# RECORD 3bbf88ea5271189e3329337f57932674d7aa341e
# TITLE: The Spatial Dimension of House Prices
# YEAR: 2017.0  VENUE: Architecture and the Built Environment  DOI: 10.59490/abe.2017.4.1747
# AUTHORS: Yun-Long Gong
# PAGES: 206  CANDIDATE TABLE PAGES (1-based): [54, 55, 56, 76, 80, 81, 83, 90, 106, 108, 118, 130, 132, 134, 135, 139, 140, 148, 168, 169]
# ABSTRACT: The economic reform in China, launched in the late 1970s, gradually promotes the free mobility of capital and labour between rural and urban areas, and between cities. The following housing market reform in the late 1990s thoroughly terminates the socialist allocation of housing and introduces market forces into the housing sector. Such institutional shifts have profound eﬀects on the evolution of the Chinese interurban housing market. Yet, little is known about the spatial behaviour of house prices across cities in the post-reform era. How do the housing markets of diﬀerent cities organise across space? What is the relationship between the house price dynamics of diﬀerent cities? To answer these questions, this research performs economic and econometric analysis of the spatial dimension of the Chinese interurban housing market. In addition, this research also concerns the construction of a reliable house price index in the presence of spatial heterogeneity and dependence in the urban housing market of China. A reliable house price index is essential to the analysis of house price dynamic behaviour. However, owing to the data problem, this part is conducted based on the housing market of a Dutch city.
This research discovers the spatial regularities of house prices across Chinese prefecture cities in an economic common area and investigates the underlying formation process. It reveals an uneven distribution of house prices across cities, with those large and/or higher-tier cities and their neighbours having signiﬁcantly higher house prices. Such an uneven pattern of house prices demonstrates the agglomeration spillovers in the interurban housing market. Two forms of spillovers are empirically examined. The ﬁrst is the urban hierarchy distance eﬀect, which is related to the position of a city in a hierarchical urban system. In general, the distance penalty of higher-tier urban centres is conﬁrmed, that is, all else being equal, the further away a city is from the higher-tier city, the lower the house price. The second form of spillovers relates to a city’s position in a city network system, in which no hierarchical structure is imposed. In such a situation, the spillovers arise from the interaction with neighbouring cities and it is found that a city that has larger neighbours tends to have higher house prices. These two forms of spillovers are somewhat correlated with each other because a higher-tier city is always associated with a larger urban size.
It i


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The Spatial Dimension of 
House Prices
Yunlong Gong
04
2017


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TABLE 2.2
Estimation of central-city house price gradient
Linear/Dist
Linear/Time
Semi-log/Dist
Semi-log/Time
Log-log/Dist
Log-log/Time
House price
House price
Ln(House price)
Ln(House price)
Ln(House price)
Ln(House price)
Constant
6699.1337***
(9.66)
6636.7919***
(8.25)
8.7021***
(82.59)
8.7009***
(73.12)
9.0524***
(45.59)
9.0272***
(48.71)
Distance to central city
-3.6720
(-1.24)
–
-0.0011***
(-2.84)
–
–
–
Ln(Distance to central city)
–
–
–
–
-0.1198**
(-2.58)
–
Travel time to central city
–
-3.5313
(-0.89)
–
-0.0012**
(-2.27)
–
–
Ln(Travel time to central city)
–
–
–
–
–
-0.1156***
(-2.62)
Dummy: Jiangsu
-2893.4657***
(-2.79)
-3018.9746***
(-3.04)
-0.4331***
(-3.08)
-0.4702***
(-3.40)
-0.4260***
(-2.93)
-0.4428***
(-3.10)
Dummy: Anhui
-3670.9179***
(-2.75)
-3917.2915***
(-3.04)
-0.6136***
(-3.56)
-0.6767***
(-4.03)
-0.7090***
(-4.73)
-0.7308***
(-5.04)
Time eﬀect
Yes
Yes
Yes
Yes
Yes
Yes
F-stats
30.27
29.51
69.17
64.73
63.22
61.92
R2
0.512
0.506
0.706
0.692
0.687
0.682
Adj.R2
0.492
0.486
0.679
0.665
0.660
0.656
AIC
3832.490
3835.200
76.90
86.64
90.04
93.03
Sample size
210
210
210
210
210
210
Notes: ** and *** denote signiﬁcance at the 5% and 1% level, respectively. The t values shown in parentheses are calculated based on clustered standard
errors that are robust to serial correlation of the same unit and heteroskedasticity over time.
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The Spatial Dimension of House Prices


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The results shown in Table 2.3 reveal that both central and subcentral cities have
signiﬁcantly negative distance eﬀects on house prices when using the semi-log/log-log
form and log-log/semi-log functional form. Those results also show that the
semi-log/log-log form is the best one in terms of adjusted R2 and AIC criteria.
However, the distance penalties are no longer signiﬁcant, even at 10% signiﬁcance
level, for the central city in the log-log/log-log model or for the subcentral city in the
semi-log/semi-log model. Thus, it is certainly correct to infer that the log-log function
is more appropriate than the semi-log function for subcentral cities. It should be kept
in mind that the central city always has a macro-eﬀect that inﬂuences a larger radius
while the subcentral city only has a local micro-eﬀect. In that light, it seems that the
choice of functional form is sensitive to the inﬂuence sphere of the centre. The log-log
function performs beɦer when the area of inﬂuence is relatively small, while the
semi-log function is more appropriate if the area is larger. These ﬁndings are in line
with those of Osland et al. (2007), who found that the exponential (semi-log) function
performs best when the estimation is based on a large area, while the power (log-log)
function performs best if the data is restricted to a small area.
TABLE 2.3
Distance eﬀects of both central and subcentral cities
Semi-log/
Semi-log
Semi-log/
Log-log
Log-log/
Semi-log
Log-log/
Log-log
Ln(House
price)
Ln(House
price)
Ln(House
price)
Ln(House
price)
Constant
8.7260***
(80.38)
8.9212***
(115.18)
8.9805***
(59.59)
9.0699***
(43.06)
Distance to central
city
-0.0008***
(-1.98)
-0.0008***
(-2.03)
–
–
Ln(Distance to
central city)
–
–
-0.0779**
(-2.06)
-0.0607
(-1.27)
Distance to
subcentral city
-0.0006
(-0.93)
–
-0.0010*
(-1.69)
–
Ln(Distance to
subcentral city)
–
-0.0701***
(-4.60)
–
-0.0746***
(-4.66)
Dummy: Jiangsu
-0.4339***
(-3.06)
-0.4142***
(-2.99)
-0.4269***
(-2.89)
-0.4196***
(-2.96)
Dummy: Anhui
-0.6733***
(-4.01)
-0.6530***
(-3.84)
-0.7598***
(-5.24)
-0.7454***
(-5.18)
Time eﬀects
Yes
Yes
Yes
Yes
F-Stats
61.78
72.32
60.03
65.76
Adj.R2
0.680
0.710
0.675
0.693
AIC
75.102
51.06
79.38
65.70
Sample size
210
210
210
210
Notes: *, ** and *** denote signiﬁcance at the 10%, 5% and 1% level,respectively. The t
values shown in parentheses are calculated based on clustered standard errors that are
robust to serial correlation of the same unit and heteroskedasticity over time.
Unlike the semi-log model that only includes the eﬀect of the central city, adding the
35
Interurban house price gradient: Eﬀect of urban hierarchy distance on house prices


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eﬀects of subcentral cities raises the adjusted R2 from 0.679 to 0.710. Their added
eﬀects explain 3% more variance in interurban house prices and decrease the
magnitude of the distance penalties of the central city by about 27%. Since we use
diﬀerent functional forms for central and subcentral cities, we cannot compare the
magnitudes of their distance penalties directly.
TABLE 2.4
House price and wage gradient after controlling for city amenities and
characteristics
House price model
Wage model
Regional
dummies
Amenities
+ Regional
dummies
Amenities +
Location variables
Amenities +
Location variables
Ln(House price)
Ln(House price)
Ln(House price)
Ln(Wage)
Constant
8.4666***
(77.39)
7.6520***
(17.77)
8.0267***
(22.26)
11.0058***
(45.97)
Dummy: Jiangsu
-0.4895***
(-3.46)
0.0946***
(0.66)
–
–
Dummy: Anhui
-0.8410***
(-6.90)
-0.1816
(-1.34)
–
–
Coastal city
–
0.3230**
(2.32)
0.3523***
(3.98)
0.0227
(0.45)
Winter temperature
–
0.0791**
(2.58)
0.0627*
(1.72)
-0.0086
(-0.40)
Smoke and dust
emissions
–
-0.0008
(-1.56)
-0.0013***
(-2.73)
0.0004
(0.60)
Doctor
–
0.0010**
(2.58)
0.0008***
(2.72)
0.0004*
(1.79)
Student/teacher ratio –
-0.0050
(-0.38)
-0.0022
(-0.15)
-0.0354***
(-3.46)
Arable land
–
-0.0003*
(-1.83)
-0.0003
(-1.33)
-0.0004**
(-2.38)
Distance to central city–
–
-0.0008***
(-2.80)
-0.0003*
(-1.67)
Ln(Distance to
subcentral city)
–
–
-0.0329*
(-1.72)
-0.0131
(-1.20)
Time eﬀects
Yes
Yes
Yes
Yes
F-Stats
61.04
85.58
89.47
63.22
Adj.R2
0.622
0.788
0.794
0.746
Sample size
210
207
207
207
Notes: *, ** and *** denote signiﬁcance at the 10%, 5% and 1% level,respectively. The t
values shown in parentheses are calculated based on clustered standard errors that are robust
to serial correlation of the same unit and heteroskedasticity over time.
§ 2.5.3
House price gradient after controlling for city amenities and characteristics
.............................................................................................................................
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The Spatial Dimension of House Prices


