m.m. fischer and a. berlin, springer. cal yardstick rbin ...€¦ · seminar is based on three...
TRANSCRIPT
J.P
aul
Elh
ors
t, U
niv
ersi
ty o
f G
ron
ing
en, th
e N
eth
erla
nd
s
Sem
inar
, S
t. A
nd
rew
s, J
anu
ary
18
, 2
010
- W
hat
are
sp
atia
l in
tera
ctio
n e
ffec
ts a
nd
wh
at i
s a
spat
ial
eco
no
met
ric
mo
del
?
- H
ow
to
est
imat
e a
spat
ial
eco
no
met
ric
mo
del
?
- A
th
eore
tica
l m
od
el t
hat
is
test
ed u
sin
g a
sp
atia
l ec
on
om
etri
c m
od
el.
Sem
inar
is
bas
ed o
n t
hre
e re
cen
t p
aper
s:
- J.
P.E
lho
rst
(20
09
) S
pat
ial
Pan
el D
ata
Mo
del
s. I
n:
M.M
. F
isch
er a
nd
A.
Get
is (
eds.
), H
and
bo
ok
of
Ap
pli
ed S
pat
ial
An
aly
sis.
Ber
lin
, S
pri
ng
er.
- E
lho
rst
J.P
.,
Fré
ret
S.
(20
09
) E
vid
ence
o
f P
oli
tica
l Y
ard
stic
k
Co
mp
etit
ion
in
Fra
nce
Usi
ng
a T
wo
-reg
ime
Sp
atia
l D
urb
in M
od
el w
ith
Fix
ed E
ffec
ts.
Jou
rna
l of
Reg
ion
al
Sci
ence
49
: 9
31
-951
.
- J.
P.E
lho
rst
(20
10
) A
pp
lied
Sp
atia
l E
con
om
etri
cs:
Rai
sin
g t
he
Bar
.
Sp
atia
l E
con
om
ic A
nal
ysi
s, F
ort
hco
min
g.
see
ww
w.r
egro
nin
gen
.nl/
elh
ors
t (s
pat
ial
eco
no
met
rics
)
J.P
.Elh
ors
t (2
01
0)
Appli
ed S
pat
ial
Eco
no
met
rics
: R
aisi
ng t
he
Bar
. S
pat
ial
Eco
no
mic
An
aly
sis,
Fo
rth
com
ing.
Th
e id
ea th
at cr
oss
-sec
tio
nal
unit
s in
tera
ct w
ith
oth
ers
has
rece
ntl
y r
ecei
ved
co
nsi
der
able
att
enti
on
, as
ev
iden
ced
in
th
e
dev
elop
men
t o
f th
eore
tica
l fr
amew
ork
s to
ex
pla
in
soci
al
ph
eno
men
a su
ch
as
soci
al
no
rms,
p
eer
infl
uen
ce,
nei
gh
bo
rhoo
d e
ffec
ts,
net
wo
rk e
ffec
ts,
con
tag
ion
, ep
idem
ics,
soci
al i
nte
ract
ion
s, i
nte
rdep
end
ent
pre
fere
nce
s, e
tc.
Inte
ract
ion
effe
ct
=
the
aver
age
beh
avio
r in
so
me
gro
up
infl
uen
ces
the
beh
avio
r o
f th
e in
div
idu
als
that
co
mp
rise
th
e
gro
up
(M
ansk
i, 1
993
).
uW
XX
WY
YN
+θ
+β
+αι
+ρ
= V
ecto
r N
ota
tion
ε+
λ=
Wu
u C
ross
-sec
tio
n d
ata
Y den
ote
s an
N
×1 vec
tor
consi
stin
g of
one
ob
serv
atio
n on th
e
dep
enden
t var
iable
for
ever
y u
nit
in t
he
sam
ple
(i=
1,…
,N),
Nι i
s an
N×1
vec
tor
of
ones
ass
oci
ated
wit
h t
he
const
ant
term
par
amet
er α
,
X den
ote
s an
N
×K
m
atri
x of
exogen
ous
exp
lanat
ory
var
iab
les,
wit
h t
he
asso
ciat
ed p
aram
eter
s β
conta
ined
in a
K×
1 v
ecto
r, a
nd
T
N1
),.
..,
(ε
ε=ε
is
a
vec
tor
of
dis
turb
ance
te
rms,
w
her
e ε i
ar
e
indep
enden
tly a
nd i
den
tica
lly d
istr
ibute
d e
rror
term
s fo
r al
l i
wit
h
zero
mea
n a
nd v
aria
nce
σ2.
S
pac
e-ti
me
dat
a: A
dd s
ub
scri
pt
t to
Y, X
, u a
nd ε
.
En
do
gen
ou
s in
tera
ctio
n e
ffec
ts (
WY
) =
th
e p
rop
ensi
ty o
f
an
ind
ivid
ual
to
b
ehav
e in
so
me
way
v
arie
s w
ith
th
e
beh
avio
r of
the
gro
up
.
Co
nsi
der
th
e ch
oic
e o
f h
igh
sch
oo
l af
ter
pri
mar
y s
cho
ol:
the
ind
ivid
ual
ch
oic
e te
nd
s to
var
y w
ith
th
e ch
oic
e m
ade
by
fri
end
s.
Ex
og
eno
us
inte
ract
ion
eff
ects
(W
X)
= t
he
pro
pen
sity
of
an
ind
ivid
ual
to
b
ehav
e in
so
me
way
v
arie
s w
ith
th
e
exo
gen
ou
s ch
arac
teri
stic
s o
f th
e g
rou
p (
mo
stly
th
ere
are
K
exo
gen
ou
s ex
pla
nat
ory
v
aria
ble
s,
and
th
us
K
exo
gen
ou
s in
tera
ctio
n e
ffec
ts).
Th
ere
are
exo
gen
ous
effe
cts
if sc
ho
ol
cho
ice
ten
ds
to
var
y w
ith
th
e ex
og
eno
us
char
acte
rist
ics
of
oth
er p
eop
le,
e.g
. th
e o
pin
ion
of
the
par
ents
of
frie
nd
s.
Co
rrel
ated
eff
ects
(d
istu
rban
ce t
erm
Wu
) =
in
div
idual
s in
the
sam
e g
roup
te
nd
to
b
ehav
e si
mil
arly
b
ecau
se th
ey
hav
e si
mil
ar
ind
ivid
ual
ch
arac
teri
stic
s o
r fa
ce
sim
ilar
inst
itu
tio
nal
en
vir
on
men
ts (
thes
e m
ay b
e u
no
bse
rved
).
Th
ere
are
corr
elat
ed e
ffec
ts i
f ch
ild
ren m
ake
the
sam
e
cho
ice
bec
ause
th
ey
hav
e si
mil
ar
bac
kg
rou
nds
(nei
gh
bo
urh
ood
) or
bec
ause
th
ey a
re t
aug
ht
by
th
e sa
me
teac
her
s.
The
rela
tionsh
ips
bet
wee
n d
iffe
rent
spat
ial
dep
enden
ce m
odel
s fo
r cr
oss
-sec
tion d
ata
λ
=0
θ=
0 ρ
=0
ρ=
0
θ
=0
λ=
0
θ=
-ρβ
ρ=
0 λ=
0
θ=
0
Up t
o 2
007
sp
atia
l ec
on
om
etri
cian
s w
ere
mai
nly
in
tere
sted
in
mo
del
s co
nta
inin
g
on
e ty
pe
of
spat
ial
inte
ract
ion
eff
ect:
the
spat
ial
lag m
odel
and t
he
spat
ial
erro
r
model
.
