Kagawa UniversityE.ιonomic Review Vol71， No..4， March1999， 253-277
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Moving
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Model
Hajime Takatsuka
1. Introduction Linear regressive models in the presence of spatially lagged dependentvariables have been not only studied from pure statistical viewpoints ( e g....
methods development for parameter estimation and diagnostic tests)， but
also increasingly app1ied to empirical studies in various fields of social
science.. Among such models， a most simple and widely-used one is the
first order mixed regressive spatial autoregressive model， .y=ρWr'y+Xs
+μ[eゎ g..Ord (1975)J， where.Yis an
N
by 1 vector of observations on thedependent variable， X is an N by K matrix of observations on the explana -tory variables，μis an N by 1 vector of disturbance terms with zero mean and variance matrix02[，
w
r
is an N by N spatial weights matrix，ρis a spatial autoregressive parameter，s
is a K by 1 vector of regression coeffi -cients and N is a number of spatial units1 . 問ycorresponds to a spatially lagged dependent variable and Anselin (1992) labeled this model the firstorder“Spatial Lag ModeF" Can (1990，1992)， and Can and Megbolugbe
(1997) used this type of model for analysis of real estate markets.. A more
1 This paper focuses onsimultaneousfy spe，αifiedψatial models.. Conditional1yψeαified
ψatial models， which are the other important spatial models， and comparison of
these two models are discussed in detail in Cressie(1993， chap.6)
-254 Kagaω'a Universi(yE，正onomicReview 1310
general model including a spatially lagged dependent variable， y
=
ρVliiy十Xβ+u，u
=
À 砂~u+ μ， has been also examined in the estimation method[e g.. Doreian (1982)， Anselin (1988a)J， where y， X，μ， Vlii， ρand βare
as before， u is an N by 1 vector of disturbance terms， which are spatially
autocorrelated via a spatial matrix Wz and a spatial autoregressive
parame-terA We might call this model the first order “Spatial Lag Model with
Spatial Autoregressive Disturbances" Case (1991，1992)， and Case， Rosen
and Hines (1993) applied this kind of model to their empirical studies in
household demand in rice， adoption of technological innovation by farmers，
and fiscal interdependence between neighboring jurisdictions， respectively，
The Spatial Autoregressive Moving Average (SARMA) model is
another general model which includes spatially lagged dependent variables，
and the first order SARMA， or SARMA(l，l)， is specified as， y =ρVliiy
十Xβ+u，u
=
θvliiμ+μ， where y， X，μ， Vlii， ρand βare as before. Itshould be noticed that this model assumes u is an N by 1 vector of
disturbance terms which are generated by the first order spatial moving
average process， with associated parameter 84 Huang (1984) defined a
high order SARMA model (se吃 thenext section) and discussed the
maxi-mum likelihood (ML) estimation of the parameters.. However， we could not find empirical studies using the SARMA model at al，5J while there are 3 Cressie (1993， pp. 442-443) prefers to specify the SARMA(l，l)modellike， (J -p W)(y - Xs)= (1-θ問)μ，because he thinks ψatial d，宅pendenceas one ofsmall-scale effeas， whi1e large-scale~併cts are perfectly expressed inXs Thus， he takes a critical attitude to using spatially lagged variables However， the author believes that spatially lagged variables are meaningful in some situations 4 In the SARMA model， a spatial weights matrix used for the disturban叩 termsis usually the same with one used for the spatiallagged variable. On the other hand， in the spatial lag model with spatial autoregressive disturbances， the two spatial weights matrix must be different (i. e 問中隔)，because the identification problem in ρand tIoccurs， 5 Re氾右nt1y，Anselin et al (1996) and Anselin (1998) have studied Lagrange Multiplier tests for spatial dependent parameters in the SARMA modeL
1311 Applicability of Spatial Autoregressive Moving Average Model -255
some applications of the spatial lag model and the spatial lag model with
spatial autoregressive disturbances 1 think there are two reasons. The
first is due to Huang's assumption that a spatial weights matrix is
symmet-ric. This assumption seems to be too strict because we often usestandard
-izedspatial weights matrices such that each row sums to one， which are
asymmetric in generaL The second reason for lack of applying the
SARMA model might be difficulty in interpretation of spatial moving
average processes6 . This study， in the following section， derives ML estimators of Huang's high order SARMA model as the solution of a system of nonlinear equations when the spatial weights matrix is asymmetric but diagonalizable In this situation， 1 will show that we can avoid to evaluate an N by N inverse matrix at every iteration in the computation process“ In section 3， asymptotic distribution of the ML estimators is also derived In section 4，
at first， 1 will show that a restriCted SARMA model is more natural and
appropriate than the spatial lag model， in the case that we convert a
deterministic relation including spatially lagged dependent variables to an
econometric one.. An empirical example is illustrated in the following step，
where parameters of a land price model [Takatsuka (1998)J are estimated..
