HEDONIC MODELLING OF HOUSING MARKETS
USING GEOGRAPHICAL INFORMATION SYSTEM
(GIS) AND SPATIAL STATISTICS:
A CASE STUDY OF GLASGOW, SCOTLAND
By
DR. SURIATINI ISMAIL*
Fakulti Kejuruteraan & Sains Geoinformasi
Universiti Teknologi Malaysia
81310 Skudai, Johor Bahru, Johor,
Malaysia
e-mail: suriatini@fksg.utm.my
PROFESSOR BRYAN D. MACGREGOR
Department of Property
University of Aberdeen Business School
Edward Wright Building
AB24 3QY Aberdeen
Scotland
e-mail: b.d.macgregor@abdn.ac.uk
ABSTRACT
This paper presents the results of a simultaneous consideration of detailed
accessibility measures and spatial autocorrelation in house price hedonic modelling.
It illustrates the application of GIS and spatial statistics in the estimation of hedonic
models for the entire housing market in Glasgow, Scotland, using 2,715 house prices
for 2002 and 61 independent variables. GIS is used in this study to construct spatial
variables including detailed accessibility measures, to help detect spatial
autocorrelation, and for map visualisation. Spatial statistics are used to test formally
and model explicitly the spatial autocorrelation. The results suggest that an
individual accessibility measure is more influential than a zonal accessibility
measure because the former is able to capture the micro effect of location on house
price. Furthermore, the application of spatial statistics can produce more accurate,
robust and reliable estimates of implicit prices.
Keywords: Hedonic modelling, house prices, GIS, spatial statistics.
________________________________________
Dr Suriatini Ismail is lecturer at UTM Skudai Malaysia and research fellow in the Aberdeen
University Business School. Professor MacGregor is Chair of Property in the University of
Aberdeen and Editor of Journal of Property Research. The authors would like to thank David R
Green of University of Aberdeen for his GIS expertise lent throughout this study. Thanks are also
due to all the data providers and individuals who have involved in this study. * corresponding
author.
1
BACKGROUND
The property market consists of residential (or housing), commercial, and
agricultural sectors. Housing markets constitute a major component of the real
estate market. Second-hand units constitute considerably larger portion than the
supply of new-build at any time (Leishman, 2003, 115). According to Hwang and
Quigley (2004), owner-occupied housing is a substantial fraction of aggregate wealth
in most advance countries. Thus, house prices provide important economic
indicators. For example, in the UK, the housing sector accounts for slightly more
than half of the nation’s fixed capital stock (Muth and Goodman, 1989). It occupies
about 40 percent of UK urban land (Field and MacGregor, 1987, 54). The
importance of house prices as a leading economic indicator (Barret and Blair, 1982,
168) nationally and locally indicates that house price analysis is an important aspect
of property economics.
House prices are an important consideration when assessing macroeconomic and
financial developments in the UK (Thwaites and Wood, 2003) and other developed
countries such as the USA. House price indices are used by the government and
private sectors in policy evaluation and implementation. Models of housing prices
are commonly estimated on national statistics (Goodman, 1998). This can be based
on time series, cross-sectional, or panel (a combination of both) data. For example, in
the current UK’s economic policy climate, house price appreciation rates are used as
a barometer for more general inflationary pressures and are thought to be one of the
important indicators consulted by the Monetary Policy Committee (Costello and
Watkins, 2002). This normally involves time series analysis. Housing prices also act
as a sensitive barometer for many social phenomena such as crime, congestion, job
opportunities, and demographics (Pace and LeSage, 2004a, 180). This normally
involves cross-sectional analysis. Thus, house price analysis is an important element
in housing economics.
The use of econometric/economic models including hedonic modelling has become
an established part of not only the policy framework employed by both the Treasury
and the Bank of England (Meen and Meen, 2003), but also of housing market
analysis. In his review of hedonic price modelling, Malpezzi (2003, 84) describes
hedonic modelling as having been applied in every permanently inhabited region of
the globe. Indicating the established state of the technique, he concludes that over the
2
past three decades, hedonic estimation has clearly matured from a new technology to
become the standard way economists deal with housing heterogeneity (Malpezzi,
2003, 87). Watkins (1998) also notes the dominance of hedonic modelling in the real
estate literature.
According to Hoesli and MacGregor (2000, 64), the hedonic method has been widely
used in the USA, and also used in other countries such as Switzerland and Taiwan
for constructing price indices. Lum (2004) also implies that this technique has been
used in several Commonwealth countries including Hong Kong and Malaysia. In the
UK, the technique is used in the creation of the Nationwide Anglia Building Society
and Halifax Bank of Scotland (HBOS) price indices (Lum, 2004; Watkins, 1998),
which are the major sources of regional and national house price data in the UK. The
Office of Deputy Prime Minister (ODPM) monthly house price index which was
launched in September 2003 is also based on hedonic price (Barker Review, 2004).
Nonetheless, Can and Megbolugbe (1997) highlight that a major limitation of
currently available house price indices constructed based on hedonic price models is
their insensitivity to the geographic location of dwellings within the metropolitan
area.
House price hedonic analysis is undertaken by regressing usually, the transaction
prices of properties against the corresponding property characteristics (Fletcher et al.,
2000b), which are categorised as structural, accessibility, and neighbourhood in this
study. The accessibility and the neighbourhood characteristics comprise mainly
location-related factors. Acknowledging the importance of location, Gallimore et al.
(1996b, 18) state that …locationally sensitive models are…statistically defensible
means of reviewing values and valuation. Given that house prices are the most
widely used measure of property values, this indicates that house price hedonic
modelling should consider spatial elements. In addition, according to Orford (2000),
if the hedonic house price function is to generate estimates that properly reflect the
implicit price of attributes, the model specification must capture sufficiently the
spatial elements at the local market level. Therefore, other than Can and Megbolugbe
(1997), Gallimore et al. (1996b) and Orford (2000) also indicate the importance of
proper consideration of spatial elements in hedonic price modelling.
3
Technically, improper consideration of spatial elements contributes to a substantial
portion of the unexplained variability of price in the hedonic model and leads to
problems. Des Rosiers et al. (2001) outline three main sources of problems to
comprise multicollinearity, heteroscedasticity and spatial autocorrelation. While the
first two can happen in both time series and cross sectional data, the last one is
specifically related to the cross sectional data. Thus, all the three problems can occur
in a cross sectional analysis of house prices. Accordingly, it is important to consider
these problems in housing market analysis if the results are not to be invalidated.
Given that a cross-sectional analysis of house prices involves geographical
information, it is important to give attention to the spatial elements. In considering
the spatial elements in house price hedonic modelling, suitable tools are required.
Two appropriate tools are Geographical Information System (GIS) and spatial
statistics.
The applications of GIS in real estate were established in the USA. These have