Man
ski
model
ε+
λ=
+θ
+β
+αι
+ρ
=
Wu
u
uW
XX
WY
YN
Kel
ejia
n-P
ruch
a m
odel
ε+
λ=
+β
+αι
+ρ
=
Wu
u
uX
WY
YN
Spat
ial
Durb
in m
odel
ε
+θ
+β
+αι
+ρ
=W
XX
WY
YN
Spat
ial
Durb
in e
rror
mod
el
ε+
λ=
+θ
+β
+αι
=
Wu
u
uW
XX
YN
Spat
ial
lag m
odel
ε+
β+
αι
+ρ
=X
WY
YN
Spat
ial
erro
r m
odel
u
XY
N+
β+
αι
=
ε+
λ=
Wu
u
(if
θ=
-ρβ
then
λ=
ρ)
OL
S m
odel
ε+
β+
αι
=X
YN
Iden
tifi
cati
on p
rob
lem
: O
ne
of
the
K+
2 i
nte
ract
ion
eff
ect
sho
uld
be
dro
pp
ed. F
oll
ow
ing
LeS
age
and
Pac
e (p
p. 15
5-
15
8),
th
e b
est
op
tio
n i
s to
ex
clu
de
the
spat
iall
y
auto
corr
elat
ed e
rro
r te
rm.
Th
e co
st o
f ig
no
rin
g s
pat
ial
dep
end
ence
in
th
e d
epen
den
t v
aria
ble
an
d/o
r in
th
e
ind
epen
den
t v
aria
ble
s is
rel
ativ
ely
hig
h s
ince
th
e
eco
no
met
rics
lit
erat
ure
has
po
inte
d o
ut
that
if
on
e o
r
mo
re r
elev
ant
exp
lan
ato
ry v
aria
ble
are
om
itte
d f
rom
a
reg
ress
ion
eq
uat
ion
, th
e es
tim
ato
r o
f th
e co
effi
cien
ts f
or
the
rem
ain
ing
var
iab
les
is b
iase
d a
nd
in
con
sist
ent
(Gre
ene
20
05
, p
p. 13
3-1
34
). I
n c
on
tras
t, i
gn
ori
ng
sp
atia
l
dep
end
ence
in
th
e d
istu
rban
ces,
if
pre
sen
t, w
ill
only
cau
se a
lo
ss o
f ef
fici
ency
.
*
*
W i
s an
N×
N m
atri
x d
escr
ibin
g t
he
spat
ial
arra
ng
emen
t o
f th
e sp
atia
l
un
its
in t
he
sam
ple
. L
ee (
20
04
) sh
ow
s th
at W
sh
ou
ld b
e a
no
nn
egat
ive
mat
rix
of
kn
ow
n c
on
stan
ts.
Th
e d
iag
on
al e
lem
ents
are
set
to
zer
o b
y
assu
mp
tio
n,
sin
ce n
o s
pat
ial
un
it c
an b
e v
iew
ed a
s it
s o
wn
nei
gh
bou
r.
Th
e m
atri
ces
I-ρW
an
d I
-λW
sh
ou
ld b
e n
on
-sin
gu
lar,
wh
ere
I re
pre
sen
ts
the
iden
tity
mat
rix
of
ord
er N
. F
or
a sy
mm
etri
c W
, th
is c
on
dit
ion
is
sati
sfie
d a
s lo
ng
as
ρ a
nd
λ a
re i
n t
he
inte
rio
r o
f (1
/ωm
in,1
/ωm
ax),
wher
e
ωm
in d
enote
s th
e sm
alle
st (
i.e.
, m
ost
neg
ativ
e) a
nd ω
max
the
larg
est
real
char
acte
rist
ic r
oot
of
W. If
W i
s ro
w-n
orm
aliz
ed s
ubse
quen
tly, th
e la
tter
inte
rval
tak
es t
he
form
(1/ω
min
,1),
sin
ce t
he
larg
est
char
acte
rist
ic r
oo
t o
f
W e
qu
als
un
ity
in
th
is s
itu
atio
n.
If W
is
an a
sym
met
ric
mat
rix
bef
ore
it
is
row
-no
rmal
ized
, it
may
hav
e co
mp
lex
char
acte
rist
ic r
oots
. L
eSag
e an
d
Pac
e (p
p. 88-8
9)
dem
onst
rate
that
in
th
at c
ase
ρ a
nd
λ a
re r
estr
icte
d t
o t
he
inte
rval
(1/r
min
,1),
wher
e r m
in e
qual
s th
e m
ost
neg
ativ
e pure
ly r
eal
char
acte
rist
ic r
oot
of
W a
fter
this
mat
rix i
s ro
w-n
orm
aliz
ed.
1
2
3
Ro
w-n
orm
aliz
ing
01
0
10
1
01
0
giv
es W
=
01
0
2/1
02/
1
01
0
.
Fin
ally
, one
of
the
foll
ow
ing t
wo c
ondit
ions
should
be
sati
sfie
d:
(a)
the
row
and c
olu
mn s
um
s of
the
mat
rice
s W
, (I
-ρW
)-1 a
nd
(I-λ
W)-1
bef
ore
W i
s ro
w-n
orm
aliz
ed s
hould
be
unif
orm
ly b
ounded
in a
bso
lute
val
ue
as N
goes
to i
nfi
nit
y, or
(b)
the
row
and c
olu
mn s
um
s of
W b
efore
W i
s ro
w-n
orm
aliz
ed
should
not
div
erge
to i
nfi
nit
y a
t a
rate
equal
to o
r fa
ster
than
the
rate
of
the
sam
ple
siz
e N
.
Condit
ion (
a) i
s ori
gin
ated
by K
elej
ian a
nd P
ruch
a (1
998, 1999),
and c
ondit
ion (
b)
by L
ee (
2004).
Both
condit
ions
lim
it t
he
cross
-
sect
ional
corr
elat
ion t
o a
man
agea
ble
deg
ree,
i.e
., t
he
corr
elat
ion
bet
wee
n t
wo s
pat
ial
unit
s sh
ould
conver
ge
to z
ero a
s th
e dis
tance
sep
arat
ing t
hem
incr
ease
s to
infi
nit
y.
When
the
spat
ial
wei
ghts
mat
rix i
s a
bin
ary c
onti
guit
y m
atri
x,
(a)
is
sati
sfie
d.
Norm
ally
, no s
pat
ial
unit
is
assu
med
to b
e a
nei
ghbour
to
more
than
a g
iven
num
ber
, sa
y q
, of
oth
er u
nit
s.
By c
ontr
ast,
when
the
spat
ial
wei
ghts
mat
rix i
s an
inver
se d
ista
nce
mat
rix,
(a)
may
not
be
sati
sfie
d.
Consi
der
an i
nfi
nit
e num
ber
of
spat
ial
unit
s th
at a
re l
inea
rly a
rran
ged
. T
he
dis
tance
of
each
sp
atia
l
unit
to i
ts f
irst
lef
t- a
nd r
ight-
han
d n
eighb
our
is d
; to
its
sec
ond
left
- an
d r
ight-
han
d n
eighb
our,
the
dis
tance
is
2d;
and s
o o
n.
When
W i
s an
inver
se d
ista
nce
mat
rix a
nd t
he
off
-dia
gonal
ele
men
ts o
f
W ar
e of
the
form
1/d
ij,
wher
e d
ij is
th
e dis
tance
b
etw
een tw
o
spat
ial
unit
s i
and
j,
each
ro
w
sum
is
...)
(2
d3
1d
21
d1
++
+×
,
rep
rese
nti
ng a
ser
ies
that
is
not
finit
e. T
his
is
per
hap
s th
e re
ason
why s
om
e em
pir
ical
appli
cati
ons
intr
oduce
a c
ut-
off
poin
t d* s
uch
that
w
ij=
0
if
dij>
d*.
Ho
wev
er,
since
th
e ra
tio
...)
