 -1.0E-01 0.9958 0.300252 0.922 No Reject
132 0.224948 -1.0E-01 0.9958 0.300252 1.082 No Reject
133 -0.71575 #DlV/O! #DlV/O #DlV/Ol #DlV/O! No Reject
134 0.0483 -1.3E-01 1.0062 0.33541 0.517 No Reject
135 -0.13056 - 1.9149 1.354006 0.642 No Reject
1.0E+00

 

 

 

 

 

 

 

 

135

 

APPENDIX H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1996 MORAN’S l
Lag Moran’sl Pairs E(l) SD s.e. Z Hypothesis
(n) score Test
1 0.092965 1954 -5.1E-04 0.9997 0.022617 4.133 Reject Null
2 0.061403 4922 -2.0E-04 0.9999 0.014252 4.323 Reject Null
3 0.03803 7022 -1.4E—04 0.9999 0.011933 3.199 Reject Null
4 0.005044 8773 -1.1E-04 0.9999 0.010676 0.483 No Reject
5 000925 10401 -9.6E-05 1.0000 0.009805 -0.934 No Reject
6 0.031717 11630 -8.6E-05 1.0000 0.009272 3.430 Rejgct Null
7 0.02418 13097 -7.6E-05 1.0000 0.008738 2.776 Reject Null
8 0.013048 13986 -7.2E-05 1.0000 0.008455 1.552 No Reject
9 0.009387 14790 -6.8E-05 1.0000 0.008222 1.150 No Reject
10 -0.00454 15723 -6.4E-05 1.0000 0.007975 -0.561 No Reject
11 0.014652 16573 -6.0E-05 1.0000 0.007768 1.894 Reject Null
12 0.005036 17841 -5.6E-05 1.0000 0.007486 0.680 No Reject
13 0.014317 18732 -5.3E-05 1.0000 0.007306 1.967 Reject Null
14 000358 19235 -5.2E-05 1.0000 0.00721 0489 No Reject
15 000163 20334 -4.9E-05 1.0000 0.007013 -0.226 No Reject
16 0.009959 21418 -4.7E-05 1.0000 0.006833 1.464 No Reject
17 -0.00601 22408 -4.5E-05 1.0000 0.00668 0893 No Reject
18 000467 23069 -4.3E-05 1.0000 0.006584 -0.702 No Reject
19 -0.00634 23738 -4.2E-05 1.0000 0.00649 -0.970 No Reject
20 0.002392 24824 -4.0E-05 1.0000 0.006347 0.383 No Reject
21 0.001356 25087 -4.0E-05 1.0000 0.006313 0.221 No Reject
22 -0.00497 26295 -3.8E-05 1.0000 0.006167 -0.799 No Reject
23 000072 26639 -3.8E-05 1.0000 0.006127 -0.112 No Reject
24 -0.01412 27115 -3.7E-05 1.0000 0.006073 -2.319 Reject Null
25 000642 28065 -3.6E-05 1.0000 0.005969 -1.070 No Reject
26 -0.01104 28608 -3.5E-05 1.0000 0.005912 -1.861 Reject Null
27 0.010427 29526 -3.4E-05 1.0000 0.00582 1.798 Reject Null
28 0.000384 30010 -3.3E-05 1.0000 0.005772 0.072 No Reject
29 0.000748 31189 -3.2E-05 1.0000 0.005662 0.138 No Reject
30 -0.02257 31460 -3.2E-05 1.0000 0.005638 -3.998 Reject Null
31 000312 32673 -3.1E-05 1.0000 0.005532 -0.559 No Reject
32 -0.00763 32618 -3.1E-05 1.0000 0.005537 -1.372 No Reject
33 -0.00965 32900 -3.0E-05 1.0000 0.005513 -1.745 Reject Null
34 -0.00237 33596 -3.0E-05 1.0000 0.005456 -0.430 No Reject
35 -0.00554 34274 -2.9E-05 1.0000 0.005401 -1.020 No Reject
36 -0.00301 33877 -3.0E-05 1.0000 0.005433 -0.549 No Reject
37 000742 33713 -3.0E-05 1.0000 0.005446 -1.358 No Reject

 

136

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

38 0.003065 33472 -3.0E-05 1.0000 0.005466 0.566 No Reject
39 -0.0083 33785 -3.0E-05 1.0000 0.00544 -1.519 NO Reject
40 0.005786 33581 -3.0E-05 1.0000 0.005457 1.066 No Reject
41 -0.00939 33972 -2.9E—05 1.0000 0.005425 -1.725 Reject Null
42 -0.00558 33758 -3.0E—05 1.0000 0.005443 -1.019 No Reject
43 -0.00659 33762 -3.0E-05 1.0000 0.005442 -1.206 No Reject
44 -0.00868 33712 -3.0E-05 1.0000 0.005446 -1.589 No Reject
45 0.013164 33266 -3.0E-05 1.0000 0.005483 2.408 Reject Null
46 -0.00429 33247 -3.0E-05 1.0000 0.005484 -0.777 No Reject
47 -0.01049 32836 -3.0E-05 1.0000 0.005518 -1.895 Reject Null
48 -0.01093 31596 -3.2E-05 1.0000 0.005626 -1.938 Reject Null
49 0.008212 30982 -3.2E-05 1.0000 0.005681 1.451 No Reject
50 -0.01107 30001 -3.3E-05 1.0000 0.005773 -1.912 Reject Null
51 -0.01277 29319 -3.4E-05 1.0000 0.00584 -2.181 Reject Null
52 -0.01896 28262 -3.5E-05 1.0000 0.005948 -3.182 Reject Null
53 -0.00213 27984 -3.6E-05 1.0000 0.005978 -0.350 No Reject
54 -0.00433 27379 -3.7E-05 1.0000 0.006043 -0.710 No Reject
55 0.001988 27094 -3.7E-05 1.0000 0.006075 0.333 No Reject
56 -0.00265 27207 -3.7E-05 1.0000 0.006062 -0.432 No Reject
57 0.011625 26516 -3.8E-05 1.0000 0.006141 1.899 Reject Null
58 0.009968 25926 -3.9E-05 1.0000 0.00621 1.611 No Reject
59 0.014226 25443 -3.9E-05 1.0000 0.006269 2.275 Reject Null
60 0.016168 25116 -4.0E-05 1.0000 0.00631 2.569 Reject Null
61 0.007464 24434 -4.1E-05 1.0000 0.006397 1.173 No Reject
62 0.021142 23872 -4.2E-05 1.0000 0.006472 3.273 Reject Null
63 0.010032 23266 -4.3E-05 1.0000 0.006556 1.537 No Reject
64 0.017329 22762 -4.4E-05 1.0000 0.006628 2.621 Reject Null
65 0.019716 21785 -4.6E-05 1.0000 0.006775 2.917 Reject Null
66 0.007459 21305 -4.7E-05 1.0000 0.006851 1.096 NO Reject
67 0.02376 21242 -4.7E-05 1.0000 0.006861 3.470 Reject Null
68 0.013156 20960 -4.8E-05 1.0000 0.006907 1.912 Reject Null
69 0.011774 20393 -4.9E-05 1.0000 0.007002 1.688 Reject Null
70 0.012056 19760 -5.1E-05 1.0000 0.007114 1.702 Reject Null
71 0.005253 19073 -5.2E-05 1.0000 0.007241 0.733 No Reject
72 -0.00331 18365 -5.4E-05 1.0000 0.007379 -0.441 No Reject
73 0.003498 17827 -5.6E-05 1.0000 0.007489 0.475 No Reject
74 0.007754 16776 -6.0E-05 1.0000 0.00772 1.012 No Reject
75 0.008128 16169 -6.2E-05 1.0000 0.007864 1.041 No Reject
76 -0.00187 15634 -6.4E-05 1.0000 0.007997 -0.226 No Reject
77 -0.01015 14941 -6.7E-05 1.0000 0.008181 -1.233 No Reject
78 0.002393 15090 -6.6E-05 1.0000 0.00814 0.302 No Re'Lect
79 -0.00252 14316 -7.0E-05 1.0000 0.008357 -0.293 No Reject
80 -0.00294 13940 -7.2E-05 1.0000 0.008469 -0.338 No Reject
81 0.001604 13579 -7.4E-05 1.0000 0.008581 0.196 No Reject
82 -0.01965 12874 -7.8E-05 1.0000 0.008813 -2.221 Reject Null