started to develop in the UK since the early 1990s. There is evidence of GIS
applications in residential, commercial and rural sectors. Residential has shown to be
the sector with most of the identified research, particularly since the late 1990s. GIS
is a relevant technology for housing markets analysis as all residential real estate
information is inherently spatial because housing is fixed in geographic space
(Belsky et al., 1998). So, spatial data1 is one of the features of residential property.
GIS has the advantages of efficient data integration and spatial analysis (Hamid,
2002).
Spatial analysis functions differentiate GIS from other data management systems.
For example, network analysis can improve the practice of distance measurement
from merely the straight-line to road network. It also offers a function to calculate
minimum travelling time via a transportation network (Des Rosiers et al., 2001). The
representation of spatial data and model results within a GIS could lead to an
improved understanding both of the attributes being examined and of the procedures
used to examine them (Fotheringham and Rogerson, 1994). Thus, GIS is relevant to
this study because it can deal with spatial elements efficiently. Nevertheless, GIS is
not yet a perfect tool for considering spatial elements in housing market analysis.
This is because, GIS is conventionally a tool for data handling and thus, in its generic
1
In this study, the term “spatial data” refers to the map and attributes describing the map.
4
form, it would not deal with spatial autocorrelation. Recent real estate studies that
use GIS revealed the involvement of spatial statistics to deal specifically with spatial
autocorrelation. So, a combination of GIS and spatial statistics can be beneficial for
effective hedonic modelling of real estate markets.
The literature shows that the study of spatial aspects of hedonic modelling falls under
the umbrella of spatial econometrics2, a sub-field of spatial statistics (Anselin,
1988). Anselin (1988, 7) defines spatial econometrics as the collection of techniques
that deal with the peculiarities caused by space in the statistical analysis of regional
science models. According to him, the emphasis on the model as the starting point
differentiates spatial econometrics from the broader field of spatial statistics,
although they share a common methodology framework. However, this study does
not differentiate the two terms and uses them interchangeably. Spatial econometrics
(and/or spatial statistics) are relevant to this study because it explicitly accounts for
the influence of space in real estate modelling (Wilhelmsson, 2002a). The spatial
effects are of two types, namely spatial autocorrelation and spatial heterogeneity3.
Spatial autocorrelation is a weaker expression for spatial dependence (Wilhelmsson,
2002a). In his review on spatial effects and real estate, Wilhelmsson (2002a) states
that before 1990, the problems of the existence of spatial effects have been ignored in
real estate analysis. However, they seem to be gaining more attention from
researchers in the past few years (Anselin, 2002).
Pace and Barry (1997a) assert that regression is perhaps the most often used
technique in statistics. Pointing towards the established state of regression analysis,
Kim et al., (2003) note that much research has been carried out to solve specific
econometric issues pertaining to hedonic regression such as functional form,
identification and statistical efficiency. However, while the applications of classical
statistics in real estate research date back to the early 1970s, spatial statistics were an
2
Anselin (1988) outlines five characteristics of spatial econometrics proposed by Paelink and
Klaaseen (1979, 5-9) as follows:
- The role of spatial interdependence in spatial models
- The asymmetry in spatial relations
- The importance of explanatory factors located in other spaces
- Differentiation between ex post and ex ante interaction
- Explicit modelling of space
3
Spatial heterogeneity refers to the variation in the relationship under study across space (Patton and
McErlean, 2003) or the systematic variation in the behaviour of a given process across space (Can,
1990). It usually leads to heteroscedastic error terms, thus violating the assumption of
homoscedasticity in the classical regression model (Can, 1990).
5
addition to the statistics literature only ten years later (Cressie, 1989). More
importantly, it was only in the late 1990s that the use of spatial statistics started to
gain the attention of many researchers4.
However, very few studies come from the UK. Similarly, the importance of spatial
dependency on the efficiency and consistency of hedonic model estimates has only
very recently started to receive some attention (Kim et al., 2003). Cressie (1989)
believes that spatial prediction is just as important as temporal prediction. However,
Anselin and Bera (1998) state that generally, econometric theory and practice have
been dominated by a focus on the time dimension. They criticise that in stark
contrast to the voluminous literature on serial dependence over time, there is scant
attention paid to its counterpart in cross sectional data, spatial autocorrelation
(Anselin and Bera, 1998, 237). In the UK, the consideration of spatial dependence in
the housing market studies is not obvious. Day (2003) considers spatial
autocorrelation in his study but provides no spatial hedonic model for the entire
market of GCC5.
On one hand, examining spatial dependency as a hedonic problem could portray it as
a methodological disadvantage. On the other hand, it can give information on spatial
pattern structure and process (Overmars et al., 2003) when explicitly specified in a
spatial model. Spatial models are generally specified as linear regression models
with spatial interdependence taking the form of a linear additive relationship of
observations on neighbours (Wilhelmsson, 2002a, 95). This is based on the first law
of geography (Tobler, 1970), which states that everything is related to everything
else, but closer things more so.
Therefore, data that are close together are usually more correlated than data that are
far apart (Cressie, 1989). Based on this, Anselin and Bera (1998, 240) suggest that
spatial dependence is a rule rather than an exception. Supporting this, Bowen et al.
4
For example, Pace et al. (forthcoming), Wilhelmsson (2004), Tu et al. (2004), Dawkins (2004), Day
(2003), Cano-Quervos et al. (2003), Brasington (2002), Besner (2002), Bowen et al. (2002), Deddis
(2002), Tse (2002), Wilhelmsson (2002a), Paez et al. (2001), Quercia et al. (2000), Gillen et al.
(2001), Pearson (2001), Carter and Haloupek (2000), Deddis et al. (2000), Figueroa (1999), Dubin et
al. (1999), Dubin (1998, 1992, 1988), Can and Megbolugbe (1997), Can (1992, 1990), Wiltshaw
(1996), Olmo (1995), Pace et al. (1998a, 1998b), Pace and Gilley (1997), Pace and Barry (1997), Pace
(1997), and Basu and Thibodeau (1998).
5
Day (2003) uses General Method of Moment (GMM), which Bell and Bockstael (2000) contend to
be less effective than the Maximum Likelihood approach adopted in this study.
6
(2001) stress that spatial diagnostics need to be included as part of the test modelfitting procedure for hedonic house price applications. Anselin6 (1998) contends that,
despite widespread recognition by both theorists and practitioners of the complex
roles of location and spatial interaction and the resulting geographically segmented
nature of real estate markets, an explicit spatial treatment of these markets in
empirical research is still in its infancy. Bowen et al. (2001, 467) note that many
applications of hedonic housing price models have not included recent advances in
spatial analysis that control for spatial dependence and heterogeneity. This provides
an opportunity for real estate research.
Realising the lack of evidence of simultaneous consideration of spatial elements in
hedonic price modelling, particularly in the UK, this paper focuses on the