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As suggested by the theoretical model, the negative house price gradient with respect
to higher-tier cities should persist after controlling for city amenities and
characteristics. Before estimating this, we ﬁrst investigate the compensating house
price diﬀerentials for urban amenities and characteristics. As noted earlier, the average
house prices in Zhejiang, Jiangsu and Anhui provinces diﬀer signiﬁcantly from each
other. This observation is further supported by the estimation results of column 1 in
Table 2.4, which only contains two regional dummy variables. We assume that the
observed house price diﬀerentials across provinces are actually proxies for the
diﬀerences in amenities. The result of testing this hypothesis is shown in column 2 of
Table 2.4. After including the variables of city amenities and characteristics, the
regional eﬀects of Jiangsu and Anhui province fall dramatically and are no longer
signiﬁcant, which oﬀers some support for our hypothesis.
The six variables of city amenities and characteristics, together with the two regional
dummy variables, account for nearly 80% of the house price variance. As a group, the
amenity and characteristic variables are statistically highly signiﬁcant at the 1%
signiﬁcance level (the joint F-statistic is 39.98 where the 1% critical value is 2.90), and
each has the anticipated sign. Among these variables, winter temperature, bordering
an ocean and number of doctors have signiﬁcantly positive eﬀects, while arable land
per capita has a negative eﬀect at a signiﬁcance level of 10% or beɦer. The unpleasant
eﬀect of smoke and dust emissions is marginally insigniﬁcant.
The third column of Table 2.4 reports the estimation results of the model with both
amenity variables and two distance measures. The two distance variables in which we
are most interested still have signiﬁcantly negative eﬀects: distance to the central city
is signiﬁcant at the 1% level, while distance to the subcentral city at the 10% level.
Compared to the semi-log / log-log model that only includes two distance variables
and two regional dummies, the magnitude of the central-city house price gradient in
this model does not change, but the distance penalties of the subcentral city decrease
by about 50%. The point estimates of city amenities and characters are quite robust as
they do not diﬀer much from the results in column 2. Perhaps the most obvious change
is that the negative eﬀect of particulate maɦer becomes highly signiﬁcant.
.............................................................................................................................
§ 2.6
Decomposition of interurban house price gradient
.............................................................................................................................
The previous section has provided estimates of the impact of urban hierarchy distances
on house prices. According to the theoretical model, the decline in the interurban
house price gradient could be aɦributed either to productivity disadvantages or
amenity disadvantages. This section will empirically decompose the interurban house
price gradient based on Equation (7) and reveal which component contributes more to
the negative price eﬀects of remoteness from higher-tier cities.
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Interurban house price gradient: Eﬀect of urban hierarchy distance on house prices


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TABLE 3.2
Estimates of nonspatial model and SLX model
Dependent variable = Ln(House prices)
Pooled model
OLS
RE
ML
SLX_G(RE)
ML
(W = W0-160
G
)
SLX_T(RE)
ML
(W = W0-150
T
)
Winter temperature
0.0617***
(5.14)
0.0262***
(2.74)
0.0258***
(2.75)
0.0260***
(2.77)
Ln(Smoke and dust
emission)
-0.1231***
(-5.24)
-0.1396***
(-4.37)
-0.1297***
(-4.12)
-0.1325***
(-4.22)
Ln(Student/Teacher
ratio)
-0.4662***
(-3.82)
-1.3482***
(-7.24)
-1.1673***
(-6.21)
-1.1815***
(-6.39)
Doctor
0.0358***
(6.01)
0.0253***
(2.69)
0.0268***
(2.93)
0.0243***
(2.66)
Coast
0.2537***
(4.78)
0.3009**
(2.54)
0.2230*
(1.93)
0.2104*
(1.78)
Ln(Arable land)
-0.0082
(-0.11)
-0.0160
(-0.13)
-0.0497
(-0.40)
-0.0856
(-0.71)
Ln(Population density)
0.1561***
(5.29)
0.1712***
(2.91)
0.1181*
(1.87)
0.1006*
(1.63)
Ln(Land)
0.1721***
(4.93)
0.1817**
(2.55)
0.1897***
(2.72)
0.1630**
(2.36)
W × Ln(Population
density)
0.2719***
(3.04)
0.2795***
(3.17)
W × Ln(Land)
0.2590**
(2.00)
0.1691
(1.35)
Constant
6.4704***
(8.23)
9.2550***
(6.48)
5.3201**
(2.38)
6.6990***
(3.21)
R-Squared
0.823
0.923
0.925
0.925
Corr-Squared
0.764
0.785
0.787
Log-likelihood
24.544
59.043
63.849
63.973
CD test
19.551***
7.6129***
7.955***
8.051***
Sample size
210
210
210
210
Notes: Corr-Squared is the squared correlation between ﬁɦed and actual value. t-values are
reported in the parentheses. W0-160
G
and W0-150
T
denote the spatial interaction structure
between a city and its neighbouring cities within the distance band 0 – 160 km and within
the travel time band 0 – 150 min, respectively. The CD test, which detects the global cross-
sectional dependence of residuals, tends to standard normal distribution under the null
hypothesis. ***, ** and * indicate a 1%, 5%, 10% signiﬁcance level, respectively.
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The Spatial Dimension of House Prices


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capita which implies no signiﬁcant inﬂuence of land supply constraint. After controlling
for random city-speciﬁc eﬀects, the results in the second column do not show any
noticeable changes compared to the results of pooled model. In general, a warmer
winter, less industrial smoke and dust emission, a beɦer education and healthcare
condition, and bordering to an ocean increases the house price of a city. Note that the
estimated eﬀect of education quality in the random eﬀect model is much higher than
that in the pooled model, while the inﬂuence of winter temperature is weakened
drastically. As expected, the two variables measuring urban size have statistically and
economically signiﬁcant eﬀects on house prices in both models. Interestingly, an
increase in urban density has almost the same eﬀect as an expansion in urban scale. A
1 percent increase of urban population of a city will drive up house prices by around
0.17%. The ﬁxed eﬀects estimation, including only the time-variant variables, also
conﬁrms the importance of climate, education quality and urban population density in
determining the house prices.
Overall, the explanatory variables we have chosen perform satisfactorily as indicated by
a relatively high Corr-Squared statistic (0.764) which represents the squared
correlation between actual and ﬁɦed value. However, the CD test (Pesaran 2004)
detects signiﬁcant global cross-section dependence in residuals, suggesting the
existence of cross-city spillovers7.
§ 3.6.2
Results of spatial models
.............................................................................................................................
Estimation of SLX model
The spatial weights matrix W is vital to measuring the city network spillovers as W
carries the underlying spatial interaction structure. In this paper, we expect that the
network externalities are only noticeable within a certain radius; at some farther
distance between the two cities, network spillovers vanish.
Such spatial interaction structure can be captured by diﬀerent weight matrixes. Based
on geographical distance, we ﬁrst divide the cities surrounding city i into three distance
bands, namely 0 – 160 km, 160 – 320 km and 320 – 480 km. A spatial weight matrix
for each distance band is then constructed. For instance, for a city j within distance
band 0 – 160 km, the spatial weight wij of W0-160
G
is deﬁned as
wij,i̸=j = d−2
ij ,
for
0 ≤dij < 160.
(7)
Geographical distance has some intrinsic pitfalls; it does not take into account the
physical obstacles, such as mountains and bays. So we have also constructed spatial
7
The CD test is constructed based on the average of pair-wise correlations of the residuals of each cross-sectional
unit. As N →∞, this test tends standard normal distribution under the null hypothesis of no cross-sectional
correlation.
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Network externalities in Chinese housing markets: A spatial econometric approach


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error correlation in the SLX_G model based on diﬀerent spatial weight matrixes.
Assuming the remaining dependence structure is still conﬁned to the neighbours in
physical distance space, the SDM model is a beɦer choice and the results will be
discussed laɦer. On the other hand, if a city is assumed to interact with the cities that
are nearby on the economic-geographical space, the LM tests are in favor of the SDEM
speciﬁcation.
TABLE 3.4
LM tests on residuals of SLX model
Residuals of SLX_G model estimated in Table 3.2
LM spatial lag
Robust LM spatial
lag
LM spatial error
Robust LM spatial
error
W0-160
G
57.784*
59.056*
18.162*
9.434*
W0-1.5
EG
0.091
1.760
17.504*
19.173*
Notes: For the deﬁnition of matrix W0-160
G
, see notes of Table 3.2. W0-1.5
EG
has the similar
deﬁnition but are constructed based on economic-geographic distance. The LM and robust
LM tests, developed by Anselin et al. (2008) and Elhorst (2010b) for the spatial panel data,
are based on the residuals of SLX_G model estimated in Table 3.2 and follow the χ2(1)
distribution under null hypothesis. * denotes the 1% signiﬁcance level.
The SDEM model is estimated by a ML procedure suggested by Elhorst (2014); the
results are reported in second column of Table 3.510. For the sake of comparison, the
ﬁrst column replicates the estimates of SLX_G model. Based on the
economic-geographical distance matrix, we ﬁnd a highly signiﬁcant spatial
autoregressive process in residuals of the SLX_G model. After controlling for the spatial
error correlation, the inﬂuence of population density on its own house prices becomes
highly signiﬁcant at 1% signiﬁcance level. The point estimates of population density
and land area as well as their spillovers eﬀects are almost in line with the estimates of
SLX_G model, showing the robustness of SLX_G model in measuring the
agglomeration spillovers. In contrast, the estimates for the local-speciﬁc
characteristics show a noticeable discrepancy between the two models. For example,
smoke and dust emission and the ratio of students to teacher no longer signiﬁcantly
aﬀect the house prices, whereas land supply constraint becomes an important house
price determinant in SDEM model. This discrepancy might be due to the fact that the
spatial paɦern of some local-speciﬁc variables is closely related to the spatial paɦern of
the residuals of SLX_G model.
Estimation of SDM and SAR model
As previously discussed, if the spatial interaction of house prices after controlling for
10
The following random eﬀects SDM and SAR model are also estimated by ML procedure. The matlab routine can
be found at hɦp://www.regroningen.nl/elhorst/.
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The Spatial Dimension of House Prices