The land price model includes a spatially lagged dependent variable， so
there are several ways in specifying the disturbance terms， which are
explained above.. This study applies both the restricted SARMA(l，l) model
and the first order spatial lag model， and compares their results..
6 Since spatial autoregressive processes may be more intuitively comprehensive， some
empirical studies use the spatial lag model with spatial autoregressive disturbances
-256- Kagawa UniversiかEωnomicReview
2
.
Maximum Likelihood Estimationof The SARMA Model
1312 Huang (1984) defined a high order Spatial Autoregressive Moving A verage (SARMA) modeF as follows : ρ q y = ~PkWky+Xß+U ， U = ~8kWkμ+μ ， μ~lN(O ， 021)， (1) k=l k=l
where y， X，μ， Wand βare as before， ρ{kH=l is a set of spatial autore
-gressive parameters in spatial lag terms and {8k}Z=1 is a set of spatial
moving average parameters in disturbance terms.. This model will be
called the SARMA (p， q) modeL
The SARMA (p， q) model includes spatially lagged dependent vari
-ables， so the Ordinary Least Squares (OLS) estimators of the parameters
are inconsistent On the other hand， the ML estimators have some desir
-able properties(i.. e.， consistency，αsymρtotic normali砂， and伺ymρtoticeffi
-ciency)under mild conditions [see Anselin (1988a， p.. 60)J. However， in
order to compute the M L estimates， we have to evaluate an N by N inverse
matrix at every iteration.. Huang showed that if the spatial weights matrix
W is symmetric， the repeated evaluation of the inverse matrix can be avoided， using orthogonal decomposition of W8. As described in the previ -ous section， however， his assumption (symmetry ofW) is too strict 7 In general， the SARMA (p， q)model is specified as: y =
志
向
Wk什 Xs+u，u = 土 BkWkμ+μ， μ~IN(O， σ21)， k=l k=l wherel.f弘 isa spatial weights matrix conesponding tok-thorder spatial proce鈴i Huang setWk isW¥ whereW is the only associated spatial weights matrix 8 Haining (1978) used the same method in estimating ML parameters of a pure spatial moving average model However， concerning the case of asymmetricW， he said， “the computational problems of obtaining exact ML estimates become severピ， and did not treat the case [Haining (1990)， p.. 126)]1313 Applicability of Spatia! Autoregressive Moving Average Mode! -257-Therefore， a more general situation is assumed here， i， e"， the spatial weights matrixW is asymmetric butdiagonalizable At first， equation (1) is expressed as follows : or where
(
1
ー
か
kWk)y=
XB+(I+品川
μ， Ay = Xs+Bμ，， P A=I-L
:
ρkWk， k=1 q B = I+~ BkWk， ゐ=1 Therefore， we have μ=
B-1(Ay-Xs)，￨
考
1
=
I(B-1A州 (2) (3) (4) (5) (6) (7) which yields the joint probability density function ofy， and the log-likeli -hood function ofy， respectively， L = (伽2加7r0り
γ2ケ
)州
一 ln L =一
千
U π一
子
lno2+
川AI-lnIBI一
歩
{B-1(Ay-Xs)}' {B-1(Ayー Xs)} Differentiating equation (9) by parametersβand 0，2， we have。
lnL 1 - 一 一 = ゴ3β (B-1X)'(B-1Ay-B-1Xs)=
0， (9)。
。
。
lnL N 1 1 f T>-1 ，，__ T>-1 vn¥'f T>-づ巧γ =ーす7
十五
;;(B-1Ay-B-1Xs)'(B-1Ay-B-1Xs)=0
"
ω
These equations give ML estimators :s
= {(B-1X)'(B-1X)}一I(B-1X)'(B-1Ay)，。
2)-258- Kagawa University Economic Review (j2 =
会
(B-1(Ay-Xs)}'{B仙 -Xs)}=
蒜
B-一→11Ay刈刈Ayy吋d
川)'川'[ト[ 1314 ( 13) Furthermore， substituting these equations into the 1og-likelihood function (9)， we can obtain the concentrated 1og-like1ihood function: 1nL* =子
(
1
n2π+ 1) --;;'ln(j2(ρ，θ)+lnIAI-1n IBI， 凶 2which is nonlinear inρ=(ρ1，…， ρρ)' and θ = (81，… ， 8q)¥ If we can find
such
p
and 8 that maximize (14)in a proper parameter space (which isdiscussed after)， all unknown parameters cou1d be estimated by substituting
p and θinto (12)and (1)3， However， equation (14)inc1udes a N by N inverse
matrix B-1 in(j2(ρ， 8) and determinants IAI and IBI， therefore from a
viewpoint of numerica1 computation it is very difficu1t to treat(14)as it is"
Here， if the spatia1 weights matrix W is diagonalizab1e9
