Discussion Paper No.348
SPATIAL AUTOREGRESSIVE CONDITIONALHETEROSCEDASTICITY MODEL AND ITSAPPLICATIONTAKAKI SATO YASUMASA MATSUDA

April 26, 2016

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANSPATIAL AUTOREGRESSIVE CONDITIONALHETEROSCEDASTICITY MODEL AND ITSAPPLICATIONTAKAKI SATO∗YASUMASA MATSUDA†

Abstract
This paper proposes spatial autoregressive conditional heteroscedasticity (SARCH) models to estimate spatial volatility in spatial data. S-ARCH model is aspatial extension of time series ARCH model. S-ARCH models specify conditionalvariances as the variances given the values of surrounding observations in spatial
data, which is regarded as a spatial extension of time series ARCH models that specify conditional variances as the variances given the values of past observations. We
consider parameter estimation for S-ARCH models by maximum likelihood method
and propose test statistics for ARCH eﬀects in spatial data. We demonstrate theempirical properties by simulation studies and real data analysis of land price datain Tokyo.

1

INTRODUCTION

In the field of finance, volatility is one of the important factor. Volatility is related to
calculating Value at Risk or pricing of derivatives. Volatility has two features. First, we
can’t observe volatility directly. One approach about this problem is to estimate volatility
from past data. Second feature of volatility is that there exists volatility clusters. Thismeans that volatility is very high for certain time periods and very low for other periods.
To estimate volatility of time series data, Engle (1982) proposed ARCH model. Bollerslev(1986) introduced GARCH model which is extension of ARCH model. These models
are widely accepted and commonly used to estimate and forecast volatility. Dolde andTirtiroglu (1997) deal with volatility of real estate. Real estate is a financial data, whileat the same time has a property of spatial data.
An important property of spatial data sample is a spatial dependency. Spatial dependency means an observation of some spatial point has similar property of observationsof near location. This property is called the first law of geography (Tobler (1970)). To
estimate or predict better, many statistician propose some spatial statistics models which
treat this spatial dependency. There are some spatial econometrics models. SAR (Spatial∗

Graduate School of Economics and Management, Tohoku University, Sendai 980-8576, Japan.
satotakananana@gmail.com†
Graduate School of Economics and Management, Tohoku University, Sendai 980-8576, Japan.
matsuda@econ.tohoku.ac.jp1Autoregressive) model which is proposed in Ord (1975) is the most basic model in spatial
econometrics. SAR model include spatial lag of dependent variables as spatial interaction. As a natural extension of SAR model, there are some models which has spatial
lag of dependent variables or disturbance terms. Kelejian and Prucha (2010) proposed
heteroscedasticity model which contains spatial lags in the dependent variable, exogenous
variables, and the disturbance terms.
In this paper, we deal with spatial volatility model. This study has two motivate.
First, estimating volatility serve to make real estate portfolio. Secondly, there is a chancethat spatial model can extend to spatiotemporal model. We could estimate spatiotemporalripple eﬀect of volatility in spatial data and also time series data.
There has been little study done concerning spatial volatility model. Robinson (2009)
apply idea of time-series stochastic volatility model to spatial model. On the other hand,
Bera and Simlai (2005) propose ARCH type spatial volatility model but little attention
has been given to statistical property of that model.
By extending ARCH model, we propose spatial econometrics model to estimate spatial
volatility. That is spatial autoregressive conditional heteroscedasticity model (S-ARCH
model). An important diﬀerence from previous study which try to apply ARCH model tospatial model is that volatility structure is described by log volatility form. This approach
has advantage. We can change model into SAR model which has a special kind of error
term. Therefore, condition that S-ARCH model can be estimated is same as SAR model.
The paper proceeds as follows. In section2, SAR model and S-ARCH model arepresented. estimation methods of these models are presented in section3. Section 4 deals
with empirical analysis of S-ARCH model. Both simulation study and real data analysis
are reported. Section 5 provides the conclusions.