simultaneous consideration of detailed accessibility measures and spatial
autocorrelation in a case study of Glasgow, Scotland. The next section of this paper
describes the study area and the hedonic data involved. This is followed by the
results of hedonic modelling and discussion. The final section concludes the paper by
highlighting the importance of individual accessibility measures and the benefits of
applying spatial statistics in hedonic price modelling.
2
THE STUDY AREA AND THE DATA
Glasgow was chosen as the main study area for its sufficient size for a meaningful
housing market study, complex accessibility conditions, availability of previous
studies based on the same area, which can serve as a guideline, and availability of
data by the time the research was scheduled to commence the empirical
investigation. The selection of the study area boundaries of Glasgow City Council
(GCC) has considered the theoretical and practical aspects 7. The theoretical aspects
include three criteria of prominent quantitative research namely reliability,
replicability and validity (Bryman, 2001), as well as the housing market economics
and evidence from real estate literature. The GCC area has a wide range of housing,
is a socially heterogeneous city and has been the area that researchers concentrate on
6
According to Anselin (1998), early efforts to implement spatial regression models in urban and real
estate analysis include Griffith (1981), and Anselin and Can (1986) which focused on urban density
functions as well as Dubin (1988; 1992) and Can (1990; 1992) in the context of hedonic models for
house prices. He follows on to state that these studies were characterised by the use of fairly small
datasets (in contrast to more "mainstream" microeconomic cross-sectional analyses) and a focus on
methodological issues.
7
The practical aspects include computing issues, location of information and familiarity with the
study area.
7
in previous studies of Glasgow housing markets. These support that GCC is a valid
and appropriate area for a housing market study that focuses on the issue of
neglected spatial elements in hedonic modelling.
Based on the GCC area, four main groups of data were gathered for this study.
These are house prices, structural characteristics, neighbourhood characteristics and
accessibility measures8. Most of the data used in this study have been obtained from
government agencies. The literature suggests that data sources reflect data quality.
Thus, this study has used data of UK government quality. Although the analysis also
involved the 1991 census based data, which are relatively outdated, this is not
thought to give an adverse effect on the whole findings because no drastic change
has been reported about the population of Glasgow City Council as per comparison
between the 1991 and 2001 censuses.
The following stage of data preparation verified, cleaned, and converted the data as
necessary into the formats suitable for further analysis 9. This stage has made ready
all the hedonic variables among which are several newly GIS constructed spatial
variables. Most importantly, the prepared data include the detailed accessibility
measures and the spatial weight matrix needed for spatial hedonic modelling. Having
the empirical data gathered and prepared, the final hedonic datasets contain 2,715
sale prices as the dependent variable and 61 independent variables. The details of the
data and their sources are as in Appendix 1.
Descriptive statistics show that structurally, the dataset is dominated by flats (75%)
followed by atttached (22%) and detached (3%) properties. Thus, there is a
possibility for flats to influence the hedonic models. The dependent variable is
normally distributed when log of selling prices are used. The independent variables
also have a reasonable variability in values based on their standard deviation
(Description of 61 variables are as in Appendix 2. Simple descriptive statistics of the
variables are as in Appendix 3 ).
8
This study considers zonal and individual accessibility measures. Data for the former were obtained
from David Simmonds Consultancy (DSC) with consent from The Scottish Executive. Data for the
latter were constructed using GIS.
9
Since further analysis were carried out in SPSS 11.5.1, ArcView 3.2 and Matlab 6.5.1 the relevant
data were to be in .sav, shapefiles and .mat formats respectively. The application of GIS and spatial
statistics software is summarised in Appendix 5.
8
Figure 1: GSPC sales by spatial and structural sub-markets overlaid upon neighbourhood mean selling price (£ per km sq) – 500 metre radius
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Glasgo w City Cou ncil - UKBORDERS.shp
Gspc2715en tire301004.sh p
U Detached
Attach ed
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Gspc2715en tire301004.sh p
#
City Cen tre
#
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No rth G lasgo w
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Moto rw ay12km.shp
Mean sellin g price (£ p er sq km) .shp
20500 - 45939
45939 - 71377
71377 - 96816
96816 - 122255
122255 - 147694
147694 - 173132
173132 - 198571
198571 - 224010
224010 - 249449
No Data
Ordnance Survey Crown Copyright. All rights reserved
9
3
HEDONIC MODELLING OF THE GLASGOW HOUSING MARKET
Using the data that have been prepared, hedonic modelling of the GCC housing
market has involved the estimation of the OLS models, detection of spatial
autocorrelation and estimation of spatial hedonic (SH) models as now presented.
3.1
Estimation of OLS models
Three OLS models of linear, semi-log and log-log functional forms were estimated
for the study area. After considering the issues of multicollinearity, functional form
and the software to be used for spatial hedonic modelling, the log-log model was
selected as the most appropriate for this study. The model has adjusted R2 of 75.8%
and F statistic of 237. For a simple evaluation of the model, Figure 2 shows the
scatter plots of observed prices against estimated prices. It indicates that the model
overestimates most of log prices under 10.5 (equivalent to under £36,316) and
underestimate almost all prices above log 12.5 (equivalent to above £268,337).
Nevertheless, it estimates the prices between 10.5 and 11.5 (equivalent to between
£36, 316 and £98,716) well. The over and underestimation is not uncommon as a
recent study by Pace et al. (forthcoming) also reports a similar observation for their
OLS models.
Figure 2: Actual log price versus estimated log price – the log-log model
13.00
Unstandardized Predicted Value
12.50
12.00
11.50
11.00
10.50
10.00
9.50
9.50
10.00
10.50
11.00
11.50
12.00
12.50
13.00
logprice
10
3.2
Detection of spatial autocorrelation
Hamid (2002) suggests two ways for detecting spatial autocorrelation. First is by
displaying the OLS residuals on the GIS map to detect the pattern that exists
graphically. Second is by using spatial statistics such as Moran’s I and Lagrange
Multiplier to test formally the existence of significant spatial autocorrelation. This
section presents the results of the spatial autocorrelation detection using GIS display
and formal testing.
3.2.1
GIS graphical display
Figure 3 depicts the GIS graphical display of the individual OLS residuals from the
model. Given the residual is calculated by subtracting the estimated value from the
actual value by SPSS, the dots representing positive residuals indicate
underestimation,
while
the
ticks
representing
negative
residuals
indicate
overestimation. Figure 3 shows that generally the dots and the ticks cluster together.
The common rule of thumb used in the graphical analysis of spatial autocorrelation is
that a positive spatial autocorrelation is present when residuals of the same sign
cluster together.
Figure 3 also shows that there are locations where both positive and negative
residuals occur indicating negative spatial autocorrelation. Using a GIS display, it is
easier to detect positive spatial autocorrelation than the negative one. The latter can