 

 

137

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

83 -0.0074 12626 -7.9E-05 1.0000 0.008899 -0.822 NO Reject
84 -0.00885 12043 -8.3E-05 1.0000 0.009112 -0.963 No Reject
85 -0.0271 12086 -8.3E-05 1.0000 0.009096 -2.970 Reject Null
86 -0.02466 11621 -8.6E-05 1.0000 0.009276 -2.649 Reject Null
87 -0.02765 10953 -9.1E-05 1.0000 0.009555 -2.885 Reject Null
88 -0.04229 10627 -9.4E-05 1.0000 0.0097 -4.350 Reject Null
89 -0.02628 9673 -1.0E-04 0.9999 0.010167 -2.575 Reject Null
90 -0.04212 9295 -1.1E-04 0.9999 0.010372 -4.051 Reject Null
91 -0.0285 8685 -1.2E-04 0.9999 0.01073 -2.645 Reject Null
92 -0.02118 7858 -1.3E-04 0.9999 0.01128 -1.866 Reject Null
93 -0.02129 7517 -1.3E-04 0.9999 0.011533 -1.835 Reject Null
94 -0.02548 7066 -1.4E-04 0.9999 0.011895 -2.130 Reject Null
95 -0.03305 6407 -1.6E-04 0.9999 0.012492 -2.633 Reject Null
96 -0.01913 6186 -1.6E-04 0.9999 0.012713 -1.492 No Reject
97 -0.00486 5851 -1.7E-04 0.9999 0.013072 -0.358 No Reject
98 -0.00172 5492 -1.8E-04 0.9999 0.013493 -0.114 No Reject
99 0.002477 5142 -1.9E-04 0.9999 0.013944 0.192 No Reject
100 0.001268 4996 ~2.0E-04 0.9999 0.014146 0.104 No Reject
101 0.001 101 4727 -2.1E-04 0.9999 0.014543 0.090 NO Reject
102 0.001538 4479 -2.2E-04 0.9999 0.01494 0.1 18 No Reject
103 0.006406 4181 -2.4E-04 0.9999 0.015464 0.430 No Reject
104 0.01 1375 3917 -2.6E-04 0.9999 0.015976 0.728 No Reject
105 0.015433 3616 -2.8E-04 0.9999 0.016627 0.945 No Reject
106 0.023872 3212 -3.1 E-04 0.9998 0.017642 1 .371 N0 Reject
107 0.033281 2861 -3.5E-04 0.9998 0.018692 1.799 Reject Null
108 0.006826 2559 -3.9E-04 0.9998 0.019764 0.365 NO Reject
109 0.025714 2201 -4.5E-04 0.9998 0.02131 1.228 No Reject
1 10 0.009923 1862 -5.4E-04 0.9997 0.023168 0.451 No Reject
111 0.042587 1682 -5.9E-04 0.9997 0.024376 1.772 Reject Null
112 0.054363 1600 -6.3E-04 0.9997 0.024992 2.200 Reject Null
113 0.050044 1379 -7.3E-04 0.9996 0.026919 1.886 Reject Null
114 0.070634 1239 -8.1E-04 0.9996 0.028398 2.516 Reject Null
115 0.066531 1060 -9.4E-04 0.9995 0.0307 2.198 Reject Null
116 0.092052 871 -1.1E-03 0.9994 0.033864 2.752 Reject Null
117 0.049087 837 -1.2E-03 0.9994 0.034545 1.456 No Reject
118 0.069078 728 -1.4E-03 0.9993 0.037037 1.902 Reject Null
119 0.048413 714 -1.4E-03 0.9993 0.037398 1.332 No Reject
120 0.065772 554 -1.8E-03 0.9991 0.042448 1.592 No Reject
121 0.044608 502 -2.0E-03 0.9990 0.044589 1.045 No Reject
122 0.086935 384 -2.6E-03 0.9987 0.050966 1.757 Reject Null
123 -0.00546 325 -3.1E-03 0.9985 0.055387 -0.043 No Reject
124 0.096965 280 -3.6E-03 0.9983 0.059658 1.685 Reject Null
125 -0.04718 226 -4.4E-03 0.9979 0.066378 -0.644 No Reject
126 0.124838 139 -7.2E-03 0.9967 0.084535 1.562 No Reject
127 -0.15894 99 -1.0E-02 0.9954 0.100046 -1.487 NO Reject

 

138

 

 

 

 

 

 

 

 

 

 

 

 

 

1 28 -0.00694 65 -1 .6E-02 0.9935 0.1 23225 No Reject
129 -0.19716 29 -3.6E-02 0.9886 0.183583 N0 Reject
130 -0.08152 21 -5.0E-02 0.9874 0.215473 No Reject
131 -0.0121 22 -4.8E-02 0.9875 0.210536 No Reject
132 -0.10838 5 -2.5E-01 1 .0992 0.491596 No Reject
133 0.153176 1 1 -1.0E-01 0.9958 0.300252 N0 Reject
134 0.18821 5 -2.5E-01 1 .0992 0.491596 N0 Reject

 

 

 

 

 

139

 

 

APPENDIX I

1997 MORAN’S |

 

E
«a

Moran’s I

Pairs

(n)

'50)

SD

score

Hypothesis
Test

 

0.069013

1618

-6.2E-04

0.9997

0.024853

2.802

Reject Null

 

0.076283

4037

-2.5E-04

0.9999

0.015737

4.863

Reject Null

 

0.036198

5958

-1 .7E-04

0.9999

0.012954

2.807

Reject Null

 

0.056766

7436

-1 .3E-04

0.9999

0.011596

4.907

Reject Null

 