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TABLE 3.5
Estimates of SDEM, SDM and SAR models
Dependent variable = Ln(House prices)
SLX_G (RE)
ML
(W = W0-160
G
)
SDEM_G (RE)
ML
(W = W0-160
G
)
(M = W0-1.5
EG
)
SDM_G(RE)
ML
(W = W0-160
G
)
(M = W0-160
G
)
SAR_G
ML
(M = W0-160
G
)
Winter temperature
0.0258***
(2.75)
0.0601***
(3.14)
0.0122
(1.52)
0.0126
(1.57)
Ln(Smoke and dust
emission)
-0.1297***
(-4.12)
-0.0219
(-0.74)
-0.0409
(-1.54)
-0.0420
(-1.59)
Ln(Student/Teacher
ratio)
-1.1673***
(-6.21)
-0.6218***
(-3.07)
-0.7090***
(-4.33)
-0.7080***
(-4.29)
Doctor
0.0268***
(2.93)
0.0026
(0.31)
0.0065
(0.85)
0.0069
(0.90)
Coast
0.2230*
(1.93)
0.1560*
(1.68)
0.1793*
(1.91)
0.1671*
(1.77)
Ln(Arable land)
-0.0497
(-0.40)
-0.2906**
(-2.40)
-0.1091
(-1.09)
-0.1178
(-1.24)
Ln(Population density)
0.1181*
(1.87)
0.1278***
(2.65)
0.2042***
(3.95)
0.1909***
(4.10)
Ln(Land)
0.1897***
(2.72)
0.1702***
(2.80)
0.1737***
(3.07)
0.1757***
(3.12)
W × Ln(Population
density)
0.2719***
(3.04)
0.2493***
(2.91)
-0.0686
(-0.85)
W × Ln(Land)
0.2590**
(2.00)
0.2490**
(2.02)
-0.0564
(-0.52)
M × Error
0.7364***
(14.25)
M × Ln(House prices)
0.5510***
(9.48)
0.5280***
(9.96)
Constant
5.3201**
(2.38)
5.5096**
(2.67)
4.4485**
(2.43)
3.8541***
(3.05)
R-Squared
0.925
0.940
0.948
0.948
Corr-Squared
0.785
0.788
0.819
0.812
Log-likelihood
63.849
82.073
95.108
94.721
Sample size
210
210
210
210
Notes: Corr-Squared is the squared correlation between ﬁɦed and actual value. t-values are
reported in the parentheses. For the deﬁnition of the various spatial matrixes, see notes of
Table 3.2. The SDEM, SDM and SAR models are estimated by the ML procedure introduced in
Elhorst (2014). ***, ** and * indicate 1%, 5%, 10% signiﬁcance level, respectively.
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network spillovers occurs based on geographical proximity, the SDM model is a beɦer
speciﬁcation. The third column of Table 3.5 shows the ML estimates of SDM model.
Again, the point estimate of spatial lag of house prices is statistically signiﬁcant,
suggesting the existence of remaining dependence arising from other channels.
Compared to the SDEM model, which models the spatial interaction in the residuals,
the magnitude of the direct eﬀect of population density in the SDM model increases
from 0.13% to 0.21%. Most importantly, the spatial lags of population density and
land area have negative signs, which ﬁnding contradicts network spillovers, but these
eﬀects are not statistically signiﬁcant. In this case, the SDM model collapses to the SAR
model which only includes the spatial lag of house prices. The SAR model estimates
presented in the fourth column of Table 3.5 are largely in line with the results of SDM
model. The results of the SDM and SAR models suggest that the spatial lag of house
prices also contains the information of network spillovers, which cannot be
distinguished from other mechanisms that can result in spatial dependence.
§ 3.6.3
Network spillovers
.............................................................................................................................
All of the four models, SLX, SDEM, SDM and SAR, can be used to model the network
spillovers in the housing markets. The ﬁrst two models directly reﬂect our theoretical
foundation and generate local network spillovers whereas the last two models generate
global spillovers which are not easy to justify11. Table 3.6 summarizes the direct and
spillovers eﬀects of the four models. For the SLX and SDEM model, the direct and
spillover eﬀects are the corresponding point estimates. On the other hand, the partial
derivative approach is needed to calculate the direct and spillover eﬀects for the SDM
and SAR models.
The direct eﬀect of land area is almost the same among the four spatial models, while
the direct eﬀect of population density estimated by the SDM and SAR models is much
more pronounced than that in the SLX and SDEM models. In contrast, the SDM and
SAR models estimate much lower network spillovers of both population density and
land area than the SLX and SDEM models do. In the SDM speciﬁcation, there is no
signiﬁcant network spillover at all. Since the spatial lag of house prices mixes various
sources of spatial interaction, and because the global spillovers assumption is not
consistent with our theoretical foundation, our interpretation is based on the SLX and
SDEM models, and in particular on the laɦer model which considers the remaining
spatial dependence in residuals.
11
Local spillovers are those spillovers occuring only between a city and its neighbouring cities connected by a
spatial weight matrix. In contrast, global spillovers are those spillovers that originate from a city and transmit
to all other cities.
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TABLE 3.6
The estimated direct eﬀects and spillovers
RE
SLX
SDEM
SDM
SAR
Direct eﬀects
Ln(Population density)
0.1712***
(2.91)
0.1181*
(1.87)
0.1278***
(2.65)
0.2148***
(3.90)
0.2059***
(3.84)
Ln(Land)
0.1817**
(2.55)
0.1897***
(2.72)
0.1702***
(2.80)
0.1832***
(2.68)
0.1914***
(3.01)
Spillovers
Ln(Population density)
0.2719***
(3.04)
0.2493***
(2.91)
0.0941
(0.66)
0.1989***
(3.07)
Ln(Land)
0.2590**
(2.00)
0.2490**
(2.02)
0.0895
(0.38)
0.1850***
(2.56)
Notes: For SLX and SDEM model, the direct eﬀect and network spillovers are
exactly the point estimates, whereas the partial derivative approach is used for
SDM and SAR model. The inferences of direct eﬀects and spillovers in SDM and
SAR model are based on 1000 simulations using the variance-covariance matrix
implied by the ML estimates (LeSage and Pace 2009; Elhorst 2010a). t-values are
reported in the parentheses. ***, ** and * indicate 1%, 5%, 10% signiﬁcance level,
respectively.
The direct eﬀect of the SDEM model shows that the inﬂuence of land area is bigger
than the eﬀect of population density, suggesting that, in current China, city growth is
likely to be characterized by an expansion of urban scale rather than an increase in
intensity. The network spillover is even more noticeable. If a city becomes 1% denser
and larger, the total house price increases of neighbouring cities are about 0.25%,
whereas its own house price only rises by 0.13% and 0.17%, respectively. Considering
that each city on average has 8 neighbours within the radius of 160 km, the network
spillover on each neighbouring city is by average around 0.03%, which is much lower
than the magnitude of the direct eﬀect.
§ 3.6.4
Discussion
.............................................................................................................................
As previously discussed, the estimation of network spillovers depends on the choice of
spatial weight matrix because this matric reﬂects the underlying interaction structure.
In this paper, the distance band used for constructing the spatial weight matrix is
somewhat arbitrarily chosen. To check the robustness of our ﬁndings, we replicated our
analysis based on 4 distance bands: 120, 200, 240 and 280 km. Table B1 in the
appendix reports the SDEM estimation of direct eﬀect and network spillovers based on
diﬀerent matrixes. Clearly, the estimation of the direct eﬀect is very robust to the
choice of spatial matrix. The estimation of network spillovers, on the other hand, shows
some variation, though small. When the deﬁnition of neighbours is restricted to the
radius of 120 km, the network spillovers decrease substantially with the spillover of
land area becoming insigniﬁcant. As we include more cities as neighbours, the
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magnitude of network spillover becomes a bit larger (see the results of distance band
240 and 280 km), which is in line with our expectation. Nevertheless, given that the
spatial weight matrix is based on a squared inverse distance function, the majority of
the network spillovers still falls into the nearby neighbours despite the total number of
neighbours being increased. In this sense, we believe that the evidence on network
spillovers presented in this paper is reliable.
The existence of network spillovers means that, all else being equal, house price in a
city surrounded by large cities are much higher than those in a city that has small
neighbours. If we see high house prices as a sign of the ‘triumph of the city’, our
ﬁndings would indicate a core-periphery structure in our study area. We checked this
implication using Moran’s I plot, which is based on a spatial weight matrix which
assumes that every city within the 160 km radius of a speciﬁc city has the same
inﬂuence on that city (a bit diﬀerent from the matrix deﬁned by equation (7))12. The
global Moran’s I plot (Figure C1 in the Appendix) clearly shows a positive correlation
between a city’s house price and the neighbouring cities’ urban population13. In
particular, almost all of the cities that are surrounded by small cities have relatively
lower house prices. The local Moran’s I map, also known as local indicators of spatial
association (LISA) (Anselin 1995), depicts a detailed clustering paɦern. In general, we
ﬁnd a ‘successful’ group (high prices – high population) in the east of the study area
and a ‘lost’ group (low price – low population) in the western part. There is one
exception: a coastal city (Nantong) in the east with large neighbours is characterized by
relatively low prices. Yet, the paɦern that a city surrounded by small cities has high
house prices is not supported. A lesson learned is that the vast number of ‘lost’ cities in
the western part will not ﬂourish in the near future, since they are unable to beneﬁt
from network spillovers of big cities.
.............................................................................................................................
§ 3.7
Concluding remarks
.............................................................................................................................
Most studies aɦributed the spatial variation of interurban house prices to local-speciﬁc
characteristics. However, the spatial clustering paɦern of house prices cannot be fully
explained by these local-speciﬁc variables, pointing to the importance of spillovers. To
account for spatial interaction, spatial econometrics is becoming the standard toolbox
in the analysis of house prices. In particular, the spatial model with spatial lag of house
prices has been widely used by researchers. Nevertheless, SAR-type models have been
heavily criticized because the endogenous interaction is diﬃcult to justify.
12
Both the global Moran’s I plot and the LISA map are calculated by the software ‘Geoda’ which is available at
hɦps://geodacenter.asu.edu.
13
The Moran’s I statistic is 0.193. Based on the distribution of 999 simulations of spatially random distributed
data, it is signiﬁcant at 5% signiﬁcance level.
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Appendix B.Tables
.............................................................................................................................
TABLE B1
Direct eﬀect and spillovers based on diﬀerent spatial weight
matrixes, SDEM
W0-120
G
W0-160
G
W0-200
G
W0-240
G
W0-280
G
Direct eﬀects
Ln(Population density)
0.1228***
(2.60)
0.1278***
(2.65)
0.1348***
(2.80)
0.1319***
(2.65)
0.1361***
(2.74)
Ln(Land)
0.1412**
(2.32)
0.1702***
(2.80)
0.1753***
(2.85)
0.1716***
(2.77)
0.1770***
(2.78)
Spillovers
Ln(Population density)
0.1198*
(1.78)
0.2493***
(2.91)
0.2484***
(2.70)
0.2765***
(2.76)
0.2894***
(2.69)
Ln(Land)
0.0589
(0.63)
0.2490**
(2.02)
0.2504*
(1.93)
0.2747*
(1.94)
0.3127*
(1.93)
Notes: The direct eﬀects and network spillovers are estimated by the spatial Durbin
error model speciﬁed in equation (3), with M = W0-1.5
EG
. The spatial weight
matrixes are deﬁned in the same way as those deﬁned in Table 3.2. t-values are
reported in the parentheses. ***, ** and * indicate 1%, 5%, 10% signiﬁcance level,
respectively.
TABLE B2
Fixed eﬀects estimation of time-variant variables
Dependent variable = Ln (House prices)
Winter
temperature
Ln (Smoke and
dust emission)
Ln (Student
/Teacher ratio)
Doctor
Ln (Population
density)
R-squared
0.0226***
(0.0065)
0.0200
(0.0337)
-1.0950***
(0.3369)
0.0088
(0.0142)
2.0576***
(0.3739)
0.730
Notes: The robust standard errors are reported in the parentheses. ***, ** and *
indicate 1%, 5%, 10% signiﬁcance level, respectively.
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Appendix C.Figures
.............................................................................................................................
FIGURE C1
Global Moran’s I plot
FIGURE C2
LISA cluster map
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structure reasonably well. The average distances between the housing markets within
Cluster 1 and Cluster 2 are 136.53 and 184.02, respectively, much smaller than the
sample average distance (221.54). The extremely low standard deviations of Cluster 1
and Cluster 2 also conﬁrm such ﬁndings.
TABLE 4.2
Cluster membership and statistics based on Euclidean distance and JKL
divergence
Euclidean distance
JKL divergence
Membership
Average
within
distance
Membership
Average
within
distance
Average
monthly
growth rate
(%)
Cluster 1 2,3,4,5,6,7,8,9,13,15,
17,18,19,20,21,22,
25,26,27,28,29,30,
31,32,33,34
9.10
(1.49)
2,3,4,5,6,7,8,9,13,15,
17,18,19,20,21,22,
25,26,27,28,29,30,
31,32,33,34
136.53
(68.24)
0.35
(0.09)
Cluster 2 1,10,11,14,16,23
10.70
(1.57)
1,11,12,14,16,23
184.02
(45.72)
0.50
(0.13)
Cluster 3 24
24
0.78
Cluster 4 12
10
0.41
Sample
10.79
(3.05)
221.54
(173.21)
0.39
(0.13)
Notes: For the cluster membership, only the ID of the city is presented. Readers can refer to
Table B1 for the names of the cities. The numbers in parentheses are the standard deviations.
The average within distance is the mean of all city-pair distances within the same cluster.
After obtaining the mean monthly growth rate for each city throughout the sample period, the
average monthly growth rate reported in the table is the average of mean growth rate across
cities in the same cluster.
In a broad sense, the classiﬁcation results can reduce to a two-cluster solution. One is
Cluster 1, which contains mainly slow-growing markets with an average monthly
growth rate of 0.35%. The other combines Clusters 2, 3 and 4, which contain markets
with relatively higher appreciation rates (see Table 4.2) and are considered to be
“red-hot” markets. There is a much higher degree of heterogeneity within these
red-hot markets, however. Compared to the diﬀerence in house price appreciation
rates, the most striking divergence between the two broad groups is that, referring back
to Figure 4.2, the red-hot markets experienced tremendous house price appreciation
in the second half of the period, whereas house price growth during the ﬁrst half of the
period makes the most important contribution for the cities in Cluster 1. One can also
easily identify a spatial paɦern in the house price dynamics of cities with red-hot
markets located in Eastern China, as well as others that are mainly in the remaining
“peripheral” regions.
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FIGURE 4.5
Spatial distribution of sample cities and their membership based on JKL
divergence
§ 4.5.3
Do structural changes maɢer?
.............................................................................................................................
The classiﬁcation analysis of the whole period relies on the premise that the housing
market structure of a city does not change. In other words, the relationship between
any two cities’ housing markets is stable throughout the sample period. This
assumption may not hold given the evolving conditions of the nascent Chinese housing
market. To explore whether structural changes aﬀect previous classiﬁcation results, I
split the sample into subsamples and perform the classiﬁcation analysis on each
subsample. If there is no structural change, the classiﬁcation results will be highly
consistent among the subsamples, as well as with the clusters presented in Table 4.2.
Otherwise, some quite diﬀerent cluster demarcations will be observed.
The house price appreciation trajectories depicted in Figure 4.1 indicate that house
price changes among sample cities are quite volatile during the ﬁrst half of the period
and the recent period. Thus, I divide the sample period into three sub-periods:
2005–2010 (66 observations), 2011–2013 (36 observations) and 2014–2016 (30
observations). For each sub-period, the JKL divergence is calculated based on an AR(3)
speciﬁcation. The average JKL distances between cities as well as the associated
standard deviations are reported in the last row of Table 4.3. The average distance
between housing markets in the period 2014–2016 (196.79) is much larger than the
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average distance in the ﬁrst (92.14) and second periods (71.23), indicating a
remarkable divergence of housing markets in the recent period. While the overall
housing market diﬀerence throughout the whole sample is approximately 220, as
measured by JKL divergence, it can be concluded that the highly divergent market
conditions in the third period account for the most important component.
The hierarchical clustering method is then applied to the three subsamples, providing
more details about the structural evolution of housing markets. To obtain comparable
clusters across the diﬀerent periods, the dendrograms of these three periods should be
cut at a common “height” (divergence threshold). To do this, I ﬁrst determine the
appropriate number of clusters (k) for the 2014–2016 sample, which should be 5
according to Figure A2. This demarcation requires a minimum height (h)
approximately 94.87. I then cut the dendrograms of the ﬁrst and second period at the
same height and obtain 3 clusters for both of the periods. According to the elbow plots
in Figure A2, the 3-cluster demarcation is not the optimal solution for the ﬁrst two
periods. But only in this way can the demarcation solutions of the three sub-periods be
directly compared with each other.
TABLE 4.3
Cluster membership and statistics for diﬀerent sample periods
2005-2010
2011-2013
2014-2016
k = 3,h = 94.87
k = 3,h = 94.87
k = 5,h = 94.87
Membership
Average
within
distance
Membership
Average
within
distance
Membership
Average
within
distance
Average
monthly
growth rate
(%)
Cluster 1 1,2,3,4,5,6,
7,8,9,11,12,
13,14,15,16,
17,18,19,22,
23,24,25,26,
27,28,29,30,
31,32,33,34
72.16
(45.72)
2,3,4,5,6,7,
8,9,11,14,
15,16,17,18,
20,21,22,25,
26,27,28,29,
30,31,32,33,
34
41.66
(24.42)
3,4,5,6,7,8,
9,19,22,25,
26,27,28,29,
30,31,32,33,
34
29.86
(11.96)
-0.12
(0.09)
Cluster 2 20,21
36.37
(20.21)
12,13,19
52.86
(30.65)
2,13,15,17,
18,20,21
52.71
(26.38)
0.21
(0.12)
Cluster 3 10
1,10,23,24
34.67
(11.24)
1,10,12,23
46.41
(24.12)
0.56
(0.27)
Cluster 4
11,14,16
47.65
(16.20)
0.89
(0.16)
Cluster 5
24
1.82
Sample
92.14
(73.90)
71.23
(66.48)
196.79
(275.54)
0.17
(0.46)
Notes: The same notes as Table 4.2.
In both the 2005–2010 period and the 2011–2013 period, one observes highly
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integrated, homogeneous cross-city housing markets. Most of the cities are classiﬁed
into the same cluster, with a few exceptions, Shanghai(10) for instance. The most
remarkable structural change between these two periods could be that a few developed
eastern cities, such as Beijing(1), Shenzhen(24), Guangzhou(23) and Hangzhou(12),
began to deviate from the main group during the 2011–2013 period.
When turning to the 2014–2016 period, the interurban housing market diverges
much more. House price growth in Shenzhen(24), where the average monthly growth
rate is 1.82%, obviously stands out from the other cities. Aside from the red-hot
markets that are already separated in the 2011–2013 period (1, 12 and 23 for
example), a few more rapid-growth markets with monthly growth rates of 0.89%
(Cluster 4), such as Hefei(11), Fuzhou(14) and Xiamen(16), are also isolated from the
main group. Furthermore, there is another cluster (Cluster 2) that deviates from the
main group but not by much; this group has a relatively low average monthly growth
rate (0.21%) and mainly contains some lower-tier centres in Eastern and Central
China, such as Ningbo(13), Fuzhou(15), Nanchang(16) and Wuhan(21). Note that a
parsimonious three-cluster solution for the 2014–2016 housing markets, which
merges Cluster 1 with Cluster 2, and fuses Cluster 3 with Cluster 4, is highly consistent
with the classiﬁcation results based on the whole period (Table 4.2). By grouping
Cluster 1 and Cluster 2 in Table 4.3, one obtains precisely Cluster 1 in Table 4.2.
Similarly, Cluster 2 in Table 4.2 is almost identical to the combination of Clusters 3 and
4 in Table 4.3.
To summarize the ﬁndings, the Chinese interurban housing market in the early periods
(2005–2013) can be considered a homogeneous market; only a few markets have
distinctive dynamic paɦerns of house prices. In the recent period (after 2014), the
interurban housing market begins to diverge; not only do the markets of the most
important centres of the three main economic regions stand out remarkably, the
markets of some lower-tier centres are also isolated. Given this sudden structural
change, it is not surprising to see that the clustering paɦern of the 2014–2016 period
almost determines the classiﬁcation results based on the whole period.
§ 4.5.4
The eﬀectiveness of geographical and economic divisions
.............................................................................................................................
The classiﬁcation results suggest that geography seems to play a role in the
homogeneous clustering of the city-level housing markets. In this section, I will test the
eﬀectiveness of the traditional geographical demarcation in describing the interurban
housing market structure. This classiﬁcation scheme was introduced by NBSC in 2011
and divides all of China into four regions: Eastern, Central, Western and Northeast
China (see Figure 4.5). In addition, I also examine whether the city-tier system of
demarcation based on economic factors can produce meaningful groups of housing
markets. There are presently several versions of the city-tier division system available,
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Appendix C.The AR(3) estimation
.............................................................................................................................
TABLE C1
The AR(3) estimation of selected cities
Lag 1
Lag 2
Lag 3
Intercept
R-squared
BG test
Tianjin
0.6423***
(0.088)
-0.0355
(0.105)
0.2180**
(0.093)
0.0922*
(0.052)
0.504
8.30**
Taiyuan
0.2663***
(0.084)
-0.0502
(0.087)
0.3104***
(0.081)
0.0847
(0.055)
0.179
27.64***
Shenyang
0.4988***
(0.089)
0.0611
(0.098)
0.1395
(0.087)
0.0922
(0.057)
0.380
3.33
Changchun
0.4762***
(0.089)
0.0576
(0.097)
-0.0414
(0.085)
0.1743**
(0.075)
0.249
26.12***
Shanghai
1.0636***
(0.090)
-0.2565*
(0.130)
0.0715
(0.090)
0.0772*
(0.040)
0.801
1.65
Ningbo
0.7572***
(0.089)
-0.0572
(0.111)
0.0777
(0.089)
0.0768
(0.051)
0.580
3.00
Hefei
0.7352***
(0.087)
0.2850***
(0.102)
-0.1992**
(0.097)
0.0982
(0.065)
0.640
13.24***
Nanchang
0.5306***
(0.089)
0.0566
(0.100)
0.1114
(0.089)
0.1336**
(0.061)
0.389
1.01
Zhengzhou
0.5888***
(0.089)
0.0147
(0.103)
0.1432
(0.090)
0.1086**
(0.047)
0.447
4.30
Changsha
0.5207***
(0.088)
0.2283**
(0.095)
-0.1019
(0.082)
0.1517**
(0.070)
0.418
6.32
Guangzhou
0.5805***
(0.089)
0.2104**
(0.102)
-0.0348
(0.091)
0.1422*
(0.080)
0.512
3.81
Chengdu
0.5154***
(0.089)
0.2242**
(0.096)
0.0024
(0.088)
0.0686*
(0.041)
0.471
2.07
Guiyang
0.5804***
(0.089)
-0.0476
(0.103)
0.0299
(0.089)
0.1588***
(0.061)
0.318
5.89
Xian
0.0792
(0.087)
0.2755***
(0.084)
0.2213**
(0.086)
0.1492*
(0.078)
0.179
2.79
Yinchuan
0.1882**
(0.090)
0.4658***
(0.081)
0.0623
(0.090)
0.0922*
(0.054)
0.369
1.38
Notes: The standard errors are reported in the parentheses. The BG test represents the
Breusch-Godfrey test, following the χ2
3 distribution under the null hypothesis of no serial
correlation of order up to 3. ***, ** and * indicate 1%, 5%, 10% signiﬁcance level, respectively.
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5
Spatial interrelations of Chinese housing markets:
Spatial causality, convergence and diﬀusion
Published as: Gong, Y., Hu, J., Boelhouwer, P., 2016. Spatial interrelations of Chinese
housing markets: Spatial causality, convergence and diﬀusion. Regional Science and
Urban Economics, 59:103-117.
Abstract: This paper comprehensively tests the spatial interrelationships of 10 housing
markets in the Pan-Pearl River Delta (Pan-PRD) in China, including the properties of
spatial causality, convergence and diﬀusion paɦerns. The pairwise Toda-Yamamoto
Granger causality tests suggest widely existing leading-lag relationships between
housing markets; a unidirectional causal ﬂow from the eastern-central area to western
China can be tentatively conﬁrmed. However, there is a lack of suﬃcient evidence
supporting pairwise long-run cointegration and convergence, indicating a diverged
interurban housing market in the Pan-PRD. In the short run, the spatial-temporal
diﬀusion model manifests the importance of the spillover eﬀect from neighbouring
cities in predicting one city’s house price changes. Furthermore, the generalized
impulse response functions (GIRFs) clearly depict the transmission paɦern of shocks to
one chosen city. The diﬀusion paɦern is characterized by the fact that the shocks ﬁrst
spread to nearby cities with cities further away taking a longer time to respond.
Keywords: Spatial causality, long-run convergence, ripple eﬀect, diﬀusion, house
prices, China
JEL: C31, C33, R21, R31
.............................................................................................................................
§ 5.1
Introduction
.............................................................................................................................
After the ﬁnancial crisis in 2008, many governments have been aɦempting to
stimulate depressed housing markets through policy interventions. However, whether
the interventions can work as expected relies heavily on our understanding of housing
markets. To provide deeper insights into house price behaviour, many scholars
advocate an investigation into a series of interrelated regional markets rather than a
single national market (Meen 1996; Yunus and Swanson 2013). Indeed, the structure
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are integrated of order one, i.e., I(1). The ADF tests provide preliminary evidence that
house prices of diﬀerent cities are integrated of diﬀerent orders.
The ADF test has very low power in distinguishing highly persistent stationary
processes from non-stationary processes; the power is lower when a deterministic
trend is included in the test. Thus, an eﬃcient unit root test, the Dickey-Fuller
generalized least square (DF-GLS) test proposed by Ellioɦ et al. (1996), is also
conducted. The results in the second and ﬁfth column of Table 5.1 are largely
consistent with the ADF test, except that the level house price determination of
Xiamen is a unit root process.
TABLE 5.1
Unit root test
level
1st diﬀerence
ADF
DF-GLS
KPSS
ADF
DF-GLS
KPSS
Guangzhou -3.886(2)** -3.361(2)** 0.080(9)
-3.467(1)** -3.494(1)** 0.056(8)
Shenzhen
-2.179(1)
-2.209(1)
0.121(9)*
-3.477(0)** -3.510(0)** 0.078(8)
Fuzhou
-2.538(2)
-1.399(2)
0.157(9)**
-3.642(1)** -3.574(1)*** 0.085(8)
Xiamen
-3.391(2)*
-2.404(2)
0.103(9)
-3.784(1)** -3.642(1)*** 0.086(8)
Nanchang
-1.411(1)
-0.957(1)
0.199(9)**
-6.037(0)*** -5.233(0)*** 0.051(8)
Changsha
-1.852(1)
-0.962(1)
0.226(9)***
-6.596(0)*** -3.899(0)*** 0.046(8)
Nanning
-2.188(1)
-0.862(1)
0.248(9)***
-4.579(0)*** -4.382(0)*** 0.083(8)
Guiyang
-1.044(1)
-0.745(1)
0.286(9)***
-6.289(0)*** -5.908(0)*** 0.042(8)
Kunming
-1.365(1)
-1.764(1)
0.148(8)**
-7.070(0)*** -5.161(0)*** 0.092(7)
Chengdu
-2.623(2)
-2.144(2)
0.127(9)*
-3.628(1)** -3.347(1)** 0.055(8)
Notes: All the models include an intercept and a deterministic trend. Numbers shown in
parentheses are the lag length or bandwidth. For the Augmented Dickey-Fuller (ADF) test
(Dickey and Fuller 1979), the lag length for level variables is selected using the Bayesian
Information Criterion (BIC) with the maximum lag length being set to 12. DF-GLS unit
root tests (Ellioɦ et al. 1996) use the same lag length chosen for ADF test. For KPSS
test (Kwiatkowski et al. 1992), the bandwidth is selected by Newey-West method. In
all the cases, the lag length for ﬁrst diﬀerence variables equals to the lag length for level
variables minus one. The null hypothesis for ADF and DF-GLS tests is having a unit root,
but stationary for the KPSS test. *, ** and *** indicate signiﬁcance at the 10%, 5% and 1%
level, respectively.
Table 5.1(column 3 and 6) also presents the results of the KPSS test (Kwiatkowski et al.
1992), which has the null hypothesis of stationary. The KPSS results support that
Xiamen’s house price series is I(0), in accordance with the ADF test but in
contradiction to the DF-GLS test. The evidence for the integration order of the Xiamen
price series appears to be mildly confusing. Given that the KPSS statistic is very close to
the 10% critical value (0.119), it is reasonable to assume Xiamen’s price process to be
I(1) for the following analysis 10.
10
One reviewer pointed out that house prices are very likely to display an asymmetric adjustment, reducing the
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Because not all the house price series are integrated of the same order, the pairwise
Toda-Yamamoto Granger causality procedure is preferable in this analysis. According
to information criteria, such as AIC and BIC, the optimal lag lengths for most of the
city-pair VARs are 2 and 3. To largely ensure that the residuals are close to white noise,
we proceed using the VAR(3 + d) system in which d is the maximum integration order
of the city pair.
The results of Toda-Yamamoto Granger causality tests are reported in Table 5.2, where
the null hypothesis is that column cities do not Granger cause row cities. Overall, in
most of the city pairs, we ﬁnd signiﬁcant evidence of bilateral or unidirectional
leading-lag relations; however, the causality paɦern is very complicated 11. Certain
literature focusing on housing market interaction, such as Clapp et al. (1995) and Chen
et al. (2011), has claimed that the house price interrelation (or causality) only occurs
between neighbouring markets. Our results cast doubt on this conclusion because
leading-lag relationships are found in many city pairs where the two cities are
separated by very long distances. Conversely, there is no spatial causality in certain
short-distance city pairs, such as Xiamen and Fuzhou. Similarly, Pollakowski and Ray
(1997) and Luo et al. (2007) also found signiﬁcant causality between non-contiguous
regions. Such complicated causality paɦerns may be largely due to economic relations
rather than behavioural reasons (Pollakowski and Ray 1997).
To further examine the results, the cities in the system are divided into two groups: the
eastern-central group including the ﬁrst six cities in Table 5.2 and the western group
containing the remaining cities. We tentatively ﬁnd a general unidirectional causal ﬂow
from the eastern-central area to western China. The historical house price information
of all eastern-central cities, except Xiamen, can signiﬁcantly contribute to predicting
the house prices of western cities. The opposite, conversely, can hardly be true given
the largely insigniﬁcant Granger causality test results in the lower-left panel. However,
among the four western cities, the housing markets of Kunming and Chengdu appear
to play a role in predicting house price behaviours in eastern-central cities. The above
ﬁndings are closely related to the socio-economic disparities between eastern, central
and western China. Considering that eastern-central cities are generally more
developed than western ones, it can be expected that their market dynamics can lead
housing market behaviours in the remaining cities. Chengdu is an exception in western
cities, given its status as the ﬁnancial and economic centre of western China.
Consequently, the mutual leading-lag relationships between Chengdu and most
power of traditional unit root tests. Thus, we employed the momentum threshold autoregressive (MTAR)
asymmetric unit root test proposed by Enders and Granger (1998) to test the integration order of 10 cities’
house price indexes. However, the results, which are available upon request, do not turn over the ﬁnding that all
the cities are I(1) process except for Guangzhou.
11
It should be noted that, throughout the paper, when we say “a market leads or causes another market in the
Granger sense”, we cannot exclude the possibility that such correlation is caused by common shocks.
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TABLE 5.2
Toda-Yamamoto Granger causality test
Guangzhou Shenzhen
Fuzhou
Xiamen
Nanchang
Changsha
Nanning
Guiyang
Kunming
Chengdu
Guangzhou
25.63***
(0.000)
6.47*
(0.091)
50.54***
(0.000)
11.13***
(0.011)
14.10***
(0.003)
11.25***
(0.010)
9.47**
(0.024)
4.41
(0.220)
9.16**
(0.027)
Shenzhen
13.82
(0.003)
9.14**
(0.027)
18.93***
(0.000)
6.05*
(0.109)
18.76***
(0.000)
8.25**
(0.041)
14.14***
(0.003)
6.22*
(0.101)
8.23**
(0.041)
Fuzhou
2.00
(0.572)
1.52
(0.678)
7.52*
(0.057)
5.04
(0.169)
7.97**
(0.047)
8.52**
(0.036)
6.49*
(0.090)
8.26**
(0.041)
6.78*
(0.079)
Xiamen
18.73***
(0.000)
2.32
(0.510)
3.38
(0.336)
4.10
(0.251)
2.82
(0.420)
5.28
(0.153)
2.01
(0.570)
0.28
(0.963)
0.94
(0.815)
Nanchang
4.47
(0.215)
7.90**
(0.048)
8.25**
(0.041)
6.42*
(0.093)
14.61***
(0.002)
12.22***
(0.007)
17.52***
(0.001)
11.44***
(0.010)
11.55***
(0.009)
Changsha
1.03
(0.794)
5.31
(0.150)
7.48*
(0.058)
6.11*
(0.106)
6.74*
(0.081)
13.36***
(0.004)
34.22***
(0.000)
13.29***
(0.004)
7.99**
(0.046)
Nanning
8.91**
(0.031)
14.22***
(0.003)
4.92
(0.178)
3.52
(0.318)
5.08
(0.166)
9.51**
(0.023)
19.42***
(0.000)
8.64**
(0.035)
8.70**
(0.034)
Guiyang
0.41
(0.938)
2.37
(0.500)
2.32
(0.509)
0.67
(0.880)
5.31
(0.150)
5.07
(0.167)
5.29
(0.152)
1.38
(0.711)
7.46*
(0.058)
Kunming
7.27*
(0.064)
11.59***
(0.009)
1.79
(0.616)
5.96
(0.113)
7.30*
(0.063)
27.03***
(0.000)
8.31**
(0.040)
11.68***
(0.009)
10.32**
(0.016)
Chengdu
5.57
(0.135)
5.24
(0.155)
16.81***
(0.001)
17.69***
(0.001)
15.25***
(0.002)
8.21**
(0.042)
6.20*
(0.102)
7.24*
(0.064)
7.19*
(0.066)
Notes: Toda-Yamamoto Granger causality test is performed under VAR(3+d) system where d is the maximum integration order of house price
processes in the city pair. Both intercept and deterministic time trend are included in the VAR system. The null hypotheses are that house prices
in column cities do not Granger cause house prices in row cities. The asymptotic Wald statistics are reported, and the corresponding p-values are
shown in parentheses. *, ** and *** indicate signiﬁcance at the 10%, 5% and 1% level, respectively.
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eastern-central cities are no surprise. Furthermore, within either the eastern-central or
western group, the signiﬁcant leading-lag relationship (at least for one direction) can
be found in every city pair.
Given the leading position of Guangdong in the Pan-PRD, we expect that the housing
markets of the two cities under its territory, namely Guangzhou and Shenzhen, will lead
the markets in other cities. Indeed, Guangzhou and Shenzhen impose a signiﬁcant
leading inﬂuence on nearly all the other housing markets, and they are less predictable
on the basis of previous information from other markets. However, these cities’
dominant role is not unique. Certain other cities appear to also have similar
‘exogenous’ properties, such as Nanchang.
.............................................................................................................................
§ 5.5
Long-run properties
.............................................................................................................................
§ 5.5.1
Cointegration and convergence test
.............................................................................................................................
The previous section reveals that the 10 cities’ housing markets in the Pan-PRD are
interrelated with each other. In this section, we go further to ask the question of
whether these markets are tied together in the long-run, i.e., if they hold a long-run
equilibrium relationship. To answer this question, the long-run cointegration and
convergence properties of house prices are investigated.Two I(1) house price series p1t
and p2t are said to be cointegrated if a linear combination of p1t and p2t is stationary.
Since we are interested in the pairwise cointegration of house price indexes of two
cities, the Engle-Granger (EG) two-step procedure (Engle and Granger 1987) is
employed in this paper, which has been applied by, for example, MacDonald and Taylor
(1993) to a similar question.
The ﬁrst step is to estimate the long-run equilibrium relationship by the following
equation
pit = D + βp2t + ut
(3)
where D is deterministic terms that may contain a constant, a deterministic trend or
both. The cointegration test involved in the second step is then based on testing the
unit root of residual series ut. If ut is stationary, we say that p1t and p2t are cointegrated
with (1,−β). Because of the spurious regression under the null hypothesis of
non-cointegration in the ﬁrst stage, the residual-based ADF test in the second stage
does not have the standard Dickey-Fuller distribution. Therefore, critical values for
cointegration test simulated by MacKinnon (1996) are used in this paper.
As noted by Holly et al.(2011) and Abboɦ and Vita (2013), conditional on
cointegration, the long-run convergence of house prices necessitates two additional
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conditions: (1) the cointegrating vector corresponding to house price series being
(1,-1), and (2) the lack of a deterministic trend being presented in cointegration space.
The long-run convergence property is tested by the so-called pair-wise approach, which
has been used by Holmes et al. (2011) and Abboɦ and Vita (2013). Speciﬁcally, the
cointegrating vector is pre-speciﬁed with form (1,-1), and then any standard unit root
test can be directly used to test the stationarity of cointegrating residuals.
TABLE 5.3
Engle-Granger pairwise cointegration results
Shenzhen Fuzhou
Xiamen
Nanchang Changsha Nanning
Guiyang
Kunming Chengdu
Shenzhen
-0.633
-3.094
-2.053
-13.706
-0.837
-4.413
-0.532
-2.092
-0.520
-1.557
-0.919
-3.967
-0.723
-2.718
-0.067
-0.280
Fuzhou
-0.221
-1.289
-0.855
-3.683
-2.094
-5.553
-1.716
-5.963
-1.981
-7.077
-2.204
-9.371
-2.239
-7.201
-2.711
-10.044
Xiamen
-1.875
-13.728
-1.129
-4.692
-0.791
-2.390
-1.362
-5.642
-1.248
-3.800
-1.482
-5.885
-1.339
-4.200
-0.831
-3.591
Nanchang -0.501
-3.266
-2.423
-6.421
-0.604
-1.918
-3.792*
-27.395**
-2.136
-6.507
-3.086*
-19.217*
-3.172
-14.187
-2.615
-14.058
Changsha -0.025
-0.129
-1.774
-6.234
-1.279
-5.748
-3.722*
-26.892**
-1.657
-3.287
-3.331*
-20.505*
-4.069**
-20.546*
-2.381
-11.213
Nanning
0.309
1.225
-1.762
-6.234
-0.693
-2.155
-1.693
-5.007
-1.162
-2.189
-1.988
-6.342
-1.938
-5.058
-1.593
-4.286
Guiyang
-0.580
-3.231
-2.387
-10.249
-1.498
-6.588
-3.086*
-19.214*
-3.435*
-21.148*
-2.453
-8.041
-2.907
-14.028
-2.680
-15.741
Kunming -0.511
-2.369
-2.793
-8.815
-1.438
-4.608
-3.470*
-15.207
-4.479**
-22.088*
-2.856
-7.390
-3.118
-14.779
-3.169
-13.768
Chengdu 0.330
1.649
-2.919
-10.753
-0.661
-2.936
-2.569
-13.714
-2.420
-11.287
-2.017
-5.444
-2.601
-14.893
-2.843
-12.465
Notes: The two statistics in each cell are, respectively the τ statistic and z statistic of ADF test with the
null hypothesis of no cointegration. The MacKinnon (1996) critical values are used. In each cointegration
equation, the row cities are deﬁned as dependent variables. The constant is included in the cointegration
space in all city pairs except the Nanchang-Guiyang pair, in which the constant is not signiﬁcant. * and **
indicate the 5% and 1% signiﬁcance level, respectively.
§ 5.5.2
Empirical results
.............................................................................................................................
Table 5.3 demonstrates the results of the pairwise Engle-Grange cointegration test.
Guangzhou is excluded from cointegration analysis because it is I(0) according to our
unit root test results. A brief view of the results indicates that cointegration
relationships rarely exist between cities, in contrast to the widely existing leading-lag
relationships. Among the city pairs that are tied together in the long-run, three cities,
Nanchang, Changsha and Guiyang, form a ‘cointegration club’ within which every city
cointegrates with each other. In addition, the signiﬁcant long-run equilibrium
relationship can also be observed in the city pair of Changsha-Kunming. We also note
that none of the three eastern cities, namely Shenzhen, Fuzhou and Xiamen, is
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The Spatial Dimension of House Prices