be confused with the random geographical distribution of the residuals. This implies
that location display for spatial autocorrelation though useful, is indefinite and
subjective. Compared to location display analysis, a neighbourhood analysis of GIS
can give a better illustration of spatial autocorrelation.
11
Figure 3: Geographical distributions of the positive and the negative residuals of the entire market OLS model
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pos itive and negative res iduals.shp
ò ne gative residual - over estima te
pos itive res idual - unde restim ate
Ú
Motorwa y12km .s hp
Gla sgow City Council - UK BO RDER S.shp
òÚ
ÚÚ
4
N
Ú
4 Miles
Ordnance Survey Crown Copyright. All rights reserved
12
Neighbourhood analysis using GIS produces a more meaningful graphic display in
terms of different segments of positive and negative mean OLS residuals as shown in
Figure 4. Comparing Figure 4 with Figure 5 that shows the neighbourhood by mean
selling price (£ per sq km within 500 metre radius) suggests that the areas of
underestimation are concentrated in the areas of higher mean selling price. The
figures also illustrate that the GIS neighbourhood analysis is useful for delineating
neighbourhood areas based on mean neighbourhood prices and OLS residuals10.
The above graphical analysis shows that GIS is a useful tool for examining spatial
autocorrelation. Nevertheless, the outcome is not definitive. Further analysis using
spatial statistics of the residuals can, however, complement GIS capabilities. Thus,
the discussion moves on to the implementation of spatial statistics tests available
from Spatial Econometric Tools (SET).
Figure 4: Neighbourhoods by over and underestimation
N
W
E
S
4
0
4 Miles
Moto rw ay12km.shp
Glasgo w City Cou ncil - UKBORDERS.shp
Nb rMean of resid uals - 500m .shp
-0.74 - -0.57 (great est o verest im ation )
-0.57 - -0.35
-0.35 - 0
0
0 - 0.35
0.35 - 0.57
0.57 - 0.78 (greatest u nd erest im ation )
No Data
Ordnance Survey Crown Copyright. All rights reserved
10
The mean residuals are within 500 metres radius of each GSPC sale. The choice of distance is
arbitrary.
13
Figure 5: Neighbourhoods by mean selling prices
N
W
E
S
4
0
4 Miles
Moto rw ay12km.shp
Glasgo w City Cou ncil - UKBORDERS.shp
Mean sellin g price (£ p er sq km) .shp
20500 - 45939
45939 - 71377
71377 - 96816
96816 - 122255
122255 - 147694
147694 - 173132
173132 - 198571
198571 - 224010
224010 - 249449
No Data
Ordnance Survey Crown Copyright. All rights reserved
3.2.2
formal testing
Moran’s I and Lagrange Multiplier (LM) are the common tests used in the literature
(some description of the tests are as in Appendix 4). Table 1 shows the results of the
tests undertaken on the OLS model. All the tests detect that spatial autocorrelation
significantly exists in the model. Moran’s I value indicates positive spatial
autocorrelation meaning that similar residuals cluster together. The higher value of
LM (error) than the LM (sar) suggests that it is more likely that the spatial
autocorrelation is of spatial error rather than of spatial lag dependence. In turn, this
means that it is more likely for the spatial autocorrelation detected to occur out of
missing variables for important property characteristics rather than for a lag variable
that captures interdependence among house prices.
14
Table 1: Results from the spatial autocorrelation tests on the OLS model
Moran’s I
LM (error)
LM (sar)
Moran I value
0.2032
Moran I-statistic
21.5287
Marginal Probability
0.0000
Mean
-0.0042
Standard deviation
0.0096
Value
1551
432
Marginal Probability
0.0000
0.0000
Chi (1) .01 value
6.635
6.635
Note: Critical value for Moran I statistic is 1.96 while the statistics computed by all the other two tests
are distributed as Chi-square (2) at 6.635 with one degree of freedom. All the spatial autocorrelation
statistic are highly significant, that is, at least at the 0.00001 level. N=2,715
3.3
Estimation of spatial hedonic models
De Koning et al. (1998) as cited in Overmars et al. (2003) suggest that, if the spatial
autocorrelation in the residuals cannot be excluded by adding regression variables a
spatial regression model is most appropriate.11 According to them, by using spatial
models, part of the variance is explained by neighbouring values. They suggest that,
this is a way to incorporate spatial interactions 12 that cannot be captured by the
independent variables. Brasington and Hite (2005) and Tse (2002) suggest that a
spatial hedonic model, through its spatial effect parameter/s, may capture spillovers,
missing variables or other forms of spatial dependence. Accordingly, this study
carried out spatial hedonic modelling to deal with the spatial autocorrelation issue.
This provides more accurate, robust and reliable hedonic models.
This study adopts the spatial weight matrix approach13 in modelling the spatial
autocorrelation explicitly. The selection is based on the literature that indicates the
suitability of this approach for real estate analysis that involves discussions of the
11
Overmars et al. (2003) note that there will be cases in which the application of a spatial model will
lead to a significant spatial autocorrelation parameter, while the Moran’s I does not show any spatial
autocorrelation. However, they do not report a further discussion on this.
12
These interactions are caused by unknown spatial processes such as social relations and market
effects (Overmars et al., 2003). Meen and Meen (2003) suggest that social relation is one of the
nonlinearity features of housing markets that should be taken into account in modelling the latter
before any model can be used for policy. Thus, Overmars et al. (2003) suggest that spatial hedonic
models are able to capture the social interaction feature out of the three housing markets features of
social interaction, nonlinearity, and segregation as proposed by Meen and Meen (2003). Leenders
(2002) also provides a discussion on modelling social influence using spatial weight matrix by using
voting data.
13
Another approach is the geostatistical approach. Ismail (2005) provides a simple discussion on the
two approaches of modelling the spatial autocorrelation of the hedonic residuals.
15
economic behaviour of the variables. The estimation using SET involved three
models namely Spatial Error Model (SEM), Spatial Autoregressive Model (SAR)
and General Spatial Model (SAC). These are the names used in the manual provided
on the website by its author. Their estimation made use of the function named
sem_d2, sar_d2 and sac_d2 (the details are included in Ismail (2005)).
Table 2 summarises the performance of the OLS and three spatial hedonic models.
The SEM model appears the best. It has the highest adjusted R2 (79.7%) and the
lowest variance (0.0513). The SAR and the SAC have 75.8% and 77.9% for adjusted
R2 respectively, while 0.0611 and 0.0588 for variance respectively.
Table 2: Four hedonic models of the entire GCC housing market
OLS
SEM
SAR
Adjusted R2 (%)
Variance
Significance of lambda ()
Significance of rho ()
N=2,715; K-1=36
3.4
75.8
0.062
79.7
0.0513
****
75.8
0.0611
0.2
SAC
77.9
0.0558
****
0.12
Comparing the OLS and the spatial hedonic models
Table 2 shows that the adjusted R2 increases from 75.8% under the OLS to 79.7%
under the SEM. Meanwhile, variance decreases from 0.062 to 0.0513 under the same
models. Thus, spatial hedonic has improved the adjusted R2 by 3.9 percentage point
(or 5.1%) and the variance by 0.011 point (or 17.3%) which is equivalent to 9%
reduction in the standard error14. These levels of improvement – from OLS to spatial
hedonic - are consistent with those reported in other studies. For example,
Wilhelmsson (2002a) and Pace and Gilley (1997) show a 2.4 and 7.2 percentage
points increase, respectively in the adjusted R2. Can and Megbolugbe (1997) report a
range of 13.9% to 17.5% reduction in variance. Meanwhile, Theebe (2004) shows
10% to 25% reduction in standard error.15The following texts discuss the results
obtained further and highlight the major changes in the level of significance and