:
W =
QAQ-¥
ω
where Q is a matrix whose co1umns are eigenvectors ofW and A is a
diagona1 matrix whose i-th diagona1 e1ement is the eigenva1ue correspond
-ing to i-th co1umn of Q， then
p p p
A=I-21ρhWh=I-zp(QAQ-l)h=I-Ep(QAhQ-l)
=Q(I-21ρ
k
A
k
)
Q
-
¥
(16)B =I+EI仇Wh=I-EF(QAQ-1)hz I-EF(QrQ-1)
=Q(I+br)Q-I， (1)7
9 An N byN matrix isdiagonalizableif and only if it hasN eigenvectors which are
linearly independent each other， This property is widely satisfied and any symmet
1315 App1icabi1ity of Spatial Autoregressive Moving Average Model -25.9ー therefore， we can obtain (A， B) = (Q.QI Q-I， Q.Q2Q-I)， (A-
，
r
B-1) = (Q.QiIQ-I， Q.QiIQ-I)， where.QI=
1一
生
M h
，ぬ=I+bAh
Substituting(19) into側， we have 山) (19) Na2 = (.Qil.QIα)'[R-R.QiIP{P' .Qil R.QiIP}-1P'.QiIR](.Qil.Qla)， (20) where R=
Q'Q， P=
Q-1X， a=
Q-ly， thus once eigenva1ues and eigenvectors ofW are eva1uated at the beginning of the optimization， we do not have to evaluateB-1 at every iteration It should be noted that.Q 2 is a diagonal matrix so evaluating.Q2-1 is easy Furthermore， we have¥
A
¥
=
¥
ω
IQい
¥Q¥¥.Qd
¥Q¥-I=
¥.Qd
=
Idi叫 - 急 凶 ) N b =日 (1-~ρklm ，。
。
¥
B
¥
=
¥Q.Q2Q-円 QIIぬ
¥¥
Q¥-I=
¥.Q2¥=
Idiag(l告側
1
)
zd(1+土
Bk}.})， 的)where ん(i= 1，… ， N) is an eigenvalue ofW， therefore we can also avoid
repeated evaluation of the determinants
¥
A
¥
and¥
B
¥
.
Substituting (21)and (22) into (14)， the concentrated log-1ikelihood function is expressed as : where lnL* =一
字
Mπ+1)子
一
{lna2一
会
せg
i
}
=
consωnt-子{lnã怯gir~}，
g
i
= (1一
室
ρk-Af)/(l+ 三 BkÀ~)
k=1 k=1 (お)。
の
260-ー Kagaωa Univers#y E
ω
nomic Revieω 1316 Maximizing仰:)is equivalent to minimizingF(ρ，め =Nð怯gir~
(25) We wi1l consider this minimizing problem， First， differentiatingF
by 向(
k
=
1，…， ρ)， we have。
IF >T~' O r -A- l-~ ， r -A- 1-~òNð23
戸
=lV5357LLIgzj+lLIgzJ7F
where ヨ rN 1-主 。rN 1-主N 万三一1
I
l
gi1
日=f7￨IIgz￨2--4L
一一， VfJk Lt=l .J "V Lt=l . J 同1-2
:
ρkAf響 =2
(
長
)Rxイ長
)RVRX， . x= .Qi1.Q la， V = .Qi1P{F'.Qi1R.Qi1P} ーlP'.Qi，I企ι
3ρh- a'.Q"IA~
U Z In these notations， equation (20) is expressed as Na2 = .x'Rx-.x'RVRx" Next， differentiatingF by 8k(k口 1，…，q)， we have oF>
T
-
'
Or
-A- l-~ ，r
-A- l-~ oNa2 一一=Na2一一I
I
I
g
i
I
N十I
I
I
R
J
一一-where 3仇 θ8kU
=
-'-rιJ
IL
t
=
.lI.5tJ
o8k ' 一ι
r
f
i
g
J
会
一
=~TI む|一切-4し
o8kL
t
:
l.5zJ -
NLt，，;¥.5zJ
(;11-f
8 kAf。
'Na2 n( OX¥ ox'RVRx θ8k “¥o8kr
明
。
θh =2lklRx-21klRm-xRimh¥o8 ， kr
"
'
"
"
¥
o8kJ
H
Y ""'^' ，^，H
¥
o8k ) ox _A
o.Qi1 ¥1'"1 百否-;-U，
o8 kF
'
&1。
。
仰) (28)。
。
(30) (3U (32) (33) ( 34) (35) (36)1317 Applicability of Spatial Autoregressive Moving Average Model -261ー
a
.Qi1 _ ，1;
n
J
7-AF ¥21
a
e
k
-
U!a忌I
(1+
2
:
:
ekÀ~1
I
IntroducingT = {P'.Qi1R.Qi1P}- we have ，Iav _
(
a
.Q-
2
1 ¥ n，.，-.t r¥-l n¥' ， t r¥-1 n¥(a
T
¥
百瓦=¥v
a
瓦
一
)PT(.Qi1
P)十(.Qi1
P)¥砿 )(.Qi1
P)'(
a~吉川+
(.Qi1
P) TP'¥v
a
e
k
，
)
θT 明[んP { t ω幻+んωi_
2
)
_
ydPT，a
e
k
-
.L.Ll
(ωi
2
ω幻y
可j ) 抑 制 (38) (39) where rijis an i，( j)element ofR，ω2iis an (i， i)element of.Q2 and {Zu} is a matrix whose i，( j)element isZijML estimators p and θlet側and(33) zero， therefore we can obtain the
ML estimates by applying a nonlinear optimization method to a system of
these nonlinear equations in a proper parameter space.. The parameter
space should ensure the regularity conditions for the log-likelihood function
(9)， therefore the following condition:
I
(
B
-
1A
)
'
1
>0 (40)is required.. This is satisfied by
I
A
I
>0 andI
B
I
>0， which means that eachterm in the product (2u and (2 i)2s positive.. For example， when p = q = 1
and the spatial weights matrix W is a standardized matrix， the bound ofρ1
is (1
刈
mln，1)and that ofe
1 is (-1， -l/Amln)， where Amln is the minimumeigenvalue of W， whose sign is negative
However， we should take care that the parameter space of the
SARMA (p， q) model is not alwaysbounded. In the cases that the parame
-ter space isnot bounded， it is difficult to obtain the global optimal parame
-ters..