2

MODEL

In this section, we briefly review the SAR model and propose S-ARCH model. SAR
model is basic spatial econometric model and can capture spatial correlation of dependent
variable. In section 2.2, S-ARCH model is defined as natural extension of time series
ARCH model.

2.1

SPATIAL AUTOREGRESSIVE MODELThe SARmodels can be expressed as shown in (1).
y = ρW y + Xβ + ϵ,ϵ ∼ N (0, σ 2 In ),(1)

where n is the total number of spatial units, y is an n × 1 vector of dependent variables,
and X is an n × k matrix of independent variables. W is a specified constant spatial
weights matrix. This matrix is based on physical distance or contiguity of spatial units.
ρ is scalar parameter which express strength of spatial dependency, β is an n × 1 vectors
whose elements are parameters, and ϵ is an error term. We assume ϵ follows multivariate
normal distribution with mean 0 and homoscedastic variance σ 2 .

2We can express equation (1) as y = (In −ρW )−1 (Xβ +ϵ). We need to check existence
condition of (In − ρW )−1 . Basically, spatial weight matrix is symmetric matrix. Underassumption that spatial weight matrix is row-normalized symmetric matrix, (In − ρW )−1
exsits if ρ ∈ (−1, 1) (Lee (2004)). Therefore, ρ is restricted to the interval(−1, 1).

2.2

SPATIAL AUTOREGRESSIVE CONDITIONALHETEROSCEDASTICITY MODEL

We propose spatial autoregressive conditional heteroscedasticity model(S-ARCH model).
S-ARCH model is natural extension of ARCH model. In ARCH model, conditional variance is described by past observations. On the other hand, conditional variance is described by surrounding area’s observations in S-ARCH model.
The S-ARCHmodels can be expressed as shown in (2).
σ11 ϵ1y =  ...  ,(2)σnn ϵnϵi ∼ IID(0, 1),
yi 2 = σii 2 ϵ2ilogyi 2 = logσii 2 + logϵ2i ,n∑2logσii = α0 + α1
wij logyj2 ,

(3)
(4)

j=1

where n is the total number of spatial units, and y is an n × 1 vector of dependent
variables. ϵ is a independent and identically distributed random variables with mean zeroand variance 1. α0 and α1 are scalar parameters, wij is an (i,j)element of spatial weightmatrix, and yj is a j-th element of y.
We get next equation by substituting equation (4) to equation (3).
2logyi = α0 + α1n∑wij logyj2 + logϵ2i ,

(5)

j=1

Equation (5) is same as SAR model. From discussion of section 2.1, α1 is restricted tothe interval(−1, 1).
We check properties of S-ARCH model. Expectation, conditional expectation, varianceconditional varianceof yi is derived ,where i means some area of spatial units.
Let ψ−i be the information set of neighborhood of area i. At first, we derive an
expectation and a conditional expectation of yi .
E(yi |ψ−i ) = E(σii ϵi |ψ−i )= σii E(ϵi )= 0.
E(yi ) = E(E(yi |ψ−i ))= 0.
3Second, we derive a variance and a conditional variance of ai .
V ar(yi |ψ−i ) = E(σii2 ϵ2i |ψ−i )= Exp(α0 + α1n∑wij logyj2 ).

j=1

V ar(yi ) = E[var(yi |ψ−i )] + V ar[E(yi |ψ−i )]n∑= E(Exp(α0 + α1
wij logyj2 )) + 0
j=1= Exp(α0 )E(Exp(α1n∑wij logyj2 ))j=1∏ 2α1 w
= Exp(α0 )E(Exp(logyj ij ))∏ 2α w= Exp(α0 )E( yj 1 ij ).
Third, we derive a covariance of yi and yj .
Cov(yi , yj ) ====E(yi yj )E(E(yi yj )|ψ−i )E(yj (E(yi |ψ−i )))0.