magnitude of the coefficients after the spatial autocorrelation is explicitly modelled.
14
Standard error is the square root of variance
Tse (2002) reports a 7% reduction in sum-of-squared errors while Pace et al. (1998) report a
37.35% reduction in median absolute errors - from OLS to SH.. This study does not calculate these
values, hence, Tse’s (2002) and Pace et al.’s (1998) levels of improvement cannot be compared with.
15
16
4
Further discussion
The effects of spatial dependence on the OLS method include biased estimation of
error variance and t-test significance levels, inefficient estimation and confidence
intervals16 and liberally-biased inference (Bell and Bockstael, 2000; Overmars et al.,
2003; Pace and LeSage, 2004b; Berg, 2005). According to Legendre (1993) and
Overmars et al. (2003), in the presence of positive spatial autocorrelation, computed
test statistics are too often declared significant under the null hypothesis..
Additionally, Bell and Bockstael (2000) note that they do not observe a specific
pattern when comparing the correlation-corrected results to the OLS results. They
report that, the cases in which significant tests produce different answers, results
suggest that the t-statistics are biased upward in some cases and downward in others
(Bell and Bockstael, 2000, 79). This is also the case in Pace and Gilley (1997) where
they find that the t value and the magnitude either increases or decreases in the
spatial hedonic model - compared to the OLS model. They regard the changes as
corrections made by the spatial hedonic modelling.
As mentioned earlier, Moran’s I test summarised in Table 1 suggests that significant
positive autocorrelation exists in the OLS model. Based on the above discussion,
some variables in the OLS model estimated may become insignificant or less
significant if the spatial autocorrelation is specifically modelled. Meanwhile, some
variables may become more significant because their true effects on price can be
revealed more accurately in the absence of spatial autocorrelation in the model.
The LM tests results summarised in Table 1 also indicate that spatial error
dependence prevails. Bell and Bockstael (2000) suggest that this type of spatial
autocorrelation can be associated with omitted variables, as well as measurement
errors arising from the spillover effects of the aggregate data (such as the census and
DSC index data), the errors of which are correlated with the errors of some other
variables included in the OLS model. Thus, spatial autocorrelation may lead to
overestimation or underestimation of some coefficients as a result of upward or
downward biased variance of OLS (Bell and Bockstael, 2000; Patton and McErleans,
2002; Wilhelmsson, 2002a; Overmars et al., 2003). In turn, it can be anticipated that
16
The confidence interval becomes narrower than it is when calculated correctly (Legendre, 1993)
17
when spatial autocorrelation is modelled explicitly, some variables may decrease or
increase in magnitude to reflect their true effects on price. Similarly, changes in the
levels of significance as well as magnitude of coefficients may also take place.
Pace and Gilley (1997) report changes in sign for three variables including an age
variable. However, no changes in sign shown by the models estimated in this study.
Nonetheless, Table 3 compares the OLS and the SEM models and highlights the
major changes in the level of significance and size of magnitude of the coefficients in
the former. The results shown in Table 3 are consistent with the expectation that
spatial autocorrelation masks the true effects of property characteristics on house
price. With spatial hedonic modelling, several structural variables and several
individual accessibility measures are shown to be more influential than the OLS can
detect. Specifically, individual accessibility measures are shown to be more
influential than a zonal accessibility measure.
The literature suggests that causes of spatial autocorrelation include the similarity in
property characteristics, price determination process and model mis-specification
(including missing variables and unsuitable functional form). The SEM is able to
capture the effect of spatial error dependence arising from the missing important
variables. Its use has provided more accurate and robust estimates for the OLS
variables, hence, more reliable inference can be made from the model. Given the
significance of structural and accessibility measures in hedonic price modelling, it is
important to test and model explicitly spatial autocorrelation to enable more accurate
and robust estimation and to produce more reliable conclusions based on them.
18
Table 3: Comparing the parameters between the OLS and the best spatial hedonic model (SEM) of the entire market
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
Ordinary Least-squares Estimates
R-squared
= 0.7612
Rbar-squared = 0.7580
sigma^2
= 0.0620
Durbin-Watson = 1.9519
Nobs, Nvars
= 2715, 37
Variable
Coefficient
t-statistic
Constant
12.796683
66.386856
detached
0.421935
11.957267
attached
0.180488
9.587471
conversion
0.094042
2.753428
ground_f
-0.085215
-5.460457
lower_f
-0.114736
-5.031696
upper_f
-0.127117
-6.029863
lgrooms
0.488471
22.715169
chxlgrms
0.137986
11.183346
garage
0.075822
4.119769
garden
0.056004
4.269845
examprim
0.002672
5.656153
examsec
0.005374
8.971248
migrant
0.001778
3.922073
more2car
0.019893
7.031586
unemploy
-0.002946
-4.805106
chil0_15
-0.003653
-7.961510
ownocc
0.006175
12.214218
profhh
0.002831
5.926224
white
-0.003857
-6.724475
ecoinact
-0.001405
-3.274278
pop_acre
-0.000340
-2.253217
feb
-0.124896
-5.363896
march
-0.080179
-3.966368
april
-0.061207
-3.258107
november
0.038997
2.525703
accbusi
-0.017117
-4.134234
lgnwund
-0.067281
-6.681671
lgnwts
-0.040643
-4.342908
lgnwcbd
-0.039150
-2.053518
lgnwprim
0.057482
7.720669
lgnwsec
-0.051086
-5.789077
lgdrdb
-0.026176
-5.111772
lgdrailw
0.017151
2.525125
g_north
-0.350239
-12.807228
g_east
-0.453537
-19.797701
g_south
-0.325119
-21.461584
t-probability
0.000000
0.000000
0.000000
0.005937
0.000000
0.000001
0.000000
0.000000
0.000000
0.000039
0.000020
0.000000
0.000000
0.000090
0.000000
0.000002
0.000000
0.000000
0.000000
0.000000
0.001073
0.024326
0.000000
0.000075
0.001136
0.011604
0.000037
0.000000
0.000015
0.040119
0.000000
0.000000
0.000000
0.011623
0.000000
0.000000
0.000000
Spatial error Model Estimates
R-squared
=
0.7995
Rbar-squared
=
0.7968
sigma^2
=
0.0513
log-likelihood =
1054.6935
Nobs, Nvars
=
2715,
37
Variable
Coefficient Asymptot t-stat
Constant
12.600434
1856.094761
Detached
0.456703
14.110263
Attached
0.210384
11.799226
Conversion
0.037743
1.160982
ground_f
-0.070901
-5.041802
lower_f
-0.069612
-3.212661
upper_f
-0.084264
-4.217802
Lgrooms
0.501707
25.353592
chxlgrms
0.116260
10.500581
Garage
0.074915
4.547272
garden
0.046258
3.936874
Examprim
0.002226
4.121607
Examsec
0.006290
8.617987
Migrant
0.001466
3.459252
more2car
0.016598
5.806313
Unemploy
-0.002733
-4.620434
chil0_15
-0.002354
-5.172502
Ownocc
0.004583
9.048814
Profhh
0.001974
4.488849
White
-0.002223
-3.608654
Ecoinact
-0.000736
-1.747672
pop_acre
-0.000213
-1.490982
Feb
-0.130125
-6.225375
march
-0.081574
-4.491666
april
-0.052259
-3.131674
november
0.034885
2.518971
Accbusi
-0.008955
-2.394681
Lgnwund
-0.088926
-7.059741
Lgnwts
-0.041090
-3.816214
Lgnwcbd
-0.064586
-2.574905
Lgnwprim
0.049703
6.313295
Lgnwsec
-0.042677
-3.881837
Lgdrdb
-0.028184
-4.508252
Lgdrailw
0.009045
1.162883
g_north
-0.315995
-9.871507
g_east
-0.420692
-15.966616
g_south
-0.308937
-16.598350
lambda
0.544972
36.565860
z-probability
0.000000
0.000000
0.000000
0.245649
0.000000
0.001315
0.000025
0.000000
0.000000
0.000005
0.000083
0.000038
0.000000
0.000542
0.000000
0.000004
0.000000
0.000000
0.000007
0.000308
0.080521
0.135966
0.000000
0.000007
0.001738
0.011770
0.016635
0.000000
0.000136
0.010027
0.000000
0.000104
0.000007
0.244877
0.000000
0.000000
0.000000
0.000000
More
significant
Increased
magnitude


