(2
d3
1d
21
d1
++
+×
/N→
0 as
N
goes
to
in
finit
y,
condit
ion (b
) is
sati
sfie
d,
whic
h i
mp
lies
that
an i
nver
se d
ista
nce
mat
rix w
ithout
a
cut-
off
p
oin
t does
not
nec
essa
rily
hav
e to
b
e ex
cluded
in
an
emp
iric
al s
tudy f
or
reas
ons
of
consi
sten
cy.
T
he
opp
osi
te s
ituat
ion o
ccurs
when
all
cro
ss-s
ecti
onal
unit
s ar
e
assu
med
to b
e nei
ghb
ours
of
each
oth
er a
nd a
re g
iven
equal
wei
ghts
. In
that
cas
e al
l off
-dia
gonal
ele
men
ts o
f th
e sp
atia
l
wei
ghts
mat
rix a
re w
ij=
1. S
ince
the
row
and c
olu
mn s
um
s ar
e N
-1,
thes
e su
ms
div
erge
to i
nfi
nit
y a
s N
goes
to i
nfi
nit
y. In
contr
ast
to
the
pre
vio
us
case
, how
ever
, (N
-1)/
N→
1 i
nst
ead o
f 0 a
s N
go
es t
o
infi
nit
y. T
his
im
pli
es t
hat
a s
pat
ial
wei
ghts
mat
rix t
hat
has
equal
wei
ghts
and t
hat
is
row
-norm
aliz
ed s
ub
sequen
tly, w
ij=
1/(
N-1
),
must
be
excl
uded
for
reas
ons
of
consi
sten
cy.
No
te:
som
e o
f th
e re
gula
rity
co
nd
itio
ns
may
ch
ang
e in
a p
anel
dat
a se
ttin
g (
Yu
et
al.
2007
).
*
*
J.P
.Elh
ors
t (2
009)
Sp
atia
l P
anel
Dat
a M
odel
s. I
n:
M.M
. F
isch
er
and A
. G
etis
(ed
s.),
Han
db
ook o
f A
pp
lied
Sp
atia
l A
nal
ysi
s. B
erli
n,
Sp
ringer
.
In r
ecen
t yea
rs,
the
spat
ial
econom
etri
cs l
iter
ature
has
exhib
ited
a
gro
win
g i
nte
rest
in t
he
spec
ific
atio
n a
nd e
stim
atio
n o
f ec
onom
etri
c
rela
tionsh
ips
bas
ed o
n s
pat
ial
pan
els.
Sp
atia
l pan
els
typic
ally
ref
er
to d
ata
conta
inin
g t
ime
seri
es o
bse
rvat
ions
of
a nu
mb
er o
f sp
atia
l
unit
s (z
ip
codes
, m
un
icip
alit
ies,
re
gio
ns,
st
ates
, ju
risd
icti
ons,
countr
ies,
etc
.). T
his
inte
rest
can
be
exp
lain
ed b
y t
he
fact
that
pan
el
dat
a off
er r
esea
rcher
s ex
tended
mod
elin
g p
oss
ibil
itie
s as
com
par
ed
to
the
single
eq
uat
ion
cross
-sec
tional
se
ttin
g,
whic
h
was
th
e
pri
mar
y f
ocu
s of
the
spat
ial
econom
etri
cs l
iter
ature
for
a lo
ng t
ime.
Sp
atia
l la
g m
odel
(E
ndogen
ous
inte
ract
ion e
ffec
ts o
nly
)
,β
xy
wy
iti
it
N
1j
jtij
itε
+µ
+∑
+δ
==
δ
is
call
ed
the
spat
ial
auto
regre
ssiv
e co
effi
cien
t an
d
wij
is
an
elem
ent
of
a sp
atia
l w
eights
m
atri
x
W
des
crib
ing
the
spat
ial
arra
ngem
ent
of
the
unit
s in
the
sam
ple
.
Note
: In
this
pap
er,
I do n
ot
use
the
vec
tor
form
nota
tion,
but
a
nota
tion i
n i
ndiv
idual
obse
rvat
ions.
Furt
her
more
, I
use
δ i
nst
ead o
f
ρ
as
the
coef
fici
ent
of
endogen
ous
inte
ract
ion
effe
cts
(WY
).
Fin
ally
, ex
ogen
ous
inte
ract
ion e
ffec
ts c
an b
e in
cluded
by r
epla
cing
xit b
y
∑=
=N
1j
jtij
itit
]x
wx[
x (
spat
ial
Durb
in m
odel
).
Rea
son t
o c
onsi
der
fix
ed e
ffec
ts/r
ando
m e
ffec
ts m
odel
s.
The
stan
dar
d r
easo
nin
g b
ehin
d s
pat
ial
spec
ific
eff
ects
is
that
they
contr
ol
for
all
spac
e-sp
ecif
ic
tim
e-in
var
iant
var
iab
les
whose
om
issi
on c
ould
bia
s th
e es
tim
ates
in a
typ
ical
cro
ss-s
ecti
onal
stu
dy
(Bal
tagi,
2005).
The
spat
ial
spec
ific
eff
ects
may
be
trea
ted a
s fi
xed
eff
ects
or
as
random
eff
ects
. In
the
fixed
eff
ects
model
, a
du
mm
y v
aria
ble
is
intr
oduce
d f
or
each
sp
atia
l unit
, w
hil
e in
the
rando
m e
ffec
ts m
odel
,
µi
is
trea
ted
as
a ra
ndo
m
var
iab
le
that
is
in
dep
enden
tly
and
iden
tica
lly
dis
trib
ute
d
wit
h
zero
m
ean
an
d
var
iance
σ
µ.
Furt
her
more
, it
is
assu
med
that
the
random
var
iab
les
µi an
d ε
it a
re
indep
enden
t of
each
oth
er.
I fi
rst
consi
der
the
fixed
eff
ects
mod
el a
nd a
ssum
e th
at t
he
dat
a ar
e
sort
ed f
irst
by t
ime
and
then
by s
pat
ial
unit
s, w
her
eas
the
clas
sic
pan
el d
ata
lite
ratu
re t
ends
to s
ort
the
dat
a fi
rst
by s
pat
ial
unit
s an
d
then
by t
ime.
A
ccord
ing t
o A
nse
lin e
t al
. (2
006),
the
exte
nsi
on o
f th
e fi
xed
effe
cts
mod
el w
ith a
sp
atia
lly l
agged
dep
enden
t var
iable
rai
ses
two
com
pli
cati
ons.
F
irst
, th
e en
dogen
eity
of
Σjw
ijy
jt
vio
late
s th
e
assu
mp
tion o
f th
e st
andar
d r
egre
ssio
n m
odel
that
E[(
Σjw
ijy
jt)ε
it]=
0.
In
model
es
tim
atio
n,
this
si
mult
anei
ty
must
b
e ac
counte
d
for.
Sec
ond,
the
spat
ial
dep
enden
ce am
ong th
e obse
rvat
ions
at ea
ch
poin
t in
tim
e m
ay a
ffec
t th
e es
tim
atio
n o
f th
e fi
xed
eff
ects
.
The
log-l
ikel
ihood fu
nct
ion of
the
spat
ial
lag m
odel
w
ith fi
xed
effe
cts
is
∑∑
µ−
−∑δ
−σ
−δ
−+
πσ−
==
==
N
1i
T
1t
2
iit
N
1j
jtij
it2
N
2,
)β
xy
wy(
2
1|
WI|
log
T)
2lo
g(
2NT
LogL
-
----
----
----
--
wher
e th
e se
cond
term
on
the
right-
han
d
side
repre
sents
th
e
Jaco
bia
n t
erm
of
the
tran
sform
atio
n f
rom
ε t
o y
tak
ing i
nto
acc
ount
the
endogen
eity
of
Σjw
ijy
jt (
Anse
lin 1
988, p
. 63).
Co
mp
uta
tional
pro
ble
ms
of
the
Jaco
bia
n t
erm
.