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cointegrated with each other or with the remaining central and western cities. This
might indicate that in the long-run, the housing market conditions in eastern cities still
signiﬁcantly diﬀer from the markets of the remaining cities, although much eﬀort has
been made to promote the integration process of Pan-PRD’s economy.
TABLE 5.4
Pairwise convergence results with pre-speciﬁed coeﬃcients (1,-1)
Shenzhen Fuzhou
Xiamen
Nanchang Changsha Nanning Guiyang Kunming Chengdu
Shenzhen
-0.919
-1.838
-1.071
-0.875
-0.667
-1.077
0.563
1.040
Fuzhou
-0.642
-1.060
-0.796
-0.940
-1.009
-1.448
0.146
1.044
Xiamen
-2.109
-0.927
-0.840
-1.047
-0.880
-1.510
0.803
1.109
Nanchang -0.853
-1.843
-0.733
-0.546
-0.612
-2.994** 0.206
0.588
Changsha -1.144
-1.607
-1.529
-2.729
-0.648
-0.465
0.348
0.614
Nanning
-0.480
-2.144
-1.132
-2.090
-0.594
-0.577
0.174
0.392
Guiyang
-0.882
-2.170
-1.468
-3.075*
-2.900*
-2.451
-0.009
-0.056
Kunming -0.480
-3.383* -1.464
-3.550** -4.012** -3.770** -2.949*
-1.154
Chengdu 0.509
-4.018** -0.491
-1.810
-2.973*
-3.443* -1.542
-2.124
Notes: The null hypothesis of no convergence is tested based on the residual from pit −pjt. The
results of ADF test (Dickey and Fuller 1979) are reported in the lower triangle, whereas the upper
triangle shows the results of DF-GLS tests (Ellioɦ et al. 1996). In the unit root test process, the
constant is included, and the lag length (not reported) is automatically selected by the Bayesian
Information Criterion (BIC). * and ** indicate the 5% and 1% signiﬁcance level, respectively.
After examining the cointegration, we proceed to test the more restricted long-run
convergence properties. The pairwise cointegration results with pre-speciﬁed
coeﬃcients (1,-1) are shown in Table 5.4. The ADF test in the lower triangle suggests
that among the four cointegration city pairs, there are three convergent pairs:
Nanchang-Guiyang, Changsha-Guiyang and Changsha-Kunming. In addition, another
seven city pairs, which are not cointegrated in the Engle-Granger cointegration test are
found to be signiﬁcantly converged. Considering that cointegration is a necessary
condition of convergence, the convergent results of these seven pairs are a surprise.
These contradictory results might be due to the low power of the ADF test in detecting
the unit root. To verify the ADF test, we also perform the more eﬃcient DF-GLS test,
the results of which are reported in the upper triangle of Table 5.4. This time we ﬁnd
only one signiﬁcantly convergent pair, the Nanchang-Guiyang pair, which is also
cointegrated. Note that regardless which unit root test we used, city pairs that are
convergent are rare. Therefore, it is reasonable to conclude that cointegration or
convergence is unlikely to widely exist among the nine cities, which indicates a diverged
interurban housing market in the Pan-PRD.
To check the robustness of the cointegration and convergence test results based on the
two-step procedure, we also conduct two additional tests: the pairwise Johansen
cointegration test and the two-step convergence test based on the momentum
threshold autoregressive (MTAR) unit root test (Enders and Granger 1998) which can
allow an asymmetric adjustment. The trace statistics of the Johansen procedure,
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computed based on a VAR(3) speciﬁcation with unrestricted intercept and no trend in
VAR, are reported in the lower triangle of Table A1. The null hypothesis is that the
column city is not cointegrated with the row city. For those cointegrated city pairs, the
upper triangle of Table A1 reports the results of the log-likelihood ratio (LR) test, which
is used to test the cointegrating vector restriction (1,-1). Table A2 displays the pairwise
cointegration results with pre-speciﬁed coeﬃcients (1,-1) based on the MTAR unit root
test. Both of these two powerful methods identify a similar cointegration and
convergence paɦern between cities with the two-step procedure does; the results are
largely in line with the lower triangle results of Table 5.4. Thus, we are conﬁdent of the
previous ﬁnding that only very few city pairs are found to be cointegrated or convergent.
For the following analysis, we mainly rely on the results of the two-step procedure.
.............................................................................................................................
§ 5.6
House price diﬀusion paɢern
.............................................................................................................................
§ 5.6.1
Spatial-temporal house price diﬀusion model
.............................................................................................................................
Previous analysis suggests that most of the city pairs do not hold a long-run
equilibrium relationship; however, a few pairs do. When modelling the house price
dynamics of a city, we should consider the interrelation with both the cointegrated
cities and the non-cointegrated cities. In other words, we should consider the inﬂuence
from the cities that can impose a long-term eﬀect and the cities that only have a
transitory eﬀect. In addition, the spatial dimension should also be considered because
it is likely that the eﬀect imposed by nearby cities is stronger than the inﬂuence of
distant cities. The spatial-temporal house price diﬀusion model adopted in this paper
can fully capture the characteristics along both spatial and temporal dimensions. This
model is a variant of spatial-temporal diﬀusion model proposed by Holly et al. (2011)
(the Holly model), which has been applied to investigating the eﬀects of language
border on the diﬀusion of house prices in Belgian markets by Helgers and Buyst (2016).
However, unlike the Holly model, we do not designate a ‘dominant’ city, which is
assumed to have contemporaneous eﬀects on non-dominant cities. The reason for
abandoning the ‘dominant’ city from our model speciﬁcation is that the general lack of
pairwise long-run cointegration relationships between cities found previously suggests
that there is no city that can be seen as the long-run forcing for other cities. In our
model speciﬁcation, the house price series in the system excluding pit is split into two
groups because of the existence of a ‘cointegration club’: one group (denoted by C)
being cointegrated with pit and the other (denoted by O) not. A ﬁrst order error
correction speciﬁcation for pit is given by
∆pit = ϕi0
(
pi,t−1 −βi¯pC
i,t−1
)
+ai + ai1∆pi,t−1 + bi1∆¯pC
i,t−1 + ci1∆¯pO
i,t−1 + ϵit,
(4)
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provide signiﬁcant evidence of cointegration for Nanchang, Changsha and Kunming. In
the case of Guiyang, we cannot reject the null hypothesis of no cointegration at the 5%
signiﬁcance level; however, the statistics are marginally signiﬁcant at the 10% level.
The third column of Table 5.5 reports the estimated long-run relationships (βi). The
estimated βi are approximately distributed around the value of unity, except for
Kunming.
TABLE 5.5
Cointegration between the four cointegrating cities and spatial
lag of their cointegrated counterparts
τ statistic
z statistic
β
Nanchang
-3.561*
-25.128*
0.843441
Changsha
-4.541**
-34.150**
1.287945
Guiyang
-3.089
-18.209
0.871291
Kunming
-4.069**
-20.546*
0.602230
Notes: The test is based on one equation regression in which we take column cities
as dependent variables and the spatial lags of their cointegrated counterparts as
independent variables. The ﬁrst two columns report the τ statistic and z statistic of
ADF test with the null hypothesis of no cointegration. The third column reports the
estimation of β. Note that the t-ratio for β is invalid in this case. * and ** indicate
the 5% and 1% signiﬁcance level, respectively.
With βi being determined, the spatial-temporal house price dynamic model for each
city can be estimated using ordinary least squares (OLS). The estimation results are
summarized in Table 5.6 where the lag-orders are set to 2. These models perform
reasonably well because the Breusch-Godfery test suggests no serial correlation in each
regression’s residuals at least at the 5% signiﬁcance level.
The error correction terms, which appear in the model speciﬁcation of four cities, are all
signiﬁcant at the 10% signiﬁcance level, three of which are signiﬁcant at the 5% level or
above. That is, the four cities’ short-run dynamics are inﬂuenced by the deviation from
the long-run equilibrium relationship. The coeﬃcient ϕi0 indicates that the house price
of Changsha responds to the disequilibrium much more rapidly than that of the other
three cities.
We now turn to the inﬂuence of short-term dynamics. Not surprisingly, the ﬁrst-lag
price changes are signiﬁcantly positive in all equations. The second-order lagged price
dynamics also play a role in the price equation for Fuzhou and Xiamen. Similarly, the
lagged price changes from neighbouring cities (either from the cointegration group or
the non-cointegration group) are also found to be statistically signiﬁcant in all
equations, except for Changsha and Kunming. This conﬁrms the existence of cross-city
spillover eﬀects from the neighbouring cities, which is in accordance with the ﬁndings
of Holly et al. (2011) for the UK market and Helgers and Buyst (2016) for Belgian
housing markets.
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TABLE 5.6
Estimation results of the spatial-temporal house price diﬀusion model
City
Constant
Error
correction
Own lag eﬀects
Lag eﬀects of
cointegration group
Lag eﬀects of non-
cointegrated group
Adjusted
R2
Serial
correlation
ϕi0
ai1
ai2
bi1
bi2
ci1
ci2
Guangzhou
0.0007
(0.001)
–
0.4326***
(0.098)
0.0765
(0.108)
–
–
0.6540***
(0.182)
-0.3067*
(0.161)
0.514
5.452
Shenzhen
0.0011
(0.001)
–
0.6464***
(0.106)
-0.0038
(0.106)
–
–
0.9324**
(0.197)
-
0.7141***
(0.200)
0.527
6.035
Fuzhou
0.0005
(0.001)
–
0.3074***
(0.100)
0.2321*
(0.097)
–
–
0.5384***
(0.140)
-0.2231
(0.141)
0.548
0.588
Xiamen
0.0008
(0.001)
–
0.1673*
(0.096)
0.1624*
(0.093)
–
–
0.7223***
(0.167)
-0.1577
(0.180)
0.477
2.344
Nanchang
0.0548*
(0.028)
-0.0755*
(0.040)
0.3530***
(0.104)
0.0723
(0.108)
0.0737
(0.098)
-0.0436
(0.096)
0.3720**
(0.171)
-0.0430
(0.174)
0.425
4.708
Changsha
-
0.2594***
(0.062)
-
0.2006***
(0.048)
0.3654***
(0.093)
0.0726
(0.092)
0.1437
(0.158)
-0.0934
(0.159)
0.1763
(0.192)
0.1628
(0.195)
0.541
1.271
Nanning
0.0003
(0.001)
–
0.4869***
(0.106)
0.0244
(0.101)
–
–
0.3624**
(0.171)
0.0928
(0.170)
0.615
3.111
Guiyang
0.0414**
(0.020)
-
0.0696**
(0.034)
0.4129***
(0.095)
-0.0258
(0.092)
0.4800***
(0.109)
-0.2064*
(0.111)
0.0976
(0.155)
-0.0413
(0.165)
0.467
7.778*
Kunming
0.1915***
(0.059)
-
0.1092***
(0.034)
0.3423***
(0.102)
-0.0546
(0.098)
0.1323
(0.109)
-0.1476
(0.106)
0.3847
(0.263)
0.0062
(0.232)
0.258
0.855
Chengdu
-0.0002
(0.000)
–
0.2419**
(0.099)
0.0420
(0.098)
–
–
0.3315***
(0.120)
0.1753
(0.124)
0.550
1.934
Notes: The estimation results are based on equation (4) with lag-orders being set to 2. For the cities except Nanchang, Changsha, Guiyang and
Kunming, we eliminate the error correction term and the term ‘lag eﬀects of cointegration group’ because these cities do not have cointegrated
counterparts. The “serial correlation” column reports the Breusch-Godfery serial correlation statistic test that distributes approximately as χ2
3 under
the null hypothesis of no residual serial correlation. *, ** and *** indicate the 10%, 5% and 1% signiﬁcance level, respectively.
120
The Spatial Dimension of House Prices