Note: Variables with increased magnitude of coefficient are put in bold. Variables that become insignificant are put in the shaded cells. Variables that become less significant
are put in italic.
19
5 CONCLUSION
House prices are importance economic indicators and hence hedonic price modelling
should produce accurate, robust and reliable results for analysis. The literature
review has revealed that there is a growing recognition from the real estate studies of
the benefits of considering micro spatial elements in hedonic price modelling for
housing markets. This is because researchers are becoming more equipped with
better data and tools such as GIS and spatial statistics. Recent studies have been able
to consider detailed accessibility measures and spatial autocorrelation in hedonic
modelling. Nonetheless, far more evidence has been reported from countries such as
the US and Sweden than from the other countries including the UK. In the UK there
has been no evidence of simultaneous consideration of detailed accessibility
measures and spatial autocorrelation in hedonic modelling for housing markets
except for Day (2003).
The discussion of the results from this study has shown that the presence of spatial
autocorrelation in the OLS models has masked the true elasticity of house price to
important variables that include structural characteristics and individual accessibility
measures. The analysis based on spatial hedonic has enabled more accurate
estimation of the implicit price of the variables. It provides more reliable statistical
inference of housing markets. Therefore, if accuracy, robustness and reliability is of
importance in urban housing market analysis, hedonic price modelling that includes
micro spatial variables (such as individual accessibility measures) and that models
explicitly spatial autocorrelation when it is detected as significant, is an effective
approach for modelling the housing market.
20
BAILEY, T., 1999. Modelling the Residential Sub-market: Breaking the Monocentricity Mould. Urban Studies,
36(7), pp. 1119-1135.
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32
Appendix-1
Table A1: Data, sources, original format, and required formats
DATA
SOURCES OF DATA
REQUIRED FORMAT
GSPC
ORIGINAL DATA
FORMAT
CSV
1) SALE PRICES
2) STRUCTURAL CHARACTERISTICS
GSPC
CSV
ArcView, SPSS, Matlab
3) ACCESSIBILITY MEASURES:
A) To geo-code using GIS:
Zonal accessibility values to business
DSC/SE
Mapinfo
ArcView, SPSS, Matlab
ArcView, SPSS, Matlab
B) To measure network distance using GIS:
Individual accessibility to several facilities:
i)
train stations
- postcodes
- x,y
www.upmystreet.com
Digimap Postcode Query, Diginap Codepoint
Textual
Textual
ArcView, SPSS, Matlab
Arcview
Arcview
ii)
underground stations
- x,y
OSMM cartography
ArcView (points)
ArcView, SPSS, Matlab
Arcview
iii)
parks
- names, postcodes
- x,y
GCVJSP, www.upmystreet.com
OSMM cartography
All textual
ArcView, SPSS, Matlab
Tectual, Excel
ArcView
iv)
Public schools: primary and secondary
- postcodes, xy
SE, Digimap Codepoint
Excel, All textual
ArcView, SPSS, Matlab
Excel, ArcView
v)
shopping centres
names, postcodes,
x,y
Trevor Wood Research Associates, SE, www.upmystreet.com,
Digimap Codepoint
Excel, textual,
textual&Excel
ArcView, SPSS, Matlab
Textual, Excel, ArcView,
vi)
CBD
OSMM
ArcView
ArcView, SPSS, Matlab
33
Appendix-1
Table A1: (Continued)
DATA
SOURCES OF DATA
ORIGINAL DATA
FORMAT
REQUIRED FORMAT
Education Service GCC, http://www.scottishschoolsonline.gov.uk/
Excel
ArcView, SPSS, Matlab
ArcView
Education Service GCC, http://www.scottishschoolsonline.gov.uk/
Digimap Codepoints
OSMM: ITN
Excel
Excel
ArcView
ArcView
Excel
ArcView
b) Quality of nearest secondary school:
year 2001 S4 exam results
postcodes
x,y
road network
SE, http://www.scottishschoolsonline.gov.uk/
SE, http://www.scottishschoolsonline.gov.uk/
Digimap Codepoints
OSMM: ITN
Excel, Textual
Excel, Textual Excel
ArcView
ArcView, SPSS, Matlab
Arcview
Excel
Arcview
Arcview
Proximity to dis-amenity:
i)
Proximity to nearest A road
ii)
Proximity to nearest B road
iii)
Proximity to motorway
iv)
Proximity to railway
v)
Proximity to Industrial and business
vi)
Size of nearest shopping centre
OSMM: ITN
OSMM: ITN
OSMM: ITN
OSMM - Topography
GCC City Plan
Trevor Wood Associates
ArcView
ArcView
ArcView
ArcView
Textual
Excel
ArcView, SPSS, Matlab
ArcView, SPSS, Matlab
ArcView, SPSS, Matlab
ArcView, SPSS, Matlab
ArcView, SPSS, Matlab
ArcView, SPSS, Matlab
4) NEIGHBOURHOOD QUALITY
To geo-code using GIS:
School quality:
a) Quality of nearest primary school:
- Year 2001 S5_14 exam results: average of reading,
writing, and mathematics
- postcodes
- x,y
- road network
34
Appendix-1
Table A1: (Continued)
DATA
SOURCES OF DATA
ORIGINAL DATA FORMAT
REQUIRED FORMAT
All from MIMAS – 1991 census
ArcView
ArcView, SPSS, Matlab
MIMAS – 1991 census
UKBORDERS
OSMM: ITN
ArcView (polygon)
ArcView (polygon)
ArcView
ArcView
ArcView
ArcView
4) NEIGHBOURHOOD QUALITY..continued
Socioeconomic:
i)
Professional HH
ii)
Higher degree persons
iii)
White residents
iv)
Unemployment
v)
HH with children aged 15 and below
vi)
More than 2 cars
vii)
Owned outright dwellings
viii) Economically inactive
ix)
Migrant
x)
Single HH
XI)
Population density of COA
5) OTHER LOCATION SPECIFIC DATA:
i)
ii)
iii)
Census Output Areas
GCC Administrative area
Road network: motorway, A, and B roads
35
Appendix-2
Table A2: List of 61 variables and their description
Variable Structural
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
DETACHED
ATTACHED
FLAT
CONVERSION
MAINDR_F
GROUND_F
FIRST_F
SECOND_F
THIRD_F
FOURTH_F
TOP_F
LOWER_F
UPPER_F
LGROOMS
CHXLGRM
GARAGE
GARDEN
Accessibility
Neighbourhood



