(6)

Therefore, we can write a covariance matrix of y as
σ11 0 · · · 0 0 σ22 · · · 0 Σ =  ..
.. . .
..  .
 .
.
.
. 
0
0 · · · σnn

3

ESTIMATION METHOD

In this section, estimation methods of two models which are introduced in section 2 is
discussed. Many estimation methods of SAR model is suggested. However, each model isestimated by maximum likelihood estimation later in this paper. Therefor likelihood ofthem is introduced.

3.1

ESTIMATION METHOD OF THE SPATIALAUTOREGRESSIVE MODEL

There are many estimation methods of SAR model. SAR model is estimated by
maximum likelihood estimation(Ord (1975)), GS2SLS(Kelejian and Prucha (1998)), andbayesian estimation(LeSage (1997)). Moreover, Kelejian and Prucha (1998) and Lee(2004) prove the consistency of respectively the GS2SLS estimator and the ML estimatorin the SAR model.
4In this article, parameters of SAR model are estimated by maximum likelihood estimation about concentrated log-likelihood. The log-likelihood function for the SAR modeltakes the form in(7).
nnϵ′ ϵlogL = − log(2π) − log(σ 2 ) + log|In − ρW | − 2222σ
ϵ = y − ρW y + Xβ

(7)where,|In − ρW |is the determinant of this n × n matrix.
For reducing a multivariate optimization problem to a univariate problem, we use
concentrated log-likelihood with respect to the parameters β, σ 2 . Working with theconcentrated log-likelihood yields exactly the same maximu likelihood estimates β̂,σˆ2 ,and ρ̂ as would arise from maximizing the full log-likelihood(LeSage and Pace (2009),p47).
The maximum likelihood estimator about parameter β and σ 2 can be written asβ̂ = (X ′ X)−1 Xy − (X ′ X)−1 XW y
= b0 − bL , (b0 = (X ′ X)−1 Xy, bL = (X ′ X)−1 XW y)(8)′[(y − Xb0 ) − ρ((W y) − XbL )] [(y − Xb0 ) − ρ((W y) − XbL )]σˆ2 =n(e0 − eρ L)′ (e0 − eρ L)
=
.
(9)
n
From above estimator, we get next concentrated log-likelihood with respect to the parameters β, σ 2 .
nn(e0 − eρ L)′ (e0 − eρ L)
logLc = − log(2π) − log(
) + log|In − ρW |.
2
2
n

(10)

The estimation process can proceed according to the following steps(Anselin (1988),p182):
1. carry out OLS of X on y: yields b0
2. carry out OLS of X on Wy: yields bL
3. compute residuals e0 and eL
4. given e0 and eL , find that maximizes logLc′

5. given ρ, compute b = b0 − ρbL and σ 2 = (e0 −ρeL )n(e0 −ρeL )3.2

ESTIMATION METHOD OF THE SPATIALAUTOREGRESSIVE CONDITIONALHETEROSCEDASTICITY MODEL

Parameters of S-ARCH model are estimated by maximum likelihood estimation about
concentrated log-likelihood.
We assume each ϵ follows independent and identically distributed standard normal
sequence. From change of variables, a probability density function of z = logϵ2i is
1 z z
1h(z) ∼ √ e− 2 e + 2 .
2π

5The log-likelihood function for the S-ARCH model takes the form in (11)

xi
fi

∑

1
1
(− efi + fi ) + log|In − α1 W |,2
2
= logyi2 ,n∑= xi − α0 − α1
wij x2j ,logL =(11)

j=1

where,|In − α1 W |is the determinant of this n × n matrix. The partial derivative of
log-likelihood function with α0 is
∑ 1 1
∂logL
=
(− + efi ).
∂α0
2 2

(12)

The maximum likelihood estimator about parameter α0 can be written as∑ xi −α1 ∑n wij xj
j=1eαˆ0 = log(
).
n

(13)

From above estimator, we get next concentrated log-likelihood with respect to theparameters α0 .
1
1logLc = (− efi + fi ) + log|In − α1 W |,22
∑ xi −α1 ∑n wij xj
n
∑
j=1
e
fi = xi − log(
) − α1wij x2j .
n
j=1

(14)

Estimation is accomplished by numerical optimize eauation (14) about α1 and α0 is gotten
by substituting estimated α1 into equation(13).