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The regression results of spatial-temporal diﬀusion models presented in Table 5.6
depict a complicated dynamic system in which the historical house price changes of a
city not only aﬀect its own price changes but also inﬂuence the price changes in other
cities directly or indirectly through their neighbouring cities, or through the long-run
equilibrium. To intuitively illustrate the diﬀuse nature of house prices in a complicated
system, we provide the generalized impulse response functions, which can trace the
time proﬁle of shocks both over time and space.
FIGURE 5.3
Generalized impulse responses of a positive unit shock (one standard
error) to Shenzhen house prices
Figure 5.3 plots the generalized impulse responses of all the cities to a positive unit
shock (one standard error) to the house prices of Shenzhen, one of the most developed
cities in the Pan-PRD. The positive shock gradually diﬀuses to the remaining cities,
signiﬁcantly raising the house prices in the whole area (being conﬁrmed by the
bootstrap conﬁdence interval in Figure B1 in the Appendix). However, the magnitude
of the spillover eﬀect diﬀers across the region. Given the one standard error shock to
Shenzhen, its own house prices soar approximately 3.5%, followed by Guangzhou,
which rises by approximately 2%. Conversely, the increases of the other cities’ house
prices are approximately 1%. This indicates a diverged interurban housing market
between developed and less-developed cities. For the sake of comparison, Figure 5.4
portrays the responses to a positive stand error shock to Changsha, a city in Central
China and cointegrating with the other three cities. It is clear that the unit shock to
Changsha generates relatively homogenous eﬀects on all other cities’ house prices
(house prices increases are approximately between 0.8% and 1%), except for Shenzhen
and Kunming. The eﬀect on Shenzhen house prices is not signiﬁcant, as indicated by
the bootstrap error bounds shown in Figure B2 (refer to the Appendix). This
information further supports our conclusion regarding the divergence of a few cities’
121
Spatial interrelations of Chinese housing markets: Spatial causality, convergence and diﬀusion