∑∑
µ−
−∑δ
−σ
−δ
−+
πσ−
==
==
N
1i
T
1t
2
iit
N
1j
jtij
it2
N
2,
)β
xy
wy(
2
1|
WI|
log
T)
2lo
g(
2NT
LogL
The
par
tial
der
ivat
ives
of
the
log-l
ikel
ihood w
ith r
esp
ect
to µ
i are
0)
βx
yw
y(1
LogL
T
1t
iit
N
1j
jtij
it2
i
=∑
µ−
−∑δ
−σ
=µ∂
∂=
=, i=
1,…
,N.
When
solv
ing f
or
µi ,
one
ob
tain
s
∑−
∑δ
−=
µ=
=
T
1t
it
N
1j
jtij
iti
)β
xy
wy(
T1, i=
1,…
,N.
∑−
∑δ
−=
µ=
=
T
1t
it
N
1j
jtij
iti
)β
xy
wy(
T1, i=
1,…
,N.
This
equat
ion s
how
s th
at t
he
stan
dar
d f
orm
ula
for
calc
ula
ting t
he
spat
ial
fixed
eff
ects
app
lies
to t
he
fixed
eff
ects
spat
ial
lag m
odel
in
a st
raig
htf
orw
ard m
anner
. C
orr
ecti
ons
for
the
spat
ial
dep
enden
ce
among
the
ob
serv
atio
ns
at
each
poin
t in
ti
me,
oth
er
than
th
e
addit
ion
of
the
spat
iall
y
lagged
dep
enden
t var
iable
to
th
ese
form
ula
s, a
re n
ot
nec
essa
ry.
Sub
stit
uti
ng t
he
solu
tion f
or
µi i
nto
the
log-l
ikel
ihood f
unct
ion,
and
afte
r re
arra
ngin
g te
rms,
th
e co
nce
ntr
ated
lo
g-l
ikel
ihood fu
nct
ion
wit
h r
esp
ect
to β
, δ a
nd σ
2 i
s ob
tain
ed
∑∑
−∑
δ−
σ−
δ−
+πσ
−=
==
=
N
1i
T
1t
2* it
N
1j
jtij
* it2
N
2,
)β
x*]
yw
[y(
2
1|
WI|
log
T)
2lo
g(
2NT
LogL
wher
e th
e as
teri
sk d
enote
s th
e dem
eanin
g p
roce
dure
∑−
==T
1t
itit
* ity
T1y
y,
∑∑
−∑
=∑
==
==
T
1t
jt
N
1j
ijjt
N
1j
ij* jt
N
1j
ijy
wT1
yw
yw
and
∑−
==T
1t
itit
* itx
T1x
x.
∑∑
−∑
δ−
σ−
δ−
+πσ
−=
==
=
N
1i
T
1t
2* it
N
1j
jtij
* it2
N
2,
)β
x*]
yw
[y(
2
1|
WI|
log
T)
2lo
g(
2NT
LogL
1. R
egre
ss y
it,
Σw
ijy
jt o
n x
it (
+*)
by O
LS
→ b
0 a
nd b
1
2. C
om
pu
te r
esid
ual
s
3. S
ub
stit
ute
res
idu
als
into
log-l
ikel
iho
od
fu
nct
ion
, co
nce
ntr
ate
it w
ith
res
pec
t
to σ
2,
and
max
imiz
e it
wit
h r
esp
ect
to δ
4. β
=b
0-δ
b1 a
nd
σ2
5. D
eter
min
e v
aria
nce
-cov
aria
nce
mat
rix
(se
e pap
er)
6. R
eco
ver
fix
ed e
ffec
ts
Go t
o htt
p:/
/ww
w.r
egro
nin
gen
.nl/
elhors
t (a
nd c
lick
on
soft
war
e) f
or
soft
war
e to
est
imat
e sp
atia
l p
anel
s.
**
Whet
her
the
random
eff
ects
model
is
an a
pp
ropri
ate
spec
ific
atio
n
in s
pat
ial
rese
arch
rem
ains
contr
over
sial
. W
hen
the
random
eff
ects
model
is
im
ple
men
ted,
the
unit
s of
obse
rvat
ion
should
b
e
rep
rese
nta
tive
of
a la
rger
p
op
ula
tion,
and
the
num
ber
of
unit
s
should
pote
nti
ally
be
able
to g
o t
o i
nfi
nit
y.
Ther
e ar
e tw
o t
yp
es o
f as
ym
pto
tics
that
are
com
monly
use
d i
n t
he
conte
xt
of
spat
ial
ob
serv
atio
ns:
(a
) T
he
‘infi
ll’
asym
pto
tic
stru
cture
, w
her
e th
e sa
mp
ling r
egio
n r
emai
ns
bounded
as
∞→
N.
In th
is ca
se m
ore
unit
s of
info
rmat
ion co
me
from
obse
rvat
ions
taken
fr
om
b
etw
een
those
al
read
y
ob
serv
ed;
and
(b)
The
‘incr
easi
ng
do
mai
n’
asy
mp
toti
c st
ruct
ure
, w
her
e th
e sa
mp
ling
regio
n g
row
s as
∞
→N
. In
this
cas
e th
ere
is a
min
imu
m d
ista
nce
sep
arat
ing a
ny t
wo s
pat
ial
unit
s fo
r al
l N
.
A
ccord
ing
to
Lah
iri
(2003),
th
ere
are
also
tw
o
typ
es
of
sam
pli
ng d
esig
ns:
(a)
The
stoch
asti
c des
ign w
her
e th
e sp
atia
l unit
s
are
random
ly d
raw
n;
and (
b)
The
fixed
des
ign w
her
e th
e sp
atia
l
unit
s li
e on a
nonra
ndom
fie
ld, p
oss
ibly
irr
egula
rly s
pac
ed.
T
he
spat
ial
econom
etri
c li
tera
ture
mai
nly
focu
ses
on i
ncr
easi
ng
do
mai
n a
sym
pto
tics
under
the
fixed
sam
ple
des
ign (
Cre
ssie
1993,
p. 100;
Gri
ffit
h a
nd L
agona
1998;
Lah
iri
2003).
Alt
hough t
he
nu
mb
er o
f sp
atia
l unit
s under
the
fixed
sam
ple
des
ign
can p
ote
nti
ally
go t
o i
nfi
nit
y,
it i
s ques
tionab
le w
het
her
they
are
rep
rese
nta
tive
of
a la
rger
pop
ula
tion.
For
a giv
en s
et o
f re
gio
ns,
such
as
al
l co
unti
es of
a st
ate
or
all
regio
ns
in a
countr
y,
the
pop
ula
tion m
ay b
e sa
id
• ‘t
o b
e sa
mp
led e
xhau
stiv
ely’
(Ner
love
and B
ales
tra
1996,
p.
4),
• ‘t
he
indiv
idual
sp
atia
l unit
s hav
e ch
arac
teri
stic
s th
at a
ctual
ly
set
them
ap
art
from
a l
arger
pop
ula
tion’
(Anse
lin 1
988, p
. 51).
• ‘t
he
crit
ical
is
sue
is th
at th
e sp
atia
l unit
s b
e fi
xed
an
d not
sam
ple
d,
and t
hat
infe
rence
be
condit
ional
on t
he
obse
rved
unit
s’ B
eck (
2001, p
. 272).
In a
ddit
ion, th
e tr
adit
ional
ass
um
pti
on o
f ze
ro c
orr
elat
ion b
etw
een
µi i
n t
he
rando
m e
ffec
ts m
odel
and t
he
exp
lanat
ory
var
iab
les,
whic
h a
lso n
eeds
to b
e m
ade,
is
par
ticu
larl
y r
estr
icti
ve.