4EMPIRICAL ANALYSIS

In this section, we report empirical analysis of S-ARCH model. First, simulation studyof S-ARCH model is shown to investigate properties of the maximum likelihood estimator.
Secondly, real data analysis of S-ARCH model is reported. We use Kanto area as spatial
scenario in both case.

4.1

SIMULATION STUDY

To investigate properties of the maximum likelihood estimator of S-ARCH model, we
do Monte Carlo simulation. We use Kanto area as spatial scenario. Spatial units used inthis simulation are ward, city and town. Therefore there are 347 observations. Spatial
weight matrix is based on the first order contiguity relations for the 347 regions and is
row-normalized such that the elements of each row sum to one. For each case, there are1000 repetitions.
<<Table 1>>
6In the first case (S-ARCH 1), the sample data are generated with α1 = 0 and α0
taken some value. We assume case that no spatial correlation exist in volatility. ME αˆ0is mean of estimated α0 . ME αˆ1 is gotten in a similar fashion. MSE αˆ0 is mean squared
error of estimated α0 . MSE αˆ1 is derived in the same way. Let λ be likelihood ratio
under null hypothesis H0 : α1 = 0. −2logλ follows chi-square distribution with degrees of
freedom 1. Last row (LR > 3.84) means percentages that likelihood ratio test statistics(i.e. −2logλ)under null hypothesis H0 : α1 = 0 is over 3.84. 3.84 is value of 5 % point of
chi-square distribution with degrees of freedom 1.
The sample data of the second case (S-ARCH2) are generated with α0 = −0.5 andα0 taken some value. We check some cases that the strength of spatial correlation ofvolatility changed. power means percentages that likelihood ratio test statistics undernull hypothesis H0 : α1 = 0 is over 3.84.
Table 1 shows estimated value of each parameter is similar to true parameters. However, lower bias may exist because mean value of estimated parameter is lower than true
parameters in each case. From last row of S-ARCH1 case, it is very likely that likelihood
ratio statistic follows chi-square distribution. Therefore, when we check spatial ARCHeﬀect in later section, we judge by whether likelihood ratio statistic is over 3.84.