0.008137

8738

-1.1E-04

0.9999

0.010697

0.771

No Reject

 

-0.0104

10112

-9.9E-05

1 .0000

0.009944

-1.036

No Reject

 

0.033149

11253

-8.9E-05

1 .0000

0.009426

3.526

Reject Null

 

-0.00692

1 2300

-8.1 E-05

1 .0000

0.009016

-0.758

No Reject

 

0.00463

1 3084

-7.6E-05

1 .0000

0.008742

0.538

No Reject

 

.5
C(OmNODU'I-hUN-A

0.016788

1 4006

-7.1 E-05

1 .0000

0.008449

1 .995

Reject Null

 

—L
—l

-0.00171

1 4839

-6.7E-05

1 .0000

0.008209

-0.200

No Reject

 

.5
N

0.000413

15108

-6.6E-05

1 .0000

0.008135

0.059

No Reject

 

.5
(A)

0.001 1

1 5745

-6.4E-05

1 .0000

0.007969

0.146

No Reject

 

.5
h

-0.00202

16698

-6.0E-05

1 .0000

0.007738

-0.253

No Reject

 

.5
01

-0.0003

1 7343

-5.8E-05

1 .0000

0.007593

-0.031

No Reject

 

.5
0)

-0.00094

18019

-5.6E-05

1 .0000

0.007449

-0.119

No Reject

 

.5
‘1

-0.01893

1 9048

-5.3E-05

1 .0000

0.007245

-2.606

Reject Null

 

.5
CD

-0.00477

19251

-5.2E-05

1 .0000

0.007207

-0.655

No Reject

 

.5
(0

0.004216

20168

-5.0E-05

1 .0000

0.007041

0.606

No Reject

 

N
0

0.008372

20998

-4.8E-05

1 .0000

0.006901

1 .220

No Reject

 

N
.5

-0.00678

21168

-4.7E-05

1 .0000

0.006873

-0.979

No Reject

 

IO
N

0.009283

21744

-4.6E-05

1 .0000

0.006781

1 .376

No Reject

 

N
GO

0.004111

22338

-4.5E-05

1 .0000

0.006691

0.621

No Reject

 

N
h

0.000992

23036

-4.3E-05

1 .0000

0.006589

0.157

No Reject

 

N
01

-0.00207

23782

-4.2E-05

1 .0000

0.006484

-0.312

No Reject

 

IO
0)

0.007415

24641

-4.1 E-05

1 .0000

0.00637

1.170

No Reject

 

N
N

-0.01078

25623

-3.9E-05

1 .0000

0.006247

-1.719

Reject Null

 

N
(D

-0.0099

26177

-3.8E-05

1 .0000

0.006181

-1.595

No Reject

 

N
‘0

-0.01575

26566

-3.8E-05

1 .0000

0.006135

-2.562

Reject Null

 

00
0

-0.01297

27529

-3.6E-05

1 .0000

0.006027

-2.145

Reject Null

 

w
.5

-0.01433

27380

-3.7E-05

1 .0000

0.006043

-2.365

Reject Null

 

0.)
'0

-0.01099

27941

-3.6E-05

1 .0000

0.005982

-1.831

Reject Null

 

(5)
00

-0.00061

28701

-3.5E-05

1 .0000

0.005903

-0.097

No Reject

 

$3

0.002767

28922

-3.5E-05

1 .0000

0.00588

0.476

No Reject

 

35

-0.00818

29258

-3.4E-05

1 .0000

0.005846

-1.394

No Reject

 

36

-0.00854

29546

-3.4E-05

1 .0000

0.005818

-1 .462

No Reject

 

 

37

 

-0.0131 5

 

29495

 

-3.4E-05

 

 

1 .0000

0.005823

 

-2.252

 

Reject Null

 

140

 

 

 

38

-0.00229

29737

-3.4E-05

1 .0000

0.005799

-0.389

No Reject

 

39

-0.01503

29441

-3.4E-05

1 .0000

0.005828

-2.572

Reject Null

 

40

0.004089

29704

-3.4E-05

1 .0000

0.005802

0.711

No Reject

 

41

0.00434

29713

-3.4E-05

1 .0000

0.005801

0.754

No Reject

 

42

-0.00331

29197

-3.4E-05

1 .0000

0.005852

-0.559

No Reject

 

43

-0.00823

29513

-3.4E-05

1 .0000

0.005821

-1.408

No Reject

 

-0.00565

28845

-3.5E-05

1 .0000

0.005888

-0.954

No Reject

 

45

0.00143

28186

-3.5E-05

1 .0000

0.005956

0.246

No Reject

 

46

0.002473

27737

-3.6E-05

1 .0000

0.006004

0.418

No Reject

 

47

0.003722

26796

-3.7E-05

1 .0000

0.006109

0.615

No Reject

 

48

0.015114

26535

-3.8E-05

1 .0000

0.006139

2.468

Reject Null

 

49

-0.00836

26091

-3.8E-05

1 .0000

0.006191

-1.343

No Reject

 

50

0.001 174

25121

-4.0E-05

1 .0000

0.006309

0.192

No Reject

 

51

0.012661

24064

-4.2E-05

1 .0000

0.006446

1.971

Reject Null

 

52

0.012

23623

-4.2E-05

1 .0000

0.006506

1.851

Reject Null

 

53

0.000485

22939

-4.4E-05

1 .0000

0.006602

0.080

No Reject

 

54

0.015464

22530

-4.4E-05

1 .0000

0.006662

2.328

Reject Null

 

55

0.006635

22223

-4.5E-05

1 .0000

0.006708

0.996

No Reject

 

56

0.006586

21869

-4.6E-05

1 .0000

0.006762

0.981

No Reject

 

57

0.0044

21363

-4.7E-05

1 .0000

0.006842

0.650

No Reject

 

58

0.00797

20415

-4.9E-05

1 .0000

0.006999

1.146

No Reject

 

59

0.008482

20379

-4.9E-05

1 .0000

0.007005

1.218

No Reject

 

60

0.00917

20039

-5.0E-05

1 .0000

0.007064

1 .305

No Reject

 

61

0.016697

19500

-5.1 E-05

1 .0000

0.0071 61

2.339

Reject Null

 

62

0.012555

19288

-5.2E-05

1 .0000

0.0072

1.751

Reject Null

 

63

0.001576

1 8949

-5.3E-05

1 .0000

0.007264

0.224

No Reject

 

64

-0.00964

1 8743

-5.3E-05

1 .0000

0.007304

-1.312

No Reject

 

65

-0.01541

1 7837

-5.6E-05

1 .0000

0.007487

-2.050

Reject Null

 

66

0.006569

1 7552

-5.7E-05

1 .0000

0.007548

0.878

No Reject

 

67

-0.00608

17286

-5.8E-05

1 .0000

0.007606

-0.791

No Reject

 

68

-0.00305

1 6435

-6.1 E-05

1 .0000

0.0078

-0.383

No Reject

 