3
.
Asymptotic Distribution of The M L Estimators-262- Kagawa Univers#y Economic Review 1318
model (1)， and letijbe the ML estimator of r;. Under mild conditions，
J
万(宇一 ηconvergesin a multivariate normal distributionN(o， N
・
I(早)→)，where 1 ( r;) is the Fisher information matrix : I o21nL l I(r;)= -
E
I
一 一 十I.
.
(4)1 L o;ror;'J That is， the asymptotic distribution of手isa multivariate normal distribu -tionN(，;rI(平り)一， and )l引 ijhas asymptotic normality and asymptotic effi -ciency 1t means t出ha抗tthe ML estimators a町sympμt叫oti比c叫a11防y achieve the Crαamer-Rαωo /，必ozωv沼ervα7ηY matrix.. The variance bound can b恥eu回s吋edfor asymptotict test for the ML parameters 1n order to obtain the Fisher information matrix， 1 will derive the second partial derivatives of log-likelihood function with respect to each parameter.. At first， we have olnL 1 -一一=ゥ(B-1X)'(B-1Ay-B-1Xs)， os。
21nL 1 -一一r =ーす(B-1X)'(B-1 X)， osos グlnL 1 一一一=ーす(B-1Wky )'(B-1 X) (k= 1，…， ρ)， O ρkOs o21nL _ 1! 0-1 A.. 0-1 VD¥'(! 0 -否瓦a
s
'
= ---;?-(B-1Ay-B-1Xs)'{(B-1 Wk )'+(B-1 Wk)}(B-1X) (k= 1，…， q)， (42) 白部 ( 44) 白5) o21nL 1 吉宗万=一石平(B-1Ay-B-1Xs)'(B-1X) = 0 [from (11)]，側。
lnL .t!， À~ 一 一 =-~一一十一一+ヲ(B-ゅう)'(B-1Ay-B-1Xs) θρk i;11_ ~ρkÀ~ ポ k;1 (k= 1，…， ρ)， 臼7) θ21nL t!，. AFt 一一一一=-~~ 一寸(
B
-
1W
ω
'
(
B
-
1W
ky) ω ρ h t=l(1-hld)2U1319 Applicability of Spatial Autoregressive Moving Average Model -263 -(k= 1，…， ρ 1 = 1，…， ρ)，
a
21nL 1 一 一 一 = ー す(B-1Ay-B-1Xs)'[B-1 W1+(B-1 W1)'](B-1 Wky) 3θtθPk (k= 1，…， ρ 1 = 1，…， q)， 旬 。 旬 。a
21nL 1 =ーァコ¥T(B-1Wky )'(B-1 Ay -B-1 Xs) (k= 1，…， ρ)， (50) 3σ3ρk σつ
z。
lnL_ ~A
f
，1 (n-17r=2
一 寸Lー十万玄(B-1Ay-B-1Xs)'(B-1 Wk)' i=11+ム叫ん x(B (k= 1，… ， q)， (51)。
21nL !，! Ak+l 一一一一 -L
1
/~今(B-1Ay -B-1 Xs)' MAh1(l+210Af)2O X [{B-1 W1+(B-1 W1)'}B-1 Wk+(B-1 Wk)(B-1 W1)] x(B→Ay-B-1X
.