**
Inte
rpre
tati
on C
oef
fici
ents
If th
e sp
atia
l D
urb
in m
odel
is
ta
ken
as
p
oin
t of
dep
artu
re an
d
rew
ritt
en a
s
ερ
−+
θ+
βρ
−+
αι
ρ−
=−
−−
11
N
1)
WI(
)W
XX(
)W
I()
WI(
Y,
the
mat
rix
of
par
tial
der
ivat
ives
of
Y
wit
h
resp
ect
to
the
kth
exp
lanat
ory
var
iab
le
of
X
in
unit
1
up
to
u
nit
N
(s
ay
xik
fo
r
i=1,…
,N, re
spec
tivel
y)
is r
elat
ivel
y e
asy t
o o
bta
in
βθ
θ
θβ
θ
θθ
β
ρ−
=
∂∂
∂∂
∂∂
∂∂
=
∂∂∂∂
−
kk
2N
k1
N
kN
2k
k2
1
kN
1k
12
k
1
Nk
N
k1N
Nk1
k11
Nk
k1
.w
w
..
..
w.
w
w.
w
)
WI(
xy.
xy.
..
xy.
xy
xY.
xY,
wher
e w
ij i
s th
e (i
,j)t
h e
lem
ent
of
W.
If W
=
01
0
w0
w
01
0
23
21
an
d
ρρ
ρ
ρρ
ρρ
ρ
ρ=
ρ−
−
2
21
21
2
23
21
23
22
23
2
1
w-
1w
w1
w
ww
-1
-
1
1
)
WI(
, w
e get
.
)w(
)w
1()
w()
w(
w)
w(w
)w(
)w
()
w()
w()
w1(
1
1
xY
xY
xY
k2
3k
2
21
kk
k2
1k
2
21
k2
3k
23
kk
k2
1k
21
k2
3k
2
23
kk
k2
1k
2
23
2
k3
k2
k1
θρ
+β
ρ−
θ+
ρβθ
ρ+
βρ
θ+
βρ
ρθ+
βθ
+β
ρ
θρ
+β
ρθ
+ρβ
θρ
+β
ρ−
ρ−
=
∂∂∂∂
∂∂
Dir
ect
effe
ct:
Mea
n d
iagonal
ele
men
t
Indir
ect
effe
ct:
Mea
n r
ow
su
m o
f no
n-d
iagonal
ele
men
ts.
Table 1.
Dir
ect
and
indir
ect
effe
cts
of
dif
fere
nt
model
spec
ific
atio
ns
[N=
3,
W a
s in
(4)]
Typ
e of
mo
del
D
irec
t ef
fect
In
dir
ect
effe
ct
Sp
atia
l D
urb
in m
odel
Man
ski
mod
el
k)
21(
3
2
k)
21(
3
)2
3(θ
+β
ρ−ρ
ρ−ρ−
k
)2
1(3
3
k)
21(
3
23
θ+
βρ−
ρ+
ρ−
ρ+ρ
Sp
atia
l la
g m
odel
Kel
ejia
n-P
ruch
a m
odel
k)
21(
3
)2
3(β
ρ−ρ−
k
)2
1(3
23
βρ−
ρ+ρ
Sp
atia
l D
urb
in e
rror
mod
el
βk
θk
OL
S m
odel
Sp
atia
l er
ror
model
βk
0
• O
LS
mo
del
: in
dir
ect
effe
cts
are
zero
by
con
stru
ctio
n.
• S
pat
ial
Du
rbin
err
or
mo
del
: it
can
sti
ll b
e se
en f
rom
th
e
coef
fici
ent
esti
mat
es
and
the
corr
esp
on
din
g
stan
dar
d
erro
rs o
r t-
val
ues
(d
eriv
ed f
rom
th
e v
aria
nce
-co
var
ian
ce
mat
rix
) w
het
her
in
dir
ect
effe
cts
are
sig
nif
ican
t.
• S
pat
ial
lag
mo
del
: li
mit
atio
n i
s th
at t
he
rati
o b
etw
een
the
ind
irec
t an
d d
irec
t ef
fect
s in
th
e sp
atia
l la
g m
od
el i
s th
e
sam
e fo
r ev
ery
ex
pla
nat
ory
var
iab
le.
• S
pat
ial
Durb
in m
od
el:
no
pri
or
rest
rict
ion
s ar
e im
po
sed
on
th
e m
agn
itud
e o
f bo
th t
he
dir
ect
and
in
dir
ect
effe
cts
and
thu
s th
at t
he
rati
o b
etw
een
the
ind
irec
t an
d t
he
dir
ect
effe
ct
may
b
e d
iffe
ren
t fo
r d
iffe
ren
t ex
pla
nat
ory
var
iab
les.
*
*
Elh
ors
t J.
P., F
rére
t S
. (2
009)
Evid
ence
of
Poli
tica
l Y
ardst
ick
Co
mp
etit
ion i
n F
rance
Usi
ng a
Tw
o-r
egim
e S
pat
ial
Durb
in M
odel
wit
h F
ixed
Eff
ects
. Jo
urn
al
of
Reg
ional
Sci
ence
49:
931-9
51.
Str
ateg
ic i
nte
ract
ion a
mong g
over
nm
ents
(m
un
icip
alit
ies,
reg
ions
or
stat
es)
has
bec
om
e a
maj
or
focu
s of
theo
reti
cal
and e
mp
iric
al
work
in
p
ub
lic
econom
ics.
M
any
st
udie
s hav
e fo
und
that
an
incr
ease
in
th
e ta
x
burd
en
of
nei
ghb
ori
ng
juri
sdic
tions
of
one
euro
/doll
ar i
s m
atch
ed b
y a
n i
ncr
ease
of
18 t
o 6
6 c
ents
per
unit
of
tax i
n a
juri
sdic
tion's
ow
n t
ax b
urd
en.
A r
elat
ed l
iter
ature
focu
ses
on e
xp
endit
ure
inte
rdep
enden
ce a
nd h
as f
ound s
imil
ar f
igure
s.
One
exp
lanat
ion i
s yar
dst
ick c
om
pet
itio
n:
vote
rs u
se i
nfo
rmat
ion
fro
m oth
er ju
risd
icti
ons
to ju
dge
the
per
form
ance
of
thei
r ow
n
gover
nors
. T
he
reas
on f
or
this
beh
avio
r is
asy
mm
etri
c in
form
atio
n;
vote
rs d
o n
ot
know
what
lev
el o
f se
rvic
es c
an b
e p
rovid
ed r
elat
ive
to a
cer
tain
tax
lev
el.
Tax
es c
over
the
min
imal
pro
duct
ion c
ost
of
pub
lic
goods
plu
s an
y
extr
a re
sourc
es l
ost
to w
aste
or
rent
seek
ing.
Thes
e lo
st r
esourc
es
cannot
be
ob
serv
ed
by
vote
rs.
Sin
ce
tax
rate
s an
d
expen
dit
ure
level
s in
nea
rby j
uri
sdic
tions
are
more
eas
ily o
bse
rved
, th
ey c
an
serv
e as
a b
ench
mar
k a
nd u
sed i
n e
lect
ions
to d
isci
pli
ne
and s
elec
t
the
typ
e of
gover
nor.
How
ever
, if
vote
rs
consi
der
re
lati
ve
per
form
ance
, ra
tional
gover
nors
wil
l do t
he
sam
e an
d (
par
tly)
mim
ic t
he
tax r
ates
and
exp
endit
ure
le
vel
s of
thei
r nei
ghb
ors
. T
his
is
ca
lled
yar
dst
ick
com
pet
itio
n.
Pre
fere
nce
s of
rep
rese
nta
tive
resi
den
t of
juri
sdic
tion i
and
indiv
idual
budget
const
rain
t ar
e giv
en b
y
U(y
i-T
i,zi;X
i)
yi=
per
ca
pit
a in
com
e,
Ti=
tax
pay
men
t p
er
cap
ita,
z i
=le
vel
of
a
pub
lic
good, X
i=ch
arac
teri
stic
s of
juri
sdic
tion i
oth
er t
han
inco
me.