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.............................................................................................................................
Appendices
.............................................................................................................................
Appendix A
.............................................................................................................................
TABLE A1
The results of pairwise Johansen cointegration test
Shenzhen Fuzhou
Xiamen
Nanchang Changsha Nanning Guiyang Kunming Chengdu
Shenzhen
Fuzhou
7.60
1.95
Xiamen
8.69
7.44
Nanchang 7.25
11.11
9.05
7.19**
0.16
1.57
Changsha 10.02
9.83
7.27
20.27**
4.45*
8.80**
Nanning
10.74
12.66
9.01
12.77
10.26
2.05
Guiyang
6.81
14.52
10.30
18.03*
18.22*
13.74
0.01
Kunming 2.40
13.20
6.46
16.89*
25.04**
18.55*
17.07*
Chengdu 6.75
16.61*
4.12
15.46
13.93
12.94
15.03
13.22
Notes: The lower triangle cells report the trace statistic of the pairwise Johansen cointegration test
under the null hypothesis H0 : r = 0. The test is based on a VAR(3) speciﬁcation, with unrestricted
intercept and no trend in VAR. For the co-integrated city pairs, the upper triangle cells report the
log-likelihood ratio (LR) test for the cointegrating vector restriction (1,-1). * and ** denote 5% and
1% signiﬁcance level, respectively.
TABLE A2
Pairwise convergence with pre-speciﬁed coeﬃcients (1,-1) base on the MTAR
unit root test
Shenzhen Fuzhou
Xiamen
Nanchang Changsha Nanning Guiyang Kunming Chengdu
Shenzhen
Fuzhou
2.08(1)
Xiamen
3.72(1)
0.70(2)
Nanchang 1.54(1)
1.43(1)
0.48(1)
Changsha 3.19(1)
3.93(2)
1.40(2)
4.87(1)
Nanning
0.18(1)
2.76(1)
1.96(2)
2.15(1)
0.32(1)
Guiyang
1.90(1)
2.38(1)
1.55(2)
4.73(1)
4.66(1)
3.03(1)
Kunming 0.91(1)
5.17*(1) 2.17(1)
7.81**(1) 8.29**(1) 6.01*(1) 4.10(1)
Chengdu 0.24(1)
4.34(1)
4.60(2)
1.67(1)
3.72(1)
5.96*(1) 1.20(1)
2.71(1)
Notes: The null hypothesis of no convergence is tested based on the residual from pit −pjt. In all
the models a constant is included, and the lag length is shown in the parentheses. The 5% and
1% critical values are 5.02 and 7.10, respectively. * and ** denote 5% and 1% signiﬁcance level,
respectively.
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The Spatial Dimension of House Prices