β (k= 1，…， q; 1 = 1，… ，q)， (52)a
2 1nL 1 ( n-1 A •. n-1 吉宗否瓦=一石'zyz(B-1Ay-B-1Xs)'(B-1 Wk)'(B-1Ay-B-1Xs) (k= 1， ...， q)， (5功。
lnL N ， l ー1A.. D - l"tTn¥'/n-万 戸
-i;2+宮 村
B-1Ay-B-1Xs)'(B-1Ay-B-1Xs)， (54)a
21nL _ N 1 (0-1 A •• 0-1
京i?Y=苛?y一夜e?r(B-1Ay-B-1Xs)'(B-1Ay-B-1Xs)
=-d
戸
from(12)] 倒Furthermore， using the following relations :
E(y) = A-1Xs， 側
E
(
ωμ)=
0， where μ=B
E
副(
M
刀川y〆y凶')=(ωA一1官
Xßめ)(は
A-一1官
Xßめ)'+~ι云(は
A一1官
B)(A一1官
B)'，
側
UE(m)=jdklB)， 側
-264- Kagawa University Economic Review 1320 each element of the information matrix is as follows : 1pp =
会
(Bーヲ)'(B-1X)， 制 1p>p =会
(B-1WkA-1Xs)'(B-1X) (k= 1， ...， ρ)， 側 ん>p=
0 (k=
1，…， ρ)， Iσ2β= 0 (k = 1，…， ρ)， 側 (64) N k + 1P1内=
2
:
:
官+
ァ
去
w
tr[ (B-1 WI)'(B-1 Wk ){o
2
(
A
-1 Xs) 4(1-klρklm2 ¥U)x
(A -1Xs)'+
(A -1B)(A -1B)')] (k=
1，…， ρ 1=
1，…， ρ)， (65) I町
内
=
合
ztr[{(B-1WI)+(B叩 )'}(B中 )(Aサ)] (k= 1，…， ρ 1 = 1，… ， q)， 側 ん い 告 す か[
A
叩(
k=
1， ....，p
)
， (67) N k + 1818>=
2
:
:
I~+
ァ
去w
tr[{(B-1WI)+(B-1 WI)'}B-1 Wk 吋 1+炉
10Af)210/ +(B-1 Wk)(B-1 Wl)] I叫=合すケ
[B-1Wk ] γ N 10'202一夜訪れ (k = 1，…， q; 1 = 1，… ， q)， (68) (k=
1，… ， q)， (69) (70) where AF Z σ y i . 、 ‘ ‘ B E E S ， f 。 u r ρ u F H A q よ . ・ ・ ん ， I l l --、 一 一 。 μ a u y i 、 、 ‘ ‘ ‘ E E E E E ， ， ， ， ， ρ P O U F -P ρ γ ・ ρ r ri.7i J ' t S E E t、 、 、 一 一 β ρ r y ' A ， ，、、 B E B E E -司 S a l t i l -1 1 1 1 1 E ﹄ 1 q -2 -叩 ゆ ， β， ρ， e σ 戸 引 ' 2 2 2 2 r h : ・ 7 h g σ σ σ ' r k yi7ir--/2511h F F 句 = 但 wo r ι 7 i r L r L ， e 2 F M V ー σ β ρ ρ 2 7 t p P 9 5 1 J Y I 7 i， ， i r t ' β 1 1 1 1 1 1 世 岬 由 2 1 H U A F A F Y L Y L r L F i 2 2 σ : ' σ -" y i : -Y F A lli--111111/ ， ， B E E s -一 -一 、 ， ， ， ， 刊 H ， ， ， a、 、 r i1321 Applicability of Spatial Autoregressive Moving Average Model -26 5-1
L
ρ山"….日.1ρMJ
pρ ρ = 1 ， 1L
ρp山 …Iρρρp q q a υ a υ 2 qo
o
v， ‘
Y ' A••
1 1 a v a υ 1 q -a v a v -y i ﹁ 1 1 1 1 1 1 1 1 -p s i l -﹄ 1 1 1 L a υ a u r， ‘
﹃ s i l l i -t t 3 a ' I l l i t -1 1 4 p q n w r a v ----q a v a v y i ・ 0 ・ 1 1 0 r A p i -q 8 9 F L y i ﹁ 1 1 1 1 1 1 6 4 B E l l i -L 一 一 A V G ur
-It should be noted again that A -1 and B-1 can be evaluated by (18)"4. The Restricted SARMA Model vs.