Let
z i
/Ti
den
ote
th
e m
inim
um
le
vel
of
pub
lic
good
pro
vis
ion
rela
tive
to
taxes
th
at
must
b
e del
iver
ed
for
juri
sdic
tion's
i's
gover
nm
ent
to r
emai
n i
n o
ffic
e. T
his
req
uir
ed l
evel
dep
ends
on
ob
serv
ed
pub
lic
good
level
s obse
rved
in
oth
er
juri
sdic
tions:
z i/T
i=φ
([z/
T] -
i).
If th
e le
vel
s of
pub
lic
good p
rovis
ion re
lati
ve
to t
axes
in oth
er
juri
sdic
tions,
[z/
T] -
i, in
crea
ses,
gover
nm
ent
i is
forc
ed t
o r
aise
zi/T
i
to re
mai
n in
off
ice.
S
ince
z i
=T
i φ
([z/
T] -
i),
we
hav
e (a
nd usi
ng
med
ian v
ote
r th
eore
m)
U(y
i-T
i,zi;X
i)=
U(y
i-T
i, T
i φ([
z/T
] -i)
;Xi)
≡V
(zi,z
-i;X
i).
Note
: In
stea
d o
f one
consu
mer
, w
e m
ay h
ave
dif
fere
nt
consu
mer
s
in e
ver
y j
uri
sdic
tion w
ith p
refe
rence
s ra
ngin
g a
long a
sp
ectr
um
on
most
p
ub
lic
serv
ices
. T
he
med
ian
vote
r th
eore
m
stat
es
that
, if
pre
fere
nce
s ar
e si
ngle
-pea
ked
and g
over
nm
ent
poli
cy i
s dec
ided
by
rep
rese
nta
tives
ele
cted
by a
maj
ori
ty v
ote
, gover
nm
ent
poli
cy w
ill
refl
ect
the
pre
fere
nce
s of
the
med
ian v
ote
r.
Fir
st-o
rder
condit
ion
)X;
z(R
z
0
VzV
ii
ii
zi
−=
⇒=
≡∂∂
,
wher
e R
re
pre
sents
a
reac
tion fu
nct
ion to
th
e ch
oic
es of
oth
er
juri
sdic
tions.
The
slop
e of
the
reac
tion f
unct
ion w
ith r
esp
ect
to z
-i
can b
e p
osi
tive
or
neg
ativ
e. A
tes
t of
the
null
hyp
oth
esis
that
the
reac
tion
funct
ion's
sl
op
e is
ze
ro
is
effe
ctiv
ely
a te
st
for
the
exis
tence
of
spil
lover
s. F
urt
her
more
, in
tera
ctio
n m
ay e
xp
ecte
d t
o
be
more
pro
nounce
d i
f gover
nors
are
poli
tica
lly s
ensi
tive
to f
isca
l
poli
cy
chan
ges
in
nei
ghb
ori
ng
juri
sdic
tions.
In
th
is
pap
er:
Dep
artm
ents
gover
ned
by
a sm
all
poli
tica
l m
ajori
ty
mim
ic
nei
ghbori
ng ex
pen
dit
ure
s on w
elfa
re to
a
gre
ater
ex
tent
than
do
Dep
artm
ents
gover
ned
by a
lar
ge
poli
tica
l m
ajori
ty. *
*
,x
wx
yw
)d
1(y
wd
yit
ti
N
1j
jtij
it
N
1j
N
1j
jtij
it2
jtij
it1
itε
+λ
+µ
+θ
∑+
β∑
+α
∑+
−δ
+δ
==
==
Reason to consider two regimes:
the
theo
reti
cal
and e
mp
iric
al
lite
ratu
re o
n p
ub
lic
econo
mic
s off
ers
two a
lter
nat
ive
exp
lanat
ions
for
the
exis
tence
of
tax a
nd e
xp
endit
ure
inte
ract
ion e
ffec
ts,
whic
h
hav
e th
e sa
me
reac
tion f
unct
ion.
They
may
als
o b
e th
e re
sult
of
spil
lover
eff
ects
, fo
r ex
amp
le,
bec
ause
exp
endit
ure
s on l
oca
l p
ub
lic
serv
ices
m
ay
hav
e b
enef
icia
l or
det
rim
enta
l ef
fect
s on
nea
rby
juri
sdic
tions
(see
Cas
e et
al.
, 1993 f
or
a th
eore
tica
l ex
pla
nat
ion),
or
be
the
resu
lt o
f ta
x o
r w
elfa
re c
om
pet
itio
n.
,x
wx
yw
)d
1(y
wd
yit
ti
N
1j
jtij
it
N
1j
N
1j
jtij
it2
jtij
it1
itε
+λ
+µ
+θ
∑+
β∑
+α
∑+
−δ
+δ
==
==
One
reas
on t
o a
dd s
pat
iall
y l
agged
indep
enden
t var
iable
s is
tak
en
from
Boar
net
and G
laze
r (2
002).
They
arg
ue
that
a n
egat
ive
gra
nt
spil
lover
ef
fect
ca
n
also
b
e in
terp
rete
d
as
a fo
rm
of
yar
dst
ick
com
pet
itio
n.
A
vote
r w
ho
sees
th
at
a nei
ghb
ori
ng
juri
sdic
tion
rece
ived
a g
rant
whic
h h
is j
uri
sdic
tion d
id n
ot
rece
ive,
may
thin
k
poorl
y o
f th
e ab
ilit
y o
f th
e lo
cal
gover
nors
and,
ther
efore
, re
duce
his
dem
and f
or
loca
l sp
endin
g.
In t
erm
s o
f m
odel
ing,
if s
pen
din
g
on p
ub
lic
serv
ices
is
taken
to d
epen
d o
n g
rants
rec
eived
by t
he
juri
sdic
tion,
the
spat
ial
aver
age
of
gra
nts
rec
eived
by n
eighb
ori
ng
juri
sdic
tions
also
aff
ects
sp
endin
g o
n p
ub
lic
serv
ices
.
LeS
age
and P
ace
(2008)
off
er a
noth
er r
easo
n t
o a
dd s
pat
iall
y l
agged
indep
enden
t var
iable
s, n
amel
y,
an o
mit
ted v
aria
ble
s m
oti
vat
ion t
o
incl
ude
the
var
iable
s in
a
regre
ssio
n
rela
tionsh
ip
that
se
eks
to
exp
lore
inte
ract
ion e
ffec
ts i
n a
sp
atia
l co
nte
xt.
If
unob
serv
ed o
r
unknow
n
but
rele
van
t var
iab
les
foll
ow
ing
a fi
rst-
ord
er
spat
ial
auto
regre
ssiv
e p
roce
ss
do
not
appea
r in
th
e m
odel
, an
d
thes
e
var
iable
s hap
pen
to b
e co
rrel
ated
wit
h i
ndep
enden
t var
iab
les
not
om
itte
d f
rom
the
model
, a
spat
ial
lag m
odel
exte
nded
to i
ncl
ude
spat
iall
y
lagged
in
dep
enden
t var
iab
les
wil
l p
roduce
unb
iase
d
coef
fici
ent
esti
mat
es,
wher
eas
a sp
atia
l la
g m
odel
wit
hout
thes
e
var
iable
s ca
nnot.
Reason to consider time-period fixed effects (and not spatially
autocorrelated error terms):
Iden
tifi
cati
on
pro
ble
m
Man
ski
(1993).
T
ime-
per
iod
fixed
ef
fect
s co
rrec
t fo
r sp
atia
l in
tera
ctio
n
effe
cts
among
the
erro
r te
rms,
su
ch
as
unob
serv
ed
shock
s
foll
ow
ing a
sp
atia
l p
atte
rn o
r var
iab
les
that
incr
ease
or
dec
reas
e
toget
her
in d
iffe
rent
juri
sdic
tions
along t
he
sam
e (b
usi
nes
s) c
ycl
e
over
tim
e.