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Appendix B: Bootstrap GIRF conﬁdence intervals
.............................................................................................................................
The methods for computing the bootstrap conﬁdence intervals of the generalized
spatial-temporal impulse response functions are borrowed from Holly et al.(2011). The
estimated model (7) is ﬁrst used to generate B bootstrap samples. The bth bootstrap
sample can be obtained by the following Data Generation Process (DGP)
p(b)
t
= ˆa + ˆΦ1p(b)
t−1 + ˆΦ2p(b)
t−2 + ˆϵ(b)
t
,
(B.1)
where ˆϵ(b)
t
= ˆΣ
1/2v∗(b)
t
. The elements of v∗(b)
t
are recursively replaced by the values that
are randomly drawn from the transformed residual matrix ˆΣ
−1/2 (ˆϵ1, ˆϵ2, . . . , ˆϵt). Note
that in equation (B.1), the ﬁrst 2 observations are replaced by the original data.
When obtaining the bootstrap sample p(b)
t
, we estimate the model (7) again and
produce the bth bootstrap GIRF
g(b)
i
(h) =
ˆΨ
(b)
h
ˆΣ
(b)ei
√
ˆσ(b)
ii
,
for
i = 1, · · · , n; h = 0, 1, · · · , H.
(B.2)
The lower and upper bands of 100(1 −α)% conﬁdence interval are equivalent to the
α/2 and 1 −α/2 quantiles of B g(b)
i
(h) for each i and h.
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When estimating the MGWR model, we used the adaptive bi-square function to
construct the weighting scheme, given that the transactions were not evenly
distributed across space. In this case, the bandwidth is generally referred to as the
window size, and the choice of window size is equivalent to the choice of the number of
nearest neighbors. To ﬁnd the optimal value, we varied the window size from 10% to
95% using a 5% interval and selected the size that yielded the lowest CV score as given
by equation (18). Each annual sample has a unique optimal window size. The CV
scores indicated that a 10% window size was preferred for most of the years, except for
1999, 2000 and 2002, with an optimal size of 15%, and 2003, with an optimal size of
30%. However, for the construction of price indexes, we would prefer using the same
window size for all years, especially since the number of sales is almost evenly spread
across the whole period. So we chose a window size of 10% for each year, leading to 60
nearest neighbors that were used in the estimation of the annual MGWR models.
TABLE 6.1
Parameter estimates for structures characteristics, 2007
OLS
OLSD
MGWR
Intercept
1480.70***
(46.93)
1405.41***
(53.71)
1395.76***
(57.51)
Building period:1960-1970
-370.48***
(25.94)
-389.50***
(36.67)
-398.40***
(41.75)
Building period:1971-1980
-311.17***
(23.36)
-261.50***
(33.96)
-323.50***
(41.75)
Building period:1981-1990
-232.93***
(23.37)
-173.08***
(32.59)
-226.14***
(42.87)
Building period:1991-2000
-58.64***
(21.64)
-49.34*
(26.55)
-115.13***
(37.26)
Terrace
-285.65***
(35.17)
-264.34***
(35.24)
-187.28***
(37.32)
Corner
-281.36***
(31.77)
-274.54***
(31.18)
-192.85***
(34.07)
Semidetached
-122.89**
(47.96)
-149.50***
(47.57)
-96.93**
(48.73)
Duplex
-151.08***
(30.60)
-147.24***
(30.17)
-104.56***
(31.03)
Notes:Standard errors are reported in parentheses; ***, ** and * denote signiﬁcance at the 1%
, 5% and 10% level, respectively.
As an illustration, Table 6.1 shows the 2007 parameter estimates for the structures
characteristics. Almost all of the estimates diﬀer signiﬁcantly from zero at the 1% level.
To some extent they vary across the diﬀerent models. For example, the OLS intercept
term is relatively high compared to the OLSD and MGWR intercepts. Note that, since
dummy variables for houses built after 2000 and for detached houses were not
included, the intercept measures the price in euros of structures per square meter of
living space for detached houses built after 2000. In accordance with a priori
expectations, detached dwellings are more expensive than other types of houses. For
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The Spatial Dimension of House Prices