The Spatial Lag Model: An Empirical Example
Suppose that we deduce a deterministicrelation between y and X from
several hypotheses as follows : p
y=
呂
ρ;W;y+Xs
，。
。
where y is an N by 1 vector of endogeneous variab1es， X is an N by K matrix of exogeneous variables， {め}f=1and s are associated parameters， and {W;}f=1 is a set of known spatial weights matrices.. Therefore， this relation includes spatially lagged dependent variables10. We can convert this deterministic relation to stochastic (econometric) one in several ways.A most simple one is the ρ-th order spatial lag model (Le t，“ he SARMA(p，
0) model):
y=
会川
，:
y
+
)
(
川 ， げN
(
O
，02J) (12)In this case， however， it is not clear where the vector of disturbance terms
μcomes.. Ifμis an vector of enors in observations ofy， it is more natural
and appropriate to specify the relation as follows : p y ー μ= 呂 PJW;(y 一 μ) 十 Xß， μ~JN(O， 021)， (73) 10 Someone might think starting from such a deterministic relation is strange But the author believes there should be several (deterministic) foundations for including spatially lagged dependent variables in a regressive model， because they express large scale effects
-266- Kagaω'a University Economic Review 1322
or
p p
y =呂ρ;W;Y+Xβ+u，u
呂
ρ;W;μ+μ，μ---IN(O，021) 仰It is clear that (74) is a SARMA model with restriction thatρ= q and
;
e
=ρ;(j = 1，…， ρ) in the ordinal SARMA (p， q) model (see section 2). We
will call (74)the restricted SARMA (p， p) modeL In this case， we should
note that a spatial moving average process is not exogenously assumed but
naturally deduced from (7]) Whether assuming the spatiallag model (72) or
the restricted SARMA model (73)for(7])is a matter for argument， because
they might give different parameter estimates.
Hear， set
w
;
= W;(j= 1，…， ρ)， where W is the only associated(standardized) spatial weights matrix and is assumed to be diagonalizable..
The estimation method forCロ';and (73)has been discussed in previous sections
but we have to take care of their parameter space“ Equation問meansB = 1 in (8)， thus the regularity conditions are ensured by
I
A
I
>0， which determines the parameter spaceわ Equation(73)means B = A in (8)， thus the regularity conditions are ensured byI
A
I
宇o
(for existence ofA -1)仲 For example， in the case of ρ = 1， the parameter space of ρ1 is {ρd1/Amln<ρ1< 1} for(72)， and it is {ρ11ρ1宇1/.ん，i
= 1，… ，N
}
for(73)， where {んlf=1is a set of eigenvalues ofW and Am1n is the minimum eigenvalue. N otice that the latter parameter space is not bounded By the way， Takatsuka (1998) derived fundamental land prices (presentvalues) when the dynamics of land rents has a constant long-run growth
rate and linear spatial dependencell， and investors are assumed to expect
11 Here， the dynamics is assumed as follows :
η=ωtro+εt，凸=ω{(l-λ)I+AW}εトl+Ut，OζAζ1，
whereUt is a white noise vector，ωis a growth rate parameter andA is a spatial
1323 Applicability of Spatial Autoregressive Moving Average Model -267
-them rationally. The fundamental land prices are expressed as : qt= pWqt十srt十o(I-W)ro， (75)
where qt is an N by 1 vector of fundamentalland prices att period， rt is an N by 1 vector of land rents att period， ro is an N by 1 vector of land rents at 0 period， W is a spatial weights matrix expressing spatial depen -dence in land rents stream， and ρ，βand δare associated parameters12.
Clearly， equation eお is a deterministic relation including a spatial lagged dependent variable，ρWqt. Therefore we can estimate parameters of(75) in two ways (L e.. the restricted SARMA(l， 1) model and the first order spatial lag model)13. The data of land prices and land rents are collected
from 40 points in Tokyo Metropolitan Area semiannually between 1976 and 1996 (i eド40 periods)l4.. For detail， see Takatsuka (1998) Septemberj
1976 is set as the initial period (t = 0) and cross-sectional models are estimated at each period between September/1986 and March/1996 (L e“20
periods) The spatial weights matrix
W
is specified in the following eight12 Using the notation in footnote 11，ρ=ω，1α{一ω(1-，1)}-¥β={α一ω(1-，1)}-1，3 = ω'+I(a一ω)-'λ{α一ω(1-，1)}-"wherea = 1 + i+ RP， i is an interest rate andRP is a risk premium Thus， if we estimate(p， s， 3)， then we can obtain estimates of (ω， ， 1 ， RP). Furthermore， the following inequalities are obtained: 0<ρ<1，β>0， 3>0. 13 Of course， we can also apply the first order spatial lag model with (high order) spatial autoregressive disturbances or the unrestricted SARMA (1， q) model to equation問 But，we can check the possibility that these model fit to仰bytesting for spatial residual autocorrelation in the first order spatial lag model [Anselin (1988b)] As a result， the tests showed that these high order models did not seem to fit to(75). 14 As a practial matter， it is impossible to get the data of land rents， because land rental markets have not been developed in Japan Thus， 1 constructed the land rents data from the data of housing rents and floor area rates (F AR).. In this case， equatione花，)becomes as follows [see Takatsuka (1998)] q， ωnstant+ρWq， + sR， + 3(I -W)Ro. where R is a vector of housing rents per unit of floor multiplied by corresponding F AR's and constant is a negative parameter
-268-ways: Kagaωα Universiiy Economic Revieω 1324 Wd(θ)
=
=
{d，
jO}，θ = 1， 2， 3， 4， where du is the Euclidean distance between i and j， Wt(θ)=
=
，{tjO}，e
= 1， 2， 3， 4"where tuis the railway-time between i andi
Furthermore， both Wd(θ) and Wt( e)are standardized such that each row
sums to one
Then， the estimation results are described below1s" First， Table 1
summarizes the performances of each models" 1 give two indexes there，
L e， (1)Average of AIC and (2) Number of periods when all parameters
satisfy the conditions (see footnote 12) and significant. The former indi.
cates statistical-based performance and the latter does eじonomictheoretical
-bαsed one， According to the first index， the restricted SARMA model is
almost always better than the spatial lag model except for the case ofW
= Wt(4) The restricted SARMA in the case of W = Wd(2) achieves the
minimum average of AI
c
.