4.2

REAL DATA ANALYSIS

In this section, we report five things. These are explanation of data, results of estimation of SAR model, Moran’s I of residuals of SARmodel , results of estimation of S-ARCH
model, and results of test of S-ARCH eﬀect.
First, data for empirical analysis is explained. Change rate of land price data of
Kanto area from 2003 to 2014 are used as dependent variable. This data consists ofwards, cities and town unit data. There are 347 observations in the data. Basically,
we use Publication of Land Prices as price data which can be gotten from National
Land Numerical information download service. However, we used Prefectural Land Price
Research as land price this time because there are towns which have no observation point
in Publication of Land Prices.
<<Table 2>>
Second, we apply SAR model to the land price data and we get residuals of SAR
model. Spatial scenario is same as section 4.1 and we use row-normalized first ordercontiguity spatial weight matrix. We assume that independent variable is only intercept
term. Table 2 shows the result of SAR model. ρ express the strength of spatial dependenceof observations. From the result, there are high spatial dependence in land price data.
This means that if change rate of land price of certain area is high, then surrounding
areas change rate is also high.
<<Table 3>>
Next, we check Moran’s I of residuals. Moran’s I is one of the index of spatial dependence. High Moran’s I means that there are strong spatial correlations in observations.
Moran’s I is defined as∑n ∑nn
j=1 wij (yi − ȳ)(yj − ȳ)
i=1
∑
.
(15)
I = ∑n ∑nn2
j=1 wij
i=1
i=1 (yi − ȳ)7Table 3 shows the Moran’s I of residuals of SAR model. From table 3, it is considered that
residuals of SAR model have little spatial dependence. However, squared residuals seems
to have spatial dependence. Especially, the values from 2009 to 2012 is higher than theothers. The cause of high Moran s I in those time is Lehman shock and The Great East
Japan Earthquake. Lehman shock occurred in 2008 and Tohoku earthquake happenedin 2011. These data are investigated in July 1 every year. The eﬀect of Lehman shock
appear in data of 2009 year.
<<Table 4>><<From Figure 1 to Figure 12>>
Then, we apply S-ARCH model to the residuals of SAR model. We use same spatial
scenario and spatial weight matrix as SAR model.Table 4 shows the results of estimationof S-ARCH model. α0 is parameter of intercept and α1 express S-ARCH eﬀect by which
the eﬀect of surrounding spatial units on given area’s conditional variance is caught.
Estimated α1 is positive in every year and the estimated value from 2006 to 2013 arebigger than those of other years. It can be presumed that we can see clusters of highvolatility and low volatility in the time that α1 is larger.
Estimated volatility is displayed on the map in each figure. It provides a visual depiction of how values of volatility diﬀer over space. As the result that estimated α1 is
low shows, each spatial unit take similar estimated volatility value in 2003 ,2004. On theother hand, we can see clusters of red and blue spatial units from 2005 to 2011. Japaneconomy was good from 2003 to 2007. As can be seen from these figures and table4, theaverage value of volatility changed significantly before and after 2007. The volatility after
2007 are small so the width of the change is also small.For this reason, it is hard to seethat we find volatility clusters in 2013 and 2014 in spite of volatilities of neighboring areas
are very similar. Lehman shock occurred in 2008. These data are investigated in July
1 every year. The eﬀect of Lehman shock appear in data of 2009 year. From figure 7,
we can find that an amount of change of land price is bigger in urban areas and west of
Kanto area. The Great East Japan Earthquake occurred in 2011. Coastal areas suﬀered
serious damage in that earthquake. we can find the width of change of land price in theseareas is bigger from figure 9.
<<Table 5>>
Finally, we test the S-ARCH eﬀect, which is to say that we test next hypothesis
H0 : α1 = 0, H1 : α1 ̸= 0. Table 4 shows the result of test of S-ARCH eﬀect. Wald, LR
and LM statistics are reported. From simulation study, we assume these statistics follow
chi-square distribution with degrees of freedom 1. These three statistics is almost samein every year and over 3.84. Therefore, estimates of α1 are statistically significant at the
5% level. It can be said that spatial ARCH eﬀect exists in land price data in the same
way as time series finance data.

8

5

CONCLUSION

We have proposed the spatial autoregressive conditional heteroscedasticity (S-ARCH)
model to estimate spatial volatility. Regarding log transformed S-ARCH models as spatial autoregressive models, we consider maximum likelihood estimators (MLE) for theparameters and test statistics for ARCH eﬀects. Empirical studies by Kanto land priceand simulated spatial data demonstrate that S-ARCH models work well to detect ARCHeﬀects with reasonable size and power and to estimate spatial volatility. In the real data
analysis, we found higher volatilities in the coastal area near Tohoku than those in otherarea, which may be considered as the quantitative evaluation of the eﬀects of the big
earthquake in Tohoku. In addition to the empirical properties, we are now considering
to establish theoretical properties of the MLE, the consistency and asymptotic normality
for them.
Finally, we will complete the paper by describing some future studies, In the empiricalanalysis for the land price, we took the first order contiguity relations of ward, city andtown of Kanto area as spatial weight matrix. As (Beck, Gleditsch, and Beardsley (2006))
shows, spatial distances diﬀerent from geographic distance can be interesting canditates toimprove our volatility analysis by S-ARCH models. This paper only evaluated volatilities
in spatial data by fixing time in the land price data, which is actually spatio-temporal
data. Spatio-temporal extensions of S-ARCH model would provide more detailed analysisof the volatility structures in the land price.

9

APPENDIX
From page 10 to 15, two types of objects is listed. They are figures and tables. Figures
shows maps of estimated spatial volatilities of each year. Tables shows five results of
estimation and simulation.