Month Type
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
continuous
interactive
dummy
dummy
36
Description
detached property
attached property
flat property
Log of total number of rooms
central heating dummy X log of total number of rooms
availability of garage
availability of garden
Appendix-2
Table A2: (continued)
Variable Structural Accessibility Neighbourhood Month Type
18 EXAMPRIM
19 EXAMSEC
20
MIGRANT
21 MORE2CAR
22 UNEMPLOY
23
CHIL0_15
24
HIGHDEG
25
OWNOCC
26 SINGLEHH
27
PROFHH
28
WHITE
29 ECOINACT
30 POP_ACRE














continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
Description
average percentage for reading, writing and mathematics (S5-14 standard assessment)
percentage of students achieving credit in year 2001 SCE standard examination
percentage of migrant household residents
percentage of households with at least three car
percentage of unemployed but economically active adult (16 and above) household residents
percentage of households with children aged 15 and below
percentage of adults (18 and above) with level A higher degree - 10% sample
percentage of owned outright tenure of households in permanent buildings
percentage of one person households with adult aged 18 and above
percentage of professional households - 10% sample
percentage of white residents
percentage of economically inactive adult (16 and above) household residents
percentage of people per acre
37
Appendix-2
Table A2: (continued)
Variable Structural Accessibility
31
32
33
34
35
36
37
38
39
40
41
42
43
44
ACCBUSI
LGNWSHP
LGNWUND
LGNWTS
LGNWCBD
LGNWPRIM
LGNWSEC
LGNWPARK
LGDINDUS
LGSHPSIZ
LGDMWAY
LGDROADA
LGDROADB
LGDRAILW