.
The second index shows that the restrictedSARMA model is more appropriate than the spatial lag model except for
the cases of W = Wd(l)， Wd(4)， Wt(1)'6， Also， the restricted SARMA
model in the case of W = Wt(3) most constant1y satisfies the required
conditions"
According to the above examination， the restricted SARMA model
15 The author used S-PLUS 4 for windows (MathSoft， Inc) for the estimation and applied 'nlminb' function， which is for nonlinear minimization subi.ect to box con. straints， to maximizing the log-likelihood十 However，the parameter space of the restricted SARMA (1， 1) model is not bounded， so 1 used 'nlminb' as folJows， At IIrst step， 1 search ρin the bound of 1/ Amln <ρ< 1 Ifthe local optimal value is found there， the value is set as ρOtherwise， 1 searchp in the bound of 1 <ρ<1μ; or 1μh <ρ<1μmln， whereんisthe second maximum eigenvalue andA.is the second minimum eigenvalue 16 In the cases ofW = Wd(1)， Wt(1)， thesignconditions are satisIIed in many periods However， in many periods， each ρexceeds one (see Figure1. and Figure 5，)
1325 Applicability of Spatial Autoregressive Moving Average Model -26::少ー
totally better than the spatial lag model in this example17， Among the
restricted SARMA models， especially， the case of W
=
Wd(2) has the best statistical-based performance and the case of W = Wt(3) achieves the best economic theoretical-based performance In these two cases(W = Wd(2)，Wt(3))， all the estimates and statistics are reported in Appendix
Table 1， Comparison in Performances between The Two Models
Restricted SARMA(l， 1) Model First order Spatial Lag Model Number of periods Number of periods when all parameters when all parameters
W θ AverAIC age of satisfy the conditions Average of satisfy the conditions and significant AIC and significant 開 level
I
切 level リ61evelI
跳 le 1 50751。
3 512“48 6 12 Wd 2 506，46 7 15 508，，87 l 11 3 508，04 2 16 509，，81 l 12 4 510，36。
5 510，，91 1 11 1 51286。
。
516，45 2 6 Wt 2 512，52 6 15 513，，92 2 8 3 512，，93 7 16 513，，10 2 10 4 513，，13 7 14 51281 4 11Next， compare parameter estimates between the restricted SARMA model and the spatiallag model Figure 1-8 show comparison between the two models in estimates of four coefficients， constant， s，δ，ρFigure 1-4 correspond to the cases of W = Wd(θ)， and figure 5 -8 conespond to the cases of W = Wt(θ)， but the results tend to be almost the same despite of the type ofW， Differences in estimates ofcons.tantare very large at
e
= 1.17 In order to check assumptions on disturbance terms， residuals of each model were also examined， As a result， we could not find significant differences between the
two models The residuals of both models seemed to satisfy norma1ity， but not to satisfy homoskedasticity
-270ー 1ぬ∞ "ω ω m mω " ω 50∞ 40∞ ，1 ω m mω '00' 0∞ Kagaωa Universily Economic Review 、?ン¥ヤポ ;~~;; ~;gg 2~~ ~g gii ~~ li Figure1..Comparison in Estimates between The Two Models [W = Wd(1)] 1326 (Note) SARMA means the restricted SARMA(l. 1) model and SLM means the first order spatiallag model " ω "ωト /，~ ".00トl' $0.00~ J m
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d V . ' ・ . "".
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事
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一 一 … 一 一 一 一 一 一 一 … ゃ 一 一 一 一 し 一 一 一 J < " ， . . 'o.;，q-.<..<$"，""'"，，，，<O<i;'""'(，<""'~'o~'"もやもや."グ.，<0"'，，;会もや<."""'"，.:}♂♂ 一 ー 一一一 一 一 台時一 一 一 ーザー ー - - ー ャ - 一 一 一 ， O ( )'Oへ
f¥
へ J :
J jj
f\~ρJ ∞脚ー E L( ; -I < l P O 別<l l l Rl < l A V l l U! A O: W <lA!SS <llll <llO~nv 1-e!~-e dS J O Á~!I! q-eコ!I dd V 6 Zt 1274 Kagaω'a University Economic Review 1330
However the differences become smaller as
e
becomes larger， and theestimates by the two models are nearly the same at θ = 3， 4 Concerning
ofδ， differences in the estimates are very small at each θexcept for the
case ofW = Wd(4).