69

-0.01286

1 6257

-6.2E-05

1 .0000

0.007843

-1.632

No Reject

 

70

0.001922

1 5472

-6.5E-05

1 .0000

0.008039

0.247

No Reject

 

71

0.004932

1 4933

-6.7E-05

1 .0000

0.008183

0.611

No Reject

 

72

0.01145

14247

-7.0E-05

1 .0000

0.008378

1 .375

No Reject

 

73

-0.00044

1 3994

-7.1 E-05

1 .0000

0.008453

-0.044

No Reject

 

74

-0.0144

1 3340

-7.5E-05

1 .0000

0.008658

-1 .655

Reject Null

 

75

0.001 135

1 2929

-7.7E-05

1 .0000

0.008794

0.138

No Reject

 

76

-0.00549

1 2353

-8.1 E-05

1 .0000

0.008997

-0.601

No Reject

 

-0.01893

11673

-8.6E-05

1 .0000

0.009255

-2.036

Reject Null

 

78

0.006715

11333

-8.8E-05

1 .0000

0.009393

0.724

No Reject

 

79

-0.02119

10759

-9.3E-05

1 .0000

0.00964

-2.188

Reject Null

 

80

0.002486

1 0228

-9.8E-05

1 .0000

0.009887

0.261

No Reject

 

81

0.00432

1 0302

-9.7E-05

1 .0000

0.009852

0.448

No Reject

 

 

82

 

-0.01514

 

9649

 

-1 .0E-04

 

 

0.9999

0.01018

 

-1 .477

 

No Reject

 

141

 

 

 

83

0.010556

9137

-1.1E-04

0.9999

0.010461

1 .020

No Reject

 

84

0.021661

8655

-1.2E-04

0.9999

0.010748

2.026

Reject Null

 

85

0.013212

8145

-1.2E-04

0.9999

0.01108

1 .204

No Reject

 

86

0.010267

7883

-1.3E-04

0.9999

0.011262

0.923

No Reject

 

87

0.00597

7408

-1 .4E-04

0.9999

0.011618

0.525

No Reject

 

88

-0.00254

7099

-1 .4E-04

0.9999

0.011868

-0.202

No Reject

 

89

-0.03446

6692

-1.5E-04

0.9999

0.012223

-2.807

Reject Null

 

90

-0.01272

6569

-1.5E-04

0.9999

0.012337

-1.019

No Reject

 

91

-0.01365

6176

-1.6E-04

0.9999

0.012724

-1.060

No Reject

 

92

-0.02012

5831

-1.7E-04

0.9999

0.013095

-1.523

No Reject

 

93

-0.00392

5502

-1.8E-04

0.9999

0.01348

-0.277

No Reject

 

94

-0.00398

5384

-1.9E-04

0.9999

0.013627

-0.279

No Reject

 

95

0.022597

4831

-2.1 E-04

0.9999

0.014386

1 .585

No Reject

 

96

-0.0161

4570

-2.2E-04

0.9999

0.014791

-1.074

No Reject

 

97

0.013369

4304

-2.3E-04

0.9999

0.015241

0.892

No Reject

 

98

-0.00548

3926

-2.5E-04

0.9999

0.015958

-0.328

No Reject

 

99

-0.02408

3634

-2.8E-04

0.9999

0.016586

-1 .435

No Reject

 

100

-0.00336

3513

-2.8E-04

0.9999

0.016869

-0.183

No Reject

 

101

-0.02516

3324

-3.0E-04

0.9999

0.017342

-1 .433

No Reject

 

102

-0.02709

3009

-3.3E-04

0.9998

0.018227

-1 .468

No Reject

 

103

-0.02065

2826

-3.5E-04

0.9998

0.018808

-1.079

No Reject

 

104

-0.00469

2562

-3.9E-04

0.9998

0.019753

-0.218

No Reject

 

105

-0.00935

2421

-4.1 E-04

0.9998

0.02032

-0.440

No Reject

 

106

-0.01108

2114

-4.7E-04

0.9998

0.021744

-0.488

No Reject

 

107

-0.04459

1 882

-5.3E-04

0.9997

0.023045

-1.912

Reject Null

 

108

-0.05373

1641

-6.1E-04

0.9997

0.024678

-2.152

Reject Null

 

109

-0.04232

1 337

-7.5E-04

0.9996

0.027338

-1.520

No Reject

 

110

-0.05051

1150

-8.7E-04

0.9996

0.029476

-1 .684

Reject Null

 

111

0.002939

1 035

-9.7E-04

0.9995

0.031069

0.126

No Reject

 

112

-0.03192

797

-1.3E-03

0.9994

0.0354

-0.866

No Reject

 

113

-0.09627

758

-1 .3E-03

0.9993

0.036298

-2.616

Reject Null

 

114

0.012108

627

-1.6E-03

0.9992

0.039905

0.343

No Reject

 

115

-0.07146

456

-2.2E-03

0.9989

0.046779

-1.481

No Reject

 

116

-0.01819

435

-2.3E-03

0.9989

0.047892

-0.332

No Reject

 

117

0.054

367

-2.7E-03

0.9987

0.05213

1 .088

No Reject

 

118

0.042165

331

-3.0E-03

0.9985

0.054884

0.823

No Reject

 

119

0.017391

345

-2.9E-03

0.9986

0.053762

0.378

No Reject

 

120

-0.02589

271

-3.7E-03

0.9982

0.060638

-0.366

No Reject

 

121

0.0431 18

221

-4.5E-03

0.9978

0.067122

0.710

No Reject

 

122

-0.04203

211

-4.8E-03

0.9977

0.068687

-0.543

No Reject

 

123

-0.00919

171

-5.9E-03

0.9972

0.076261

-0.043

No Reject

 

124

0.06923

146

-6.9E-03

0.9968

0.082496

0.923

No Reject

 

125

0.031726

120

-8.4E-03

0.9962

0.090938

0.441

No Reject

 

126

-0.07505

66

-1.5E-02

0.9935

0.1 22298

-0.488

No Reject

 

 

127

 

-0.06887

 

51

 

-2.0E-02

 

0.9921

 

0.13892

 

-0.352

 

No Reject

 

 

142

I! . ‘1’
.4. .
\

 

 

 

 

 

 

 

 

 

128 0.091303 44 -2.3E-02 0.9912 0.149425 0.767 No Reject
129 -0.00868 30 -3.4E-02 0.9888 0.180532 0.143 No Reject
130 0.009625 44 -2.3E-02 0.9912 0.149425 0.220 NO Reject
131 0.039202 33 -3.1E-02 0.9894 0.172228 0.409 No Reject
132 -0.09827 26 -4.0E-02 0.9881 0.193778 -0.301 No Reject
133 0.006994 21 -5.0E-02 0.9874 0.215473 0.265 No Reject
134 -0.02469 10 -1.1E-01 1.0000 0.316228 0.273 No Reject
135 -0.10592 8 -1.4E-01 1.0157 0.359122 0.103 No Reject

 

 

 

 

 

 

 

 