Neighbourhood
Month
Type
Description
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
continuous
DSC zonal accessibility index to business
log of shortest network distance to a shopping centre
log of shortest network distance to underground station
log of shortest network distance to a train station
log of shortest network distance to CBD (George Square)
log of shortest network distance to a primary school
log of shortest network distance to a secondary school
log of shortest network distance to a park
log of straight distance to the nearest industrial and business centre
log of size of the nearest shopping centre
log of straight line distance to the nearest motorway
log of straight line distance to the nearest A road
log of straight line distance to the nearest B road
log of straight line distance to the nearest railway line







38
Appendix-2
Table A2: (continued)
Variable
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
JANUARY
FEBRUARY
MARCH
APRIL
MAY
JUNE
JULY
AUGUST
SEP
OCTOBER
NOVEMBER
DECEMBER
G_CITY
G_WEST
G_NORTH
G_EAST
G_SOUTH
Structural
Accessibility
Neighbourhood







Month













Type
Description
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
dummy
City centre
West End
North Glasgow
East End
South Side
39
Appendix-2
Table A2: (continued)
Variable Structural Accessibility Neighbourhood
Month Type
Description
32
NWSHP

continuous
shortest network distance to a shopping centre

33
NWUND
continuous
shortest network distance to underground station

34
NWTS
continuous
shortest network distance to a train station

35
NWCBD
continuous
shortest network distance to George Square

36
NWPRIM
continuous
shortest network distance to a primary school

37
NWSEC
continuous
shortest network distance to a secondary school

38
NWPARK
continuous
shortest network distance to a park


39
DROADB
continuous
straight line distance to the nearest B road


40
DROADA
continuous
straight line distance to the nearest A road


41
DMWAY
continuous
straight line distance to the nearest motorway


42
DINDUS
continuous
straight distance to the nearest industrial and business centre

43
DRAILW
continuous
straight line distance to the nearest railway line

44
SHPSIZ
continuous
size of the nearest shopping centre (sq ft)

45
CHXRM
Interactive
central heating dummy X total number of rooms
Note: The shaded cells contain 13 non-transformed variables and 1 interactive non-transformed variable.
40
Appendix-3
Table A3: Descriptive statistics of the dependent and the independent variables
0
0
Sellingp (£)
logprice
Minimum
20,000
9.9
Maximum
360,000
12.79
Mean
75,932
11.1033
Std. Deviation
43,668
0.5061
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
FLAT
DETACHED
ATTACHED
CONVERSION
MAINDR_F
GROUND_F
FIRST_F
SECOND_F
THIRD_F
FOURTH_F
TOP_F
LOWER_F
UPPER_F
LGROOMS
CHXLGRM
GARAGE
GARDEN
EXAMPRIM
EXAMSEC
MIGRANT
MORE2CAR
UNEMPLOY
CHIL0_15
HIGHDEG
OWNOCC
SINGLEHH
PROFHH
WHITE
ECOINACT
POP_ACRE
ACCBUSI
LGNWSHP
LGNWUND
LGNWTS
LGNWCBD
LGNWPRIM
LGNWSEC
LGNWPARK
LGSHPSZ
LGDINDUS
LGDRDA
LGDRDB
LGDMWAY
LGDRAILW
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
37
4
0
0
0
0
0
0
0
0
15
3
0.2
49
4.9768
2.8831
2.4849
5.4482
1.4633
3.7544
1.1878
10.8198
4.3307
2.6334
2.5769
3.1586
2.5161
1
1
1
1
1
1
1
1
1
1
1
1
1
2.0794
2.0794
1
1
99
46
114
26
73
72
67
68
88
100
100
94
371
69
8.5484
9.3859
8.5011
9.3643
7.9054
8.4553
8.0860
13.5924
8.6225
7.6556
8.0301
8.7426
8.0161
0.7492
0.0295
0.2214
0.0250
0.0136
0.1260
0.1448
0.1223
0.0074
0.0015
0.1241
0.0715
0.0825
1.1069
0.9383
0.1090
0.5359
75
24
15
1
14
22
1
14
37
5
95
39
42
54.1775
7.5590
7.8012
6.8004
8.4643
6.1596
7.0672
6.5347
11.8637
7.7150
5.5619
6.3121
7.2670
5.8830
0.4336
0.1691
0.4152
0.1563
0.1160
0.3319
0.3519
0.3277
0.0855
0.0384
0.3298
0.2576
0.2752
0.3364
0.5230
0.3117
0.4988
12
12
13
2
10
14
5
12
17
11
10
16
37
2.0081
0.5512
0.9054
0.7454
0.4931
0.7011
0.6032
0.7669
0.7372
0.4906
1.1043
1.1944
0.8426
0.9352
41
Appendix-3
Table A3: (continued)
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
JANUARY
FEBRUARY
MARCH
APRIL
MAY
JUNE
JULY
AUGUST
SEP
OCTOBER
NOVEMBER
DECEMBER
CITY
G_WEST
G_NORTH
G_EAST
G_SOUTH
Minimum
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Maximum
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Mean
0.0239
0.0453
0.0615
0.0726
0.0917
0.0902
0.0947
0.1160
0.1013
0.1381
0.1127
0.0519
0.04
0.38
0.06
0.15
0.37
Std. Deviation
0.1529
0.2080
0.2403
0.2595
0.2887
0.2866
0.2928
0.3203
0.3018
0.3451
0.3163
0.2219
0.20
0.49
0.24
0.36
0.48
32
33
34
35
36
37
38
39
40
41
42
43
44
NWSHP
NWUND
NWTS
NWCBD
NWPRIM
NWSEC
NWPARK
DINDUS
SHPSZ (sq ft)
DMWAY
DROADA
DROADB
DRAILW
145
18
12
232
4
43
3
76
50,000
24
14
13
12
5,158
11,919
4,920
11,664
2,712
4,700
3,249
5,555
800,000
6,264
2,113
3,072
3,029
2,170
3,325
1,129
5,257
581
1,366
867
2,473
195,869
1,877
415
921
511
947
2,207
716
2,167
358
706
541
977
191,862
1,176
355
773
398
Note: The shaded cells contain 13 non-transformed variables. All distances are in metres.
42
Appendix-4
TEST
Moran’s I
Table A4: Common tests of spatial autocorrelation
DESCRIPTION
Moran’s I is a test that measures the spatial correlation in the residuals of a
regression model. It checks for similarities among the housing price and
attribute data in relation to the spatial relationships in the spatial contiguity
matrix (Bowen et al., 2001). If Moran’s I is larger than the critical value, the
hypothesis of no correlation is rejected (Anselin, 1988). Moran’s I takes the
form:
I = eWe/ ee
where e is the OLS residuals (If the spatial weight matrix is row standardised).
If the Moran’s I show that the existence of spatial autocorrelation cannot be
rejected, it indicates that the spatial error model (SEM) is an appropriate way to
proceed.
Lagrange
Multiplier for
spatial error
model (LM(sem))
LM error statistics measure the correlation in residuals of a regression model. If
LM statistics are larger than the critical value, the null hypothesis of no spatial
correlation is rejected. The LM takes the form (Burridge, 1980 in Wilhelmson,
2002b)
LM = (1/T) (eWe)/2)2  2 (1)
T = tr (W + W )W
where e is the OLS residuals,
2 is equal to the OLS variance and tr is equal to the trace.
The test statistic is asymptotically distributed as 2 with one degree of freedom.
According to Wilhelmsson (2002a), this LM-test is a restricted version of a
more general test statistics presented in Anselin et al. (1996). It is also a
simplification of the LM (sar) below.
Lagrange
Multiplier for
spatial lag model
(LM (sar))
A test based on the residuals from the spatial lag (LM(sar)) model can be used
to examine whether inclusion of the spatial lag term eliminates spatial
dependency in the residuals of the model. That is the test for spatial dependence
( = 0) is conditioned on  not equal to zero
Lagrange multiplier test (Anselin, 1988) is preferred over a simpler association
test (i.e. Moran's I) applied to the error terms, because it aims at specific
sources of model specification and is therefore more revealing. When
significant, the LM tests, which are 2 distributed with 1 degree of freedom,
indicate that the model is incomplete. Critical value is 6.63 (Wilhelmsson,
2002a). According to Patton, (2002) it is used in preference to the Wald and
Likelihood ratio testing procedures as it is based on the null hypothesis and
thus, can be computed using the results of ordinary least squares estimation
(Anselin and Bera, 1998).
43
Appendix-5
Figure A5: The application of ArcView 3.2, SPSS 11.5.1 and Matlab 6.5.1 (SET) in this study
GSPC data





1.
2.
ACCESSIBILITY data
NEIGHBOURHOOD data
Selling price
Location (x,y)
Structural characteristics
sub-market dummies
month of sale


Zonal (DSC index)
Individual
accessibility based
on network distance
To create shapefiles for
GSPC sales from x, y coordinates
To measure straightline
distance to create proximity
variables
1.
To measure network
distance to create
individual accessibility
measures to 7 facilities
To match DSC index with
GSPC sales locations
2.



Census 1991
Schools quality
Proximity to dis-amenity
1.
2.
To match property
characteristics data with
GSPC sales locations
To match census data and
schools data with GSPC
sales locations
ARCVIEW 3.2
Spatial data preparation

Geocode to Create Spatial Data

Spatial Join for Matching the Data

Network Distance Calculation for constructing individual accessibility measures
Visualisation

Location Display of Map Overlays

Density Analysis

Neighbourhood Mean Analysis
To geocode OLS
residuals to enable their
spatial analysis in GIS
To create .sav
files for stepwise
regression
To geocode OLS residuals for graphical
analysis for detecting heteroscedasticity
and spatial autocorrelation
SPSS
11.5.1
OLS models
OLS residuals


To create .mat files for
spatial hedonic modelling
Matlab
6.5.1
(for SET)


Note: Boxes with the thin line represent tasks involving the spatial data
44
Formal testing for
heteroscedasticity using
B-Pagan
Formal testing for
spatial autocorrelation
using Moran’s I and LM
tests
Estimating OLS for a
check with the SPSS
outputs
Estimating spatial
hedonic models: SEM,
SAR and SAC