Figure 1: 2003 estimated volatilities map

Figure 2: 2004 estimated volatilities map

Figure 3: 2005 estimated volatilities map

Figure 4: 2006 estimated volatilities map

10Figure 5: 2007 estimated volatilities map

Figure 6: 2008 estimated volatilities map

Figure 7: 2009 estimated volatilities map

Figure 8: 2010 estimated volatilities map

11Figure 9: 2011 estimated volatilities map

Figure 10: 2012 estimated volatilities map

Figure 11: 2013 estimated volatilities map

Figure 12: 2014 estimated volatilities map

12

Table 1: S-ARCH simulation
S-ARCH1α0(α1 = 0)
0.5
1.5
-0.5
-1.5
ME αˆ00.491 1.488 -0.513 -1.523
ME αˆ1-0.005 -0.003 -0.004 -0.007
MSE αˆ00.008 0.006 0.018 0.037
MSE αˆ1
0.004 0.004 0.004 0.004
LR > 3.84 0.052 0.050 0.046 0.050
S-ARCH2α1(α0 = −0.5) 0.1
0.2
0.5
0.8
ME αˆ0-0.524 -0.514 -0.530 -0.561
ME αˆ1
0.091 0.195 0.494 0.793
MSE αˆ00.022 0.022 0.032 0.069
MSE αˆ1
0.004 0.004 0.002 0.000
power
0.320 0.833
1
1

13

14

σ2

β

ρ

Table 2: The result of SAR model.
2006
2007
2008
2009
2010
0.830
0.933
0.848
0.880
0.700
(0.028) (0.016) (0.027) (0.024) (0.046)-0.096 0.037-0.002 -0.270 -0.452(0.050) (0.046) (0.031) (0.065) (0.076)
0.781
0.689
0.333
0.380
0.276
(0.097) (0.110) (0.228) (0.202) (0.282)
2011
0.820
(0.030)-0.250(0.048)0.236(0.322)

2012
2013
2014
0.831
0.880
0.892
(0.028) (0.023) (0.021)-0.174 -0.083 -0.037(0.037) (0.026) (0.022)
0.215
0.179
0.156
(0.352) (0.424) (0.486)

Table 3: The result of Moran’s I of residuals of SAR model.
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
-0.088 -0.078 -0.056 -0.063 -0.049 -0.043 -0.010 -0.036 -0.075 -0.082 -0.076
0.055 0.142 0.062 0.090 0.048 0.152 0.322 0.295 0.246 0.101 0.072

2004
2005
0.836
0.805
(0.033) (0.033)-0.423 -0.359(0.094) (0.071)
0.607
0.524
(0.144) (0.151)

2003
noraml -0.075
squared 0.076

2003
0.865
(0.031)-0.408(0.104)0.668(0.136)

15

α1

α0

2004
2005
-0.222 -0.136(0.021) (0.015)
0.147
0.268
(0.004) (0.002)

Table 4: The result of S-ARCH model.
2006
2007
2008
2009
2010
0.507
0.234
-0.295 -0.146 -0.423(0.015) (0.014) (0.030) (0.021) (0.019)
0.428
0.341
0.301
0.351
0.363
(0.002) (0.002) (0.003) (0.002) (0.002)
2011
2012
-0.309 -0.647(0.026) (0.020)
0.432
0.334
(0.002) (0.001)

2014
22.48921.36422.27420132014-0.244 -0.974(0.036) (0.041)
0.491
0.267
(0.003) (0.003)

Table 5: The result of S-ARCH eﬀect test.
2003 2004 2005
2006
2007
2008
2009
2010
2011
2012
2013
Wald 4.735 5.621 38.494 98.027 67.700 30.570 56.431 83.730 83.756 77.226 88.585
LR
4.585 5.384 35.535 83.519 52.915 28.654 49.763 69.206 75.321 66.006 77.388
LM
4.738 5.612 38.644 93.177 67.147 30.039 55.529 78.791 81.962 77.158 86.408

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