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all models, there is a clear tendency for the structures to become less expensive as they
are geɦing older.
TABLE 6.2
Summary statistics for estimated land prices
OLS
OLSD
MGWR
Mean
S.D.
Min
Max
Median
Mean
S.D.
1998
116.80
131.50
31.14
72.30
231.03
122.66
125.49
28.66
1999
154.64
178.50
34.85
105.95
223.66
174.07
167.77
30.39
2000
239.77
239.41
36.24
138.53
319.32
251.34
241.83
44.27
2001
214.54
235.58
47.59
110.41
295.01
229.52
226.70
48.77
2002
234.77
245.11
38.41
156.15
323.63
255.05
242.23
40.89
2003
166.07
185.11
44.23
82.12
248.23
179.93
172.26
44.55
2004
186.40
197.19
29.75
104.95
254.20
197.70
195.41
33.78
2005
226.13
224.11
36.55
127.53
299.74
214.19
205.89
35.17
2006
202.84
195.77
30.85
125.90
274.24
207.43
201.27
32.05
2007
214.87
236.73
27.96
141.46
286.91
235.07
229.25
30.99
Notes: For OLS, the land price estimates are reported. For OLSD, the columns show the
weighted mean and standard deviation of the estimated land prices for 9 postcode areas where
the weights are equal to the share of transactions within each postcode area. For MGWR, the
columns provide summary statistics for the land price estimates of all transacted properties.
Table 6.2 contains summary statistics for the estimated price per square meter of land
from the three models. The three average land price series exhibit a similar paɦern
over time, which diﬀers substantially from the changes in the average transaction price
of the properties (see Table A1 in the Appendix). After a sharp increase in 1999, the
estimated average land price ﬂuctuated during a couple of years, experienced a
dramatic drop in 2003, and then increased again.
As mentioned earlier, a virtue of MGWR is that it allows us to plot a continuous map
with estimated prices of land per square meter. For the year 2007, such a map is
depicted in Figure 6.1 for the city of “A”, where the land prices have been rescaled to
the range [0,1]. The postcode areas are indicated as well. While the spatial paɦern in
Figure 6.1 is largely consistent with the paɦern found using the OLSD model (shown in
Figure A1 in the appendix), the MGWR land prices estimates do vary within some of
the areas. This suggests that the use of postcode dummies, as in the OLSD model, is a
rather crude strategy to incorporate spatial variation of land prices.
To formally compare the performance of the three hedonic models, two statistics were
calculated, the Corrected Akaike Information Criterion (AICc) and the Root Mean
Square Error (RMSE). The AICc takes into account the trade-oﬀbetween goodness of ﬁt
and degrees of freedom. The AICc expressions for the OLS and OLSD models can be
found in Hurvich and Tsai (1989). And for MGWR models, it is deﬁned by
AICc = 2nln(ˆσ) + nln(2π) + n
(
n + tr(S)
n −2 −tr(S)
)
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Accounting for spatial variation of land prices in hedonic imputation house price indexes: A semi-parametric
approach


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FIGURE 6.1
Price of land per square meter, 2007
where ˆσ is the estimated standard deviation of the error term and tr(S) the trace of the
hat matrix described in section 6.3.2. The RMSE measures the variability of the
absolute prediction errors of the models and is given by
RMSE = 1
n
√∑
i
(pi −ˆpi)2.
Table 6.3 shows the AICc and RMSE and their diﬀerences for the three models. A rule
of thumb states that if the diﬀerence in the AICc for two models is larger than 3, a
signiﬁcant diﬀerence exists in their performance (Fotheringham et al. 2002). It can be
seen that the OLSD model performs much beɦer than the OLS model in all years, as we
would expect, and in turn that the MGWR model outperforms the OLSD model (except
for 2003, when the diﬀerence is insigniﬁcant). The same ranking is found if the RMSE
is used to assess the various models. These results conﬁrm the earlier ﬁnding that land
prices vary across space, both across and within postcode areas.
Although MGWR is obviously beɦer suited to model the variation of land prices and to
predict property prices, the OLSD model does a surprisingly good job. In several years,
150
The Spatial Dimension of House Prices