The
mat
hem
atic
al e
xp
lanat
ion i
s th
at t
ime-
per
iod f
ixed
eff
ects
are
iden
tica
l to
a
spat
iall
y
auto
corr
elat
ed
erro
r te
rm
wit
h
a sp
atia
l
wei
ghts
mat
rix w
hose
ele
men
ts a
re a
ll e
qual
to 1
/N,
incl
udin
g t
he
dia
gonal
el
emen
ts.
Wh
en
this
sp
atia
l w
eights
m
atri
x
would
be
adop
ted,
one
ob
tain
s e.
g.
∑−
=∑
−=
=
N
1j
jtit
N
1j
jtij
ity
N1y
yw
y
whic
h
is
equiv
alen
t to
the
dem
eanin
g p
roce
dure
of
Eq.
(6)
but
then
for
fixed
effe
cts
in t
ime.
,x
wx
yw
)d
1(y
wd
yit
ti
N
1j
jtij
it
N
1j
N
1j
jtij
it2
jtij
it1
itε
+λ
+µ
+θ
∑+
β∑
+α
∑+
−δ
+δ
==
==
Pre
vio
us
stu
die
s o
n y
ard
stic
k c
om
pet
itio
n
- B
esle
y a
nd
Cas
e (1
995
), R
evel
li (
200
6)
Tw
o-e
qu
atio
ns
spat
ial
lag
mo
del
est
imat
ed b
y I
V b
ased
on
pan
el d
ata
Obje
ctio
n:
Co
effi
cien
ts o
f co
ntr
ol
var
iab
les
no
t id
enti
cal;
res
ult
s m
ay a
lso
co
ver
dif
fere
nce
s o
ther
th
an t
he
po
liti
cal
pro
cess
- B
ord
ignon
et
al.
(2003
), A
ller
s an
d E
lho
rst
(2005
)
Tw
o r
egim
es s
pat
ial
lag
/err
or
mo
del
est
imat
ed b
y M
L b
ased
on
cro
ss-s
ecti
on
al
dat
a. O
bje
ctio
n:
No c
on
tro
l fo
r sp
atia
l fi
xed
eff
ects
- C
ase
(199
3),
Sch
alte
gg
er a
nd
Kü
ttel
(20
02)
and
So
llé-
Oll
é (2
00
3)
Sp
atia
l la
g m
od
el
wit
h c
ross
-pro
du
ct v
aria
ble
s es
tim
ated
by I
V b
ased
on
pan
el
dat
a. O
bje
ctio
n:
Non
stat
ion
arit
y;
Jaco
bia
n t
erm
no
t d
efin
ed f
or
all
ob
serv
atio
ns.
∑η
−δ
−=J1
jj
jN
|W)
Pdia
g(
WI|
ln*
T
- O
ne
maj
or
sho
rtco
min
g o
f al
l st
ud
ies:
No
sp
atia
l D
urb
in m
od
el.
,x
wx
yw
)d
1(y
wd
yit
ti
N
1j
jtij
it
N
1j
N
1j
jtij
it2
jtij
it1
itε
+λ
+µ
+θ
∑+
β∑
+α
∑+
−δ
+δ
==
==
ML
est
imat
ion
(IV
wo
uld
igno
re t
he
Jaco
bia
n t
erm
, in
stru
men
tal
var
iab
les?
)
∑∑
λ−
µ−
θ∑
−β
−α
−∑
−δ
−∑
δ−
σ−
∑−
δ−
δ−
+πσ
−=
==
==
=
=
N
1i
T
1t
2
ti
N
1j
jtij
it
N
1j
jtij
it2
N
1j
jtij
it1
it2
T
1t
tN
2t
1N
2
,]
xw
xy
w)
d1(
yw
dy[
2
1
|W)
DI(
WD
I|lo
g)
2lo
g(
2NT
Lo
gL
Solv
e fo
r in
terc
ept
and s
pat
ial
and t
ime-
per
iod f
ixed
eff
ects
∑∑
θ∑
−β
−∑
−δ
−∑
δ−
σ−
∑−
δ−
δ−
+πσ
−=
==
==
=
=
N
1i
T
1t
2N
1j
* jtij
* it
N
1j
jtij
it2
N
1j
jtij
it1
* it2
T
1t
tN
2t
1N
2
,]
xw
x*)
yw
)d
1((
*)y
wd(
y[2
1
|W)
DI(
WD
I|lo
g)
2lo
g(
2NT
Lo
gL
∑∑
θ∑
−β
−∑
−δ
−∑
δ−
σ−
∑−
δ−
δ−
+πσ
−=
==
==
=
=
N
1i
T
1t
2N
1j
* jtij
* it
N
1j
jtij
it2
N
1j
jtij
it1
* it2
T
1t
tN
2t
1N
2
,]
xw
x*)
yw
)d
1((
*)y
wd(
y[2
1
|W)
DI(
WD
I|lo
g)
2lo
g(
2NT
Lo
gL
1. R
egre
ss y
it,
ditΣ
wijy
it a
nd
(1-d
it)Σ
wijy
jt o
n x
it (
+*)
by O
LS
→ b
0, b
1 a
nd
b2
2. C
om
pu
te r
esid
ual
s
3. S
ub
stit
ute
res
idu
als
into
log-l
ikel
iho
od
fu
nct
ion
, co
nce
ntr
ate
it w
ith
res
pec
t
to σ
2,
and
max
imiz
e it
wit
h r
esp
ect
to δ
1 a
nd δ
2.
4. β
=b
0-δ
1b
1-
δ2b
2 a
nd
σ2.
5. D
eter
min
e v
aria
nce
-cov
aria
nce
mat
rix
. N
ot
rep
ort
ed i
n t
he
lite
ratu
re b
efo
re.
6. R
eco
ver
fix
ed e
ffec
ts
Tab
le 2
Est
imat
ion
res
ult
s: W
elfa
re s
pen
din
g b
y F
ren
ch D
epar
tmen
ts
Ex
pla
nat
ory
var
iab
les
Tw
o-
way
spat
ial
Du
rbin
mo
del
(1)
On
e-
way
spat
ial
Du
rbin
mo
del
(2)
Tw
o-w
ay
spat
ial
Du
rbin
mo
del
,
two
reg
imes
*
(3)
Op
erat
ion
al g
ran
t 0
.00
0
(0.0
3)
0.0
31
(7.7
3)
0.0
01
(0.0
2)
W*
Op
erat
ion
al
gra
nt
-0.0
35
(-2
.07
)
0.0
31
(4.0
0)
-0.0
32
(-1
.88
)
δ
0.0
83
(1.6
1)
0.2
82
(6.2
4)
0.1
67
0.0
34
(7.0
7)
(1.5
2)
int.
eff.
=0
.18
/ 0
.20 i
n s
tud
ies
wit
h c
on
tro
ls f
or
spat
ial
fix
ed e
ffec
ts, 0
.20
/ 0
.66
in
stu
die
s w
ith
ou
t co
ntr
ols
for
spat
ial
fix
ed e
ffec
ts.