 

143

I .
I

APPENDIX J

1998 MORAN’S I

 

E
on

Moran’s l

Pair
3 (n)

50)

SD

score

Hypothesis
Test

 

-0.30732

43

-2.4E-02

0.9910

0.151131

-1 .876

Reject Null

 

0.023046

161

-6.3E-03

0.9971

0.078581

0.373

No Reject

 

0.154391

172

-5.8E-03

0.9973

0.07604

2.107

Reject Null

 

0.083922

246

-4.1 E-03

0.9980

0.063633

1 .383

No Reject

 

0.014157

280

-3.6E-03

0.9983

0.059658

0.297

No Reject

 

-0.07067

292

-3.4E-03

0.9983

0.058424

-1.151

No Reject

 

-0.10698

319

-3.1 E-03

0.9985

0.055904

-1 .857

Reject Null

 

-0.0212

361

-2.8E-03

0.9987

0.052561

-0.350

No Reject

 

0.018491

365

-2.7E-03

0.9987

0.052273

0.406

No Reject

 

.5
ocooouoamhwm-s

-0.09171

383

-2.6E-03

0.9987

0.051033

-1 .746

Reject Null

 

_L
—L

-0.0801

436

-2.3E-03

0.9989

0.047838

-1.626

No Reject

 

.5
N

0.031669

504

-2.0E-03

0.9990

0.0445

0.756

No Reject

 

.5
0)

0.036761

506

-2.0E-03

0.9990

0.044412

0.872

No Reject

 

.5
h

-0.08724

511

-2.0E-03

0.9990

0.044195

-1 .930

Reject Null

 

.5
01

-0.04177

555

-1.8E-03

0.9991

0.04241

-0.942

No Reject

 

.5
O)

-4.9E-05

591

-1 .7E-03

0.9992

0.0411

0.040

No Reject

 

.5
\I

0.006067

617

-1.6E-03

0.9992

0.040226

0.191

No Reject

 

.5
CD

0.055809

657

-1.5E-03

0.9993

0.038984

1.471

No Reject

 

.5
(0

0.000165

650

-1.5E-03

0.9992

0.039194

0.044

No Reject

 

N
0

0.027171

699

-1.4E-03

0.9993

0.037797

0.757

No Reject

 

N
.5

0.060449

648

-1.5E-03

0.9992

0.039254

1.579

No Reject

 

N
N

0.008086

681

-1.5E-03

0.9993

0.038292

0.250

No Reject

 

N
00

-0.03045

658

-1.5E-03

0.9993

0.038955

-0.743

No Reject

 

N
&

-0.06821

700

-1.4E-03

0.9993

0.03777

-1 .768

Reject Null

 

N
01

0.027359

699

-1 .4E-03

0.9993

0.037797

0.762

No Reject

 

N
0)

-0.05681

748

-1.3E-03

0.9993

0.036539

-1.518

No Reject

 

N
\I

-0.0021 8

810

-1.2E-03

0.9994

0.035115

-0.027

No Reject

 

N
00

0.018382

792

-1.3E-03

0.9994

0.035511

0.553

No Reject

 

N
(0

0.006612

844

-1.2E-03

0.9994

0.034401

0.227

No Reject

 

00
O

-0.06125

962

-1 .0E-03

0.9995

0.032225

-1 .868

Reject Null

 

(IO
.5

0.006059

950

-1.1E-03

0.9995

0.032427

0.219

No Reject

 

00
10

0.005373

934

-1.1E-03

0.9995

0.032704

0.197

No Reject

 

00
OD

-0.04708

941

-1.1E-03

0.9995

0.032582

-1.412

No Reject

 

Q
h

0.053097

935

-1.1 E-03

0.9995

0.032686

1 .657

Reject Null

 

00
01

-0.02796

879

-1.1E-03

0.9994

0.03371

-0.796

No Reject

 

00
0)

0.025814

946

-1.1E-03

0.9995

0.032496

0.827

No Reject

 

(5)
\1

0.003247

909

-1.1E-03

0.9995

0.03315

0.131

No Reject

 

 

38

 

-0.03238

 

907

 

-1.1E-03

 

0.9995

 

0.033186

 

-0.943

 

No Reject

 

 

144

. “'55“ :5!"—
~ ' 'I

 

39

0.007373

914

-1.1E-03

0.9995

0.033059

0.256

No Reject

 

40

0.036672

919

-1.1E-03

0.9995

0.032969

1.145

No Reject

 

41

0.022617

881

-1.1E-03

0.9994

0.033672

0.705

No Reject

 

42

-0.00014

925

-1.1E-03

0.9995

0.032862

0.029

No Reject

 

43

-0.00815

870

-1.2E-03

0.9994

0.033884

-0.207

No Reject

 

-0.05502

841

-1.2E-03

0.9994

0.034462

-1.562

No Reject

 

45

0.00319

908

-1.1E-03

0.9995

0.033168

0.129

No Reject

 

46

0.032541

901

-1.1E-03

0.9995

0.033297

1.011

No Reject

 

47

-0.01167

899

-1.1E-03

0.9994

0.033334

-0.317

No Reject

 

48

-0.04249

820

-1.2E-03

0.9994

0.0349

-1.182

No Reject

 

49

-0.03057

708

-1 .4E-03

0.9993

0.037556

-0.776

No Reject

 

50

-0.01061

737

-1 .4E-03

0.9993

0.036811

-0.251

No Reject

 

51

-0.05794

700

-1 .4E-03

0.9993

0.03777

-1 .496

No Reject

 

52

-0.01676

697

-1 .4E-03

0.9993

0.037851

-0.405

No Reject

 

53

0.006491

690

-1.5E-03

0.9993

0.038042

0.209

No Reject

 

54

0.009332

646

-1.6E-03

0.9992

0.039314

0.277

No Reject

 

55

0.016761

629

-1.6E-03

0.9992

0.039841

0.461

No Reject

 

56

0.026188

684

-1.5E-03

0.9993

0.038208

0.724

No Reject

 

57

0.062292

702

-1.4E-03

0.9993

0.037716

1 .689

Reject Null

 

58

-0.00771

667

-1.5E-03

0.9993

0.038692

-0.160

No Reject

 

59

-0.00795

657

-1.5E-03

0.9993

0.038984

-0.165

No Reject

 

60

0.041778

616

-1.6E-03

0.9992

0.040259

1 .078

No Reject

 

61

-0.02844

572

-1.8E-03

0.9991

0.041776

-0.639

No Reject

 

62

-0.01319

587

-1.7E-03

0.9992

0.04124

-0.279

No Reject

 

63

-0.00407

588

-1.7E-03

0.9992

0.041205

-0.057

No Reject

 

-0.05184

581

-1.7E-03

0.9992

0.041452

-1.209

No Reject

 

65

-0.04433

517

-1.9E-03

0.9991

0.043938

-0.965

No Reject

 

66

-0.01619

574

-1 .7E-03

0.9991

0.041703

-0.346

No Reject

 