On the other hand， differences in estimates of s and ρare significant
at each θConcerning ofβ， the estimates by the spatial model are almost
always lager than those by the restricted SARMA mode.l Conversely，
estimates ofρby the restricted SARMA model are almost always larger
than those by the spatial lag model. Especially， atθ= 1， differences in
the estimates are very large， which seems to correspond the large differ
-ences in estimates ofconstant.
Such differences in estimates of regression coefficients between the two
model are of critical importance because they also bring differences in
estimates of original parameters，ω， ，1.，RP， which are more meaningful
from economic viewpoints (see footnote 11 and 12). Therefore， if there
exist significant differences in estimates ofs and ρnot only in this example
but also in general， we indeed have to notice which model we should use.
Are there generally these differences in the estimates? In order to answer
this question， we will have to examine the differences via some
computational experiments
5
.
ConclusionLinear regressive models with spatially lagged dependent variables
have been increasingly studied and applied“ Among these modles， the
SARMA model is a general one， but it has not been used in emprical
studies.. One of the reason is that Huang' s ML estimation method
assumes a spatial weights matrix is symmetric， and the other is that
-1331 Applicability of Spatial Autoregressive Moving Average Model
-275-texts
In the situation， this study gave high applicability to the SARMA
model in the following two ways.. First， this study derived a useful
computation method for estimating parameters of the SARMA model when
a spatial weights matrix is asymmetric but diagonalizable.. Second， 1
showed that if a deterministic relation includes spatially lagged dependent
variables， a stochastic expression of the relation is more naturally specified
by the restricted SARMA model than by the spatial lag modeL Further
-more， 1 estimated parameters of Takatsuka's land price model using both
these models. In this exapmle， these models seemed to give a part of the
parameters significant different estimates each other. This means that the
specification problem is of critical importance.
In order to make applicability of the SARMA model higher and higher，
we have to overcome computational difficulties based on non-bounded
parameter space“ References Anselin， L (1988a) Spatial Econometrics.: Methods and Models， Dordrecht: Klwer Academic Anselin， L (1988b) Lagrange Multiplier TestDiagnostics for Spatial Dependence and Spatial Heterogeneity， Geographical Analysis， 20， 1-17 Anselin， L (1992)めaceStatTutorialれ A Workbook for Using SpaceStat in the Ana(ysis ofめαtialData， Regional Research Institute， West Virginia University Anselin， L ，..A K.Bera， R Florax， M J . Yoon (1995) SimpleDiagnostic Tests for Spatial Dependence， Regional Science and Urban Economics， 26， 77-104 Anselin， L (1998) Rao's Score Test in Spa均1Econometrics， BrutonCenter for Development Studies， School of Social Sciences， University of Texas at Dallas(mimeo) Can， A. (1990) The Mesurement of Neighborhood Dynamics in Urban House Prices， B乙onomicGeography， 66， 254-272
-276ー Kagaωa University Economic Revi，仰 1332
Can， A (1992) Specification and Estimation of Hedonic Housing Price Models，
Regional Science and Uγban Economics， 22， 453-474
Can， A and L Megbolugbe (1997) Spatial Dependence and House Price Index Construction， !ournal0/Real Estate Finance and Economics， 14， 203-222
Case， A C (1991)Spatial Patterns in Household Demand， Eωnometガca，59， 953-965 Case， A (1992) Neighborhood Influence and Technological Change， Regional Scienα
and Urban Economics， 22， 491-508
Case， A C ， H S Rosen and J， R Hines (1993)Budget Spillovers and Fiscal Policy
Interdependence: Evidence from the States， Jou門zal0/Public Economiω，， 52， 285
-307，
Cressie， N A C (1993) Statistics for5.抑tilData [Revised Edition]， New York: John Wiley
Doreian， P (1980) Linear Models with Spatially Distributed Data: Spatial Distur -bances or Spatial Effects 人SociologicalMethods& Research， 9， 29-60
Doreian， P (1982) Maximum Likelihood Methods forLinear Models: Spatial Effects and Spatial Disturbance Terms， Sociological Methods& Research， 10， 243-269 Haining， R (1978) The Moving Average Model for Spatial Interaction， Transactions，
Insti似た0/British Geogr，aρher， 3， 202-225
Haining， R (1990) Spatial Data Ana(ysis in the SociaJ and Environmental Scienc札 Cambridge: Cambridge University Press
Huang， J S (1984)The Autoregressive Moving Average Model for Spatial Analysis，
Au宮tralianJournal0/Statistics， 26， 169-178
Takatsuka， H (1998) A Present V訟lue Model with Interdψ，endence0/Profits ' Estimation0/Spatio-Tem，ρoral Structure0/Land Rents， Ph， D Dissertation in Social Engineering， Tokyo Institute of Technology (in Japanese)
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