Sp
atia
l F
E
yes
n
o
yes
Tim
e-p
erio
d F
E
yes
yes
yes
Reg
ime
du
mm
y
yes
Lo
gL
1
30
5.8
3
63
8.2
2
13
14
.15
R2
0.9
41
0
.70
3
0.9
42
t-v
alu
es i
n p
aren
thes
is, *
Go
ver
no
rs b
ack
ed b
y m
ajo
rity
les
s th
an o
r eq
ual
to
75
% a
nd
g
reat
er t
han
75
%, re
spec
tiv
ely
Tab
le 3
Yar
dst
ick
co
mp
etit
ion
an
d v
oti
ng
mar
gin
5
5%
6
0%
6
5%
7
0%
7
5%
8
0%
8
5%
Nu
mb
er o
f o
bse
rvat
ion
s
and
δ1 w
hen
mar
gin
is
less
th
an o
r eq
ual
to
.%
10
5
0.1
02
(3.6
1)
20
3
0.1
14
(3.5
0)
31
5
0.1
27
(5.4
8)
43
0
0.1
42
(5.3
5)
54
3
0.1
67
(7.0
7)
65
9
0.1
68
(3.7
3)
73
9
0.2
06
(2.3
5)
Nu
mb
er o
f o
bse
rvat
ion
s
and
δ2 w
hen
mar
gin
is
gre
ater
th
an .
%
73
2
-0.0
16
(0.2
7)
63
4
0.0
29
(0.6
2)
52
2
0.0
20
(0.7
0)
40
7
0.0
33
(1.2
9)
29
4
0.0
34
(1.5
2)
17
8
0.0
56
(1.9
2)
98
0.0
75
(2.5
0)
δ1 -
δ2
T-v
alu
e o
f d
iffe
rence
0.1
18
(1.7
8)
0.0
85
(1.5
1)
0.1
08
(3.0
1)
0.1
10
(3.0
0)
0.1
34
(4.2
6)
0.1
12
(2.1
0)
0.1
30
(1.4
0)
Dif
fere
nce
in
terc
epts
0
.62
2
0.4
62
0
.58
2
0.5
88
0
.71
1
0.5
76
0
.68
0
Lo
gL
1
30
9.2
3
13
09
.77
13
11
.54
13
11
.96
13
14
.15
13
11
.61
13
09
.05
Tab
le 3
. B
ias
and
RM
SE
of
δ1 a
nd
δ2 o
f d
iffe
ren
t ex
per
imen
tal
par
amet
er c
om
bin
atio
ns
wh
en u
sin
g t
he
ML
est
imat
or
and
wh
en u
sin
g I
V*
ML
est
imat
or
Inst
rum
enta
l v
aria
ble
s (I
V)
Bia
s in
δ1
Bia
s in
δ1
δ1 \
δ2
-0.0
66
-0
.01
6
0.0
34
0
.08
4
0.1
34
-0
.06
6
-0.0
16
0
.03
4
0.0
84
0
.13
4
0.0
67
-0
.02
4
-0.0
22
-0
.00
3
-0.0
05
-0
.00
2
0.0
82
0
.02
6
0.0
04
-0.0
02
0
.00
0
0.1
17
-0
.01
2
-0.0
39
-0
.02
9
-0.0
03
-0
.00
2
0.0
05
0
.11
6
0.0
00
0.0
04
0
.00
1
0.1
67
-0
.00
1
-0.0
10
-0
.03
1
-0.0
10
-0
.00
2
0.0
06
0
.01
0
0.1
44
0.0
07
0
.00
3
0.2
17
-0
.00
2
-0.0
05
-0
.01
8
-0.0
29
-0
.00
7
0.0
01
0
.00
1
0.0
05
0.1
30
0
.00
9
0.2
67
0
.00
1
0.0
00
-0
.00
5
-0.0
17
-0
.03
3
0.0
02
0
.00
2
0.0
02
0.0
22
0
.07
1
R
MS
E o
f δ
1
RM
SE
of
δ1
δ1 \
δ2
-0.0
66
-0
.01
6
0.0
34
0
.08
4
0.1
34
-0
.06
6
-0.0
16
0
.03
4
0.0
84
0
.13
4
0.0
67
0
.05
2
0.0
45
0
.03
2
0.0
19
0
.01
4
0.1
01
0
.06
0
0.0
36
0.0
20
0
.01
4
0.1
17
0
.03
2
0.0
51
0
.03
9
0.0
28
0
.01
7
0.0
38
0
.12
8
0.0
52
0.0
30
0
.01
7
0.1
67
0
.02
1
0.0
38
0
.06
4
0.0
36
0
.02
5
0.0
22
0
.04
6
0.1
51
0.0
43
0
.02
7
0.2
17
0
.01
6
0.0
22
0
.03
6
0.0
57
0
.03
0
0.0
16
0
.02
3
0.0
43
0.1
21
0
.03
3
0.2
67
0
.01
3
0.0
15
0
.02
2
0.0
37
0
.04
6
0.0
13
0
.01
5
0.0
23
0.0
46
0
.09
8
* B
ased
on
10
0 r
epet
itio
ns
Co
ncl
usi
on
s
Str
ong e
vid
ence
in f
avor
of
poli
tica
l yar
dst
ick c
om
pet
itio
n:
If D
epar
tmen
ts a
re
gov
ern
ed b
y a
po
liti
cal
maj
ori
ty i
n t
he
cou
nci
l le
ss t
han
or
equ
al t
o 7
5%
, th
ey
wil
l ch
ang
e th
eir
spen
din
g o
n w
elfa
re b
y s
even
teen
cen
ts i
n r
eact
ion
to
a c
han
ge
in s
pen
din
g o
n w
elfa
re o
f one
euro
by n
eigh
bo
rin
g D
epar
tmen
ts.
By c
on
tras
t, i
f
Dep
artm
ents
ar
e g
ov
ern
ed
by
a po
liti
cal
maj
ori
ty
gre
ater
th
an
75
%,
this
inte
ract
ion e
ffec
t d
ecre
ases
to t
hre
e ce
nts
.
Bo
th t
he
inte
ract
ion
eff
ect
of
sev
ente
en c
ents
and
th
e d
iffe
ren
ce b
etw
een t
hes
e
two
in
tera
ctio
n e
ffec
ts o
f fo
urt
een
cen
ts a
pp
eare
d t
o b
e si
gnif
ican
t. I
n c
on
tras
t to
pre
vio
us
stud
ies,
we
can
be
sure
that
th
is s
ign
ific
ant
dif
fere
nce
do
es n
ot
stem
fro
m i
gn
ori
ng s
pat
iall
y l
agg
ed i
nd
epen
den
t v
aria
ble
s o
r a
spat
iall
y c
orr
elat
ed
erro
r te
rm (
Man
ski,
1993
), s
ince
th
e fo
rmer
wer
e ex
pli
citl
y t
aken
in
to a
cco
un
t,
wh
ile
the
latt
er w
as c
ov
ered
by t
ime-
per
iod
fix
ed e
ffec
ts.
Fu
rth
erm
ore
, w
e fo
und
no e
mp
iric
al e
vid
ence
of
any a
dd
itio
nal
sp
atia
l p
atte
rns
in t
he
erro
r te
rms
of
the
fin
al m
od
el.
Co
ncl
usi
on
s
Th
e m
od
el d
evel
op
ed i
n t
his
pap
er i
s a
two
-reg
ime
spat
ial
Du
rbin
mo
del
wit
h
spat
ial
and
tim
e-p
erio
d f
ixed
eff
ects
. W
e d
emo
nst
rate
d t
he
ML
est
imat
or
of
this
mo
del
an
d f
oun
d t
hat
th
is e
stim
ato
r p
erfo
rms
as w
ell
as,
if n
ot
bet
ter
than
, it
s
coun
terp
art
bas
ed o
n i
nst
rum
enta
l v
aria
ble
s. S
ince
we
exp
ect
that
th
is m
od
el a
nd
the
app
lica
tio
n o
f th
is e
stim
atio
n t
ech
niq
ue
wil
l al
so p
rov
e b
enef
icia
l to
oth
er
emp
iric
al a
pp
lica
tio
ns,
a M
atla
b r
ou
tin
e h
as b
een
dev
elo
ped
wh
ich
can
be
free
ly
dow
nlo
aded
fro
m t
he
firs
t au
thor'
s w
ebsi
te.
*
*