67

-0.05008

557

-1.8E-03

0.9991

0.042334

-1.141

No Reject

 

68

0.020336

525

-1.9E-03

0.9991

0.043603

0.510

No Reject

 

69

0.047994

484

-2.1 E-03

0.9990

0.045409

1.103

No Reject

 

70

0.054733

479

-2.1 E-03

0.9990

0.045644

1 .245

No Reject

 

71

-0.01964

434

-2.3E-03

0.9989

0.047947

-0.361

No Reject

 

72

0.096656

411

-2.4E-03

0.9988

0.049268

2.011

Reject Null

 

73

0.013296

425

—2.4E-03

0.9989

0.048451

0.323

No Reject

 

74

-0.01862

351

-2.9E-03

0.9986

0.053302

-0.296

No Reject

 

75

0.102741

383

-2.6E-03

0.9987

0.051033

2.065

Reject Null

 

76

-0.05773

381

-2.6E-03

0.9987

0.051166

-1.077

No Reject

 

77

0.011362

368

-2.7E-03

0.9987

0.05206

0.271

No Reject

 

78

-0.00376

350

-2.9E-03

0.9986

0.053378

-0.017

No Reject

 

79

0.006808

375

-2.7E-03

0.9987

0.051573

0.184

No Reject

 

80

-0. 07051

323

-3.1 E-03

0.9985

0.055558

-1.213

No Reject

 

81

-0.03654

280

-3.6E-03

0.9983

0.059658

-0.552

No Reject

 

82

0.051508

283

-3.5E-03

0.9983

0.059342

0.928

No Reject

 

 

83

 

-0.01786

 

287

 

-3.5E-03

 

0.9983

 

0.058929

 

-0.244

 

No Reject

 

145

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

84 0.043182 264 -3.8E-03 0.9982 0.061433 0.765 No Reject
85 0.043408 279 -3.6E-03 0.9983 0.059765 0.787 No Reject
86 -0.00244 280 -3.6E-03 0.9983 0.059658 0.019 No Reject
87 0.014153 235 -4.3E-03 0.9980 0.0651 0.283 No Reject
88 0.0446 220 -4.6E-03 0.9978 0.067274 0.731 NO Reject
89 -0.12742 225 -4.5E—03 0.9979 0.066525 -1.848 Reject Null
90 -0.02758 224 -4.5E-03 0.9979 0.066673 -0.346 No Reject
91 -0.00696 194 -5.2E-03 0.9976 0.07162 -0.025 No Reject
92 -0.0725 169 -6.0E-03 0.9972 0.076709 -0.868 No Reject
93 -0.01423 171 -5.9E-03 0.9972 0.076261 -0.109 No Reject
94 0.033785 175 -5.7E-03 0.9973 0.075389 0.524 NO Reject
95 0.051661 164 -6.1E-03 0.9971 0.077863 0.742 NO Reject
96 0.020284 164 -6.1E-03 0.9971 0.077863 0.339 No Reject
97 0.131212 146 -6.9E-03 0.9968 0.082496 1.674 Reject Null
98 0.01185 126 -8.0E-03 0.9963 0.088761 0.224 No Reject
99 -0.04232 97 -1.0E-02 0.9954 0.101064 -0.316 No Reject
100 -0.06618 108 -9.3E-03 0.9958 0.09582 -0.593 No Reject
101 0.087114 99 -1.0E-02 0.9954 0.100046 0.973 No Reject
102 -0.10381 112 -9.0E-03 0.9959 0.094106 -1.007 No Reject
103 0.117165 87 -1.2E-02 0.9949 0.106664 1.207 No Reject
104 0.034414 92 -1.1E-02 0.9951 0.103751 0.438 No Reject
105 -0.17875 78 -1.3E-02 0.9944 0.112593 -1.472 No Reject
106 -0.0328 52 -2.0E-02 0.9922 0.137593 -0.096 No Reject
107 0.06158 46 -2.2E-02 0.9915 0.146182 0.573 No Reject
108 #029661 43 -2.4E-02 0.9910 0.151131 -1.805 Reject Null
109 0.066982 44 -2.3E-02 0.9912 0.149425 0.604 No Reject
110 -0.2221 35 -2.9E-02 0.9897 0.167296 -1.152 No Reject
111 -0.04213 32 -3.2E-02 0.9892 0.174866 -0.056 No Reject
112 0.103687 24 -4.3E-02 0.9878 0.201624 0.730 No Reject
113 0.347942 14 -7.7E-02 0.9897 0.264506 1.606 No Reject
114 -0.07501 22 -4.8E-02 0.9875 0.210536 -0.130 No Reject
1 15 0.16843 15 -7.1E-02 0.9888 0.255301 0.940 No Reject
116 -0.00323 19 -5.6E-02 0.9874 0.22653 0.231 No Reject
117 -0.11062 29 -3.6E-02 0.9886 0.183583 -0.408 NO Reject
118 -0.26904 15 -7.1E-02 0.9888 0.255301 -0.774 No Reject
119 0.024517 22 -4.8E-02 0.9875 0.210536 0.343 No Reject
120 0.561431 18 -5.9E-02 0.9875 0.232765 2.665 Reject Null
121 -0.1349 14 -7.7E-02 0.9897 0.264506 -0.219 No Reject
122 0.131594 9 -1.3E-01 1.0062 0.33541 0.765 No Reject
123 0.315564 13 -8.3E-02 0.9910 0.274863 1.451 No Reject
124 0.067272 6 -2.0E-01 1.0556 0.430946 0.620 No Reject
125 0 1.0E-l-00 #NUM! #NUM! #NUM! No Reject
126 0.209432 6 -2.0E-01 1.0556 0.430946 0.950 No Reject
127 0.019853 4 -3.3E-01 1.1832 0.591608 0.597 No Reject
128 -0.23952 2 -1.0E+00 1.9149 1.354006 0.562 No Reject

 

146

 

 

 

129

-0.13529

-1.0E+00

1.9149

1 .354006

0.639

No Reject

 

 

130

 

0.008527

 

 

-1.0E+00

 

1.9149

 

1 .354006

 

0.745

 

No Reject

 

147

 

 

Pearson’s Bivariate Correlations at the Census Tract Scale n = 41

APPENDIX K

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AV LNAV AR S LNS WHITE BLACK

AV r 1.000 0.975 -0.537 0.996 0.975 0.433 -0.252

P . 0.000 0.000 0.000 0.000 0.005 0.112

LNAV r 0.975 1.000-0.552 0.963 0.995 0.472 -0.278

P 0.000. 0.000 0.000 0.000 0.002 0.079

AR r -0.537 -0.552 1.000 -0.588 -0.622 -0.382 0.273

P 0.000 0.000. 0.000 0.000 0.014 0.084

S r 0.996 0.963 -0.588 1.000 0.971 0.420 -0.241

P 0.000 0.000 0.000. 0.000 0.006 0.129

LNS r 0.975 0.995 -0.622 0.971 1.000 0.460 -0.263

P 0.000 0.000 0.000 0.000. 0.002 0.097

WHITE r 0.433 0.472 -0.382 0.420 0.460 1.000 -0.955

P 0.005 0.002 0.014 0.006 0.002. 0.000

BLACK r -0.252 -0.278 0.273 -0.241 -0.263 -0.955 1.000
P 0.112 0.079 0.084 0.129 0.097 0.000.

HISPANIC r -0.680 -0.731 0.480 -0.680 -0.748 -0.287 0.022

P 0.000 0.000 0.001 0.000 0.000 0.069 0.892

VACANT -0.575 -0.631 0.425-0.553-0.618 -0.441 0.265

P 0.000 0.000 0.006 0.000 0.000 0.004 0.094

RENTOCC r -0.546 -0.566 0.334 -0.527—0.548 -0.345 0.219

P 0.000 0.000 0.033 0.000 0.000 0.027 0.169

LNINC r 0.808 0.860 -0.528 0.793 0.851 0.461 -0.269

P 0.000 0.000 0.000 0.000 0.000 0.002 0.089

INC r 0.889 0.899 -0.531 0.879 0.894 0.421 -0.225

P 0.000 0.000 0.000 0.000 0.000 0.006 0.156

PUBLIC r -0.724 0.798 0.555-0.710 -0.798 -0.640 0.453

P 0.000 0.000 0.000 0.000 0.000 0.000 0.003

YEAR r 0.378 0.375 -0.027 0.354 0.362 0.055 0.052

P 0.015 0.016 0.869 0.023 0.020 0.733 0.745

 

 

148

 

CensusTract Scale Pearson Bivariate Correlations contd.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HISPANIC VACANT RENTOCC LNINC INC PUBLIC YEAR

AV r -0.680 -0.575 -0.546 0.808 0.889 -0.724 0.378

P 0.000 0.000 0.000 0.000 0.000 0.000 0.015

LNAV r -0.731 -0.631 -0.566 0.860 0.899 -0.798 0.375

P 0.000 0.000 0.000 0.000 0.000 0.000 0.016

AR r 0.480 0.425 0.334 -0.528 -0.531 0.555 -0.027

P 0.001 0.006 0.033 0.000 0.000 0.000 0.869

S r -0.680 -0.553 -0.527 0.793 0.879 -0.71 0 0.354

P 0.000 0.000 0.000 0.000 0.000 0.000 0.023

LNS r -0.748 0.618 -0.548 0.851 0.894 -0.798 0.362

P 0.000 0.000 0.000 0.000 0.000 0.000 0.020

WHITE r -0.287 -0.441 -0.345 0.461 0.421 -0.640 0.055

P 0.069 0.004 0.027 0.002 0.006 0.000 0.733

BLACK r 0.022 0.265 0.219 -0.269 -0.225 0.453 0.052

P 0.892 0.094 0.169 0.089 0.156 0.003 0.745

HISPANIC r 1.000 0.501 0.312 -0.599 -0.638 0.654 -0.433

P . 0.001 0.047 0.000 0.000 0.000 0.005

VACANT r 0.501 1.000 0.775 -0.815 -0.778 0.720 -0.219

P 0.001 . 0.000 0.000 0.000 0.000 0.169

RENTOCC r 0.312 0.775 1.000 -0.774 -0.770 0.590 0.143

P 0.047 0.000 . 0.000 0.000 0.000 0.374

LNINC r -0.599 -0.815 -0.774 1 .000 0.967 -0.888 0.187

P 0.000 0.000 0.000 . 0.000 0.000 0.241

INC r -0.638 -0.778 -0.770 0.967 1 .000 -0.832 0.263

P 0.000 0.000 0.000 0.000 . 0.000 0.096

PUBLIC r 0.654 0.720 0.590 -0.888 -0.832 1.000 -0.290

P 0.000 0.000 0.000 0.000 0.000 . 0.066

YEAR r -0.433 -0.219 0.143 0.187 0.263 -0.290 1 .000
P 0.005 0.169 0.374 0.241 0.096 0.066 .

 

 

 

 

 

 

 

 

 

149

 

 

Pearson Bivariate Correlations at the Block Group Scale 11 = 117

APPENDIX L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AV LNAV AR S LNS WHITE BLACK

AV r 1.000 0.976 -0.364 0.993 0.967 0.395 -0.220

P . 0.000 0.000 0.000 0.000 0.000 0.017

LNAV r 0.976 1.000 -0.395 0.966 0.991 0.437 -0.243

P 0.000 . 0.000 0.000 0.000 0.000 0.008

AR r -0.364 -0.395 1.000 -0.451 -0.503 -0.341 0.252

P 0.000 0.000 . 0.000 0.000 0.000 0.006

S r 0.993 0.966 -0.451 1.000 0.972 0.391 -0.217

P 0.000 0.000 0.000 . 0.000 0.000 0.019

LNS r 0.967 0.991 -0.503 0.972 1.000 0.438 -0.243

P 0.000 0.000 0.000 0.000 . 0.000 0.008

WHITE r 0.395 0.437 -0.341 0.391 0.438 1.000 -0.948

P 0.000 0.000 0.000 0.000 0.000 . 0.000

BLACK r -0.220 -0.243 0.252 -0.217 -0.243 -0.948 1.000
P 0.017 0.008 0.006 0.019 0.008 0.000 .

HISPANIC r -0.679 -0.746 0.380 -0.680 -0.754 -0.396 0.120

P 0.000 0.000 0.000 0.000 0.000 0.000 0.196

VACANT r -0.478 -0.533 0.400 -0.475 -0.537 -0.418 0.261

P 0.000 0.000 0.000 0.000 0.000 0.000 0.005

RENTOCC r -0.442 -0.469 0.292 -0.437 -0.465 -0.382 0.250

P 0.000 0.000 0.001 0.000 0.000 0.000 0.007

LNINC r 0.702 0.736 -0.436 0.701 0.736 0.438 -0.271

P 0.000 0.000 0.000 0.000 0.000 0.000 0.003

INC r 0.758 0.769 -0.420 0.758 0.769 0.383 -0.215

P 0.000 0.000 0.000 0.000 0.000 0.000 0.020

PUBLIC r -0.650 -0.710 0.421 -0.650’-0.715 -0.612 0.461

P 0.000 0.000 0.000 0.000 0.000 0.000 0.000

YEAR r 0.483 0.462 -0.033 0.432 0.437 0.046 0.059

P 0.000 0.000 0.727 0.000 0.000 0.621 0.526

 

 

 

 

 

 

 

 

 

150

 

 

Block group scale Pearson Bivariate Correlation contd.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HISPANIC VACANT RENTOCC LNINC INC PUBLIC YEAR

AV r -0.679 -0.478 -0.442 0.702 0.758 -0.650 0.463

P 0.000 0.000 0.000 0.000 0.000 0.000 0.000

LNAV r -0.746 -0.533 -0.469 0.736 0.769 -0.710 0.462

P 0.000 0.000 0.000 0.000 0.000 0.000 0.000

AR r 0.380 0.400 0.292 -0.436 -0.420 0.421 -0.033

P 0.000 0.000 0.001 0.000 0.000 0.000 0.727

S r -0.680 -0.475 -0.437 0.701 0.758 -0.650 0.432

P 0.000 0.000 0.000 0.000 0.000 0.000 0.000

LNS r -0.754 -0.537 -0.465 0.736 0.769 -0.715 0.437

P 0.000 0.000 0.000 0.000 0.000 0.000 0.000

WHITE r -0.396 -0.418 -0.382 0.438 0.383 -0.612 0.046

P 0.000 0.000 0.000 0.000 0.000 0.000 0.621

BLACK r 0.120 0.261 0.250 -0.271-0.215 0.461 0.059

P 0.196 0.005 0.007 0.003 0.020 0.000 0.526

HISPANIC r 1.000 0.475 0.369 -0.555 -0.561 0.584 -0.364

P. 0.000 0.000 0.000 0.000 0.000 0.000

VACANT r 0.475 1.000 0.739 -0.742 -0.714 0.665 -0.222

P 0.000. 0.000 0.000 0.000 0.000 0.016

RENTOCC r 0.369 0.739 1.000 -0.753 -0.740 0.615 0.074

P 0.000 0.000. 0.000 0.000 0.000 0.428

LNINC r -0.555 -0.742 -0.753 1.000 0.973 -0.804 0.241

P 0.000 0.000 0.000. 0.000 0.000 0.009

INC r -0.561 -0.714 -0.740 0.973 1.000 -0.760 0.259

P 0.000 0.000 0.000 0.000. 0.000 0.005

PUBLIC r 0.584 0.665 0.615 -0.804 -0.760 1.000 -0.235

P 0.000 0.000 0.000 0.000 0.000. 0.011

YEAR r -0.364 -0.222 0.074 0.241 0.259 -0.235 1.000
P 0.000 0.016 0.428 0.009 0.005 0.011 .

 

 

 

 

 

 

 

 

 

151

 

APPENDIX M

Pearson Bivariate Correlations - Parcel Scale n = 8056

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AV LNAV AR S LNS WHITE BLACK

AV r 1.000 0.949 0.083 0.926 0.868 0.281 -0.158

p . 0.000 0.000 0.000 0.000 0.000 0.000

LNAV r 0.949 1 .000 0.088 0.876 0.913 0.309 -0.171

p 0.000 . 0.000 0.000 0.000 0.000 0.000

AR r 0.083 0.088 1.000 -0.254 -0.322 -0.063 0.045

p 0.000 0.000 . 0.000 0.000 0.000 0.000

S r 0.926 0.876 -0.254 1.000 0.940 0.282 -0.159

p 0.000 0.000 0.000 . 0.000 0.000 0.000

LNS r 0.868 0.913 -0.322 0.940 1.000 0.317 -0.180

p 0.000 0.000 0.000 0.000 . 0.000 0.000

WHITE r 0.281 0.309 -0.063 0.282 0.317 1.000 -0.952

p 0.000 0.000 0.000 0.000 0.000 . 0.000

BLACK r -0.158 -0.171 0.045 -0.159 -0.180 -0.952 1.000
p 0.000 0.000 0.000 0.000 0.000 0.000 .

HISPANIC r -0.499 -0.548 0.077 -0.506 -0.552 -0.418 0.160

p 0.000 0.000 0.000 0.000 0.000 0.000 0.000

VACANT r -0.362 -0.405 0.067 -0.362 -0.407 -0.381 0.235

p 0.000 0.000 0.000 0.000 0.000 0.000 0.000

RENTOCC r -0.329 -0.355 0.057 -0.332 -0.357 -0.414 0.293

p 0.000 0.000 0.000 0.000 0.000 0.000 0.000

LNINC r 0.552 0.576 -0.093 0.561 0.582 0.396 -0.235

p 0.000 0.000 0.000 0.000 0.000 0.000 0.000

INC r 0.597 0.603 -0.087 0.606 0.606 0.353 -0.196

p 0.000 0.000 0.000 0.000 0.000 0.000 0.000

PUBLIC r -0.501 -0.546 0.087 -0.507 -0.553 -0.652 0.509

p 0.000 0.000 0.000 0.000 0.000 0.000 0.000

YEAR r 0.395 0.400 0.003 0.372 0.376 -0.019 0.1 17

p 0.000 0.000 0.810 0.000 0.000 0.082 0.000

 

152

 

Parcel Scale Pearson Bivariate Correlations contd.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HISPANIC VACAN RENTOCC LNINC INC PUBLIC YEAR
T

AV r -0.499 -0.362 -0.329 0.552 0.597 -0.501 0.395
p 0.000 0.000 0.000 0.000 0.000 0.000 0.000
LNAV r -0.548 -0.405 -0.355 0.576 0.603 -0.546 0.400
p 0.000 0.000 0.000 0.000 0.000 0.000 0.000
AR r 0.077 0.067 0.057 -0.093 -0.087 0.087 0.003
p 0.000 0.000 0.000 0.000 0.000 0.000 0.810
S r -0.506 -0.362 -0.332 0.561 0.606 -0.507 0.372
p 0.000 0.000 0.000 0.000 0.000 0.000 0.000
LNS r -0.552 -0.407 -0.357 0.582 0.606 -0.553 0.376
p 0.000 0.000 0.000 0.000 0.000 0.000 0.000
WHITE r -0.418 -0.381 -0.414 0.396 0.353 -0.652 -0.019
p 0.000 0.000 0.000 0.000 0.000 0.000 0.082
BLACK r 0.160 0.235 0.293 -0.235 -0.196 0.509 0.117
p 0.000 0.000 0.000 0.000 0.000 0.000 0.000
HISPANIC r 1.000 0.491 0.389 -0.575 -0.570 0.631 -0.327
p . 0.000 0.000 0.000 0.000 0.000 0.000
VACANT r 0.491 1.000 0.674 -0.701 -0.674 0.602 -0.271
p 0.000 . 0.000 0.000 0.000 0.000 0.000
RENTOCC r 0.389 0.674 1.000 -0.721 -0.711 0.566 0.056
p 0.000 0.000 . 0.000 0.000 0.000 0.000
LNINC r -0.575 -0.701 -0.721 1.000 0.972 -0.744 0.333
p 0.000 0.000 0.000 . 0.000 0.000 0.000
INC r -0.570 -0.674 -0.71 1 0.972 1.000 -0.714 0.336
p 0.000 0.000 0.000 0.000 . 0.000 0.000
PUBLIC r 0.631 0.602 0.566 -0.744 -0.714 1.000 -0.285
p 0.000 0.000 0.000 0.000 0.000 . 0.000
YEAR r -0.327 -0.271 0.056 0.333 0.336 -0.285 1 .000

p 0.000 0.000 0.000 0.000 0.000 0.000 .

 

 

 

 

 

 

 

 

 

153

 

 

BIBLIOGRAPHY

154

'r-' ‘

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