2SIV Estimation of A Dynamic Spatial Panel Data Model with
Endogenous Spatial Weight Matrices
Xi Qu
Antai College of Economics and Management
Shanghai Jiao Tong University
xiqu@sjtu.edu.cn
Lung-fei Lee
Department of Economics
The Ohio State University
lee.1777@osu.edu
Xiaoliang Wang
Antai College of Economics and Management
Shanghai Jiao Tong University
April 8, 2015
Abstract
The spatial panel data model is a standard tool for analyzing data with both spatial correlation
and dynamic dependences among economic units. Conventional estimation methods rely on the key
assumption that the spatial weight matrix is strictly exogenous, which would likely be violated in some
empirical applications where spatial weights are determined by economic factors. This paper studies
the estimation method of a dynamic spatial panel model with individual …xed e¤ects when the time
dimension is short. The spatial weight matrices are constructed by some economic variables and can be
endogenous and time varying. We establish the consistency and asymptotic normality of the two-stage
instrumental variable (2SIV) estimator and investigate their …nite sample properties by a Monte Carlo
study. This model is applied to study the …scal interactions among the 91 non-oil countries.
JEL classi…cation: C31; C51
Keywords: Spatial panel models; Endogenous spatial weight matrices; Fixed e¤ects
1
Introduction
Spatial panel data models are standard tools to analyze data with both cross-sectional and dynamic dependences among economic units. They are generalized from a cross-sectional spatial autoregressive (SAR)
model proposed by Cli¤ & Ord (1973). Recently, there is much progress in empirical and theoretical works
on spatial panel data models. Static spatial panel data models can be applied to agricultural economics
(Druska & Horrace, 2004), transportation research (Frazier & Kockelman, 2005), public economics (Egger
et al., 2005), consumer demand (Baltagi & Li 2006), to name a few. Dynamic spatial panel data model
can be applied to the growth convergence of countries and regions (Ertur & Koch, 2007), regional markets
(Keller & Shiue, 2007), labor economics (Foote, 2007), public economics (Revelli, 2001; Tao, 2005; Franzese,
2007; Michela, 2007), and some other …elds. For the estimation and statistical inference, random e¤ects
and …xed e¤ects spatial panel models are most commonly used. For the random e¤ects model, Baltagi et
al. (2003, 2007a, 2007b), Mutl (2006) and Kapoor et al. (2007) investigate various speci…cations with error
1
components. For the …xed e¤ects model, Elhorst (2005), Korniotis (2010), Su and Yang (2007), Yu et al.
(2008, 2012) and Lee and Yu (2010a) study static or dynamic models under various spatial structures. Mutl
and Pfa¤ermayr (2010) and Lee and Yu (2010b) consider the estimation of spatial panel data models with
both …xed and random e¤ects speci…cations, and propose Hausman-type speci…cation tests.
In the current literature of spatial panel data models, the spatial weights matrix is usually speci…ed
to be exogenous and time invariant. This is plausible if spatial weight matrices are based on contiguity
or geographic distances among regions. But there are plenty of cases that spatial weights are constructed
with economic/socioeconomic distances. For example, Aiello and Cardamone (2008) construct their spatial
weights by a variable that re‡ects …rms technological similarity and geographical proximity to study an
R&D spillover in Italy. In Crabb and Vandenbussche (2008), where in addition to the physical distance,
spatial weight matrices are constructed by inverse trade share and inverse distance between GDP per capita.
When elements of a spatial weights matrix are constructed from economic/socioeconomic characteristics of
regions (or districts) in a panel setting, these characteristics might be endogenous and changing over time.
Qu and Lee (2015) study a cross-sectional SAR model with endogenous spatial weights and …nd ignoring the
endogeneity in spatial weights matrices would have substantial consequences on estimates. Lee and Yu (2012)
consider spatial dynamic panel data models with time varying spatial weights matrices, but they assume the
weights are still exogenous. One may wonder whether ignoring the endogeneity in spatial weights matrices
would have severe consequences in panel setting, and whether spatial panel models with endogenous time
varying spatial weights can be easily handled and estimated. These motivate our investigation on the spatial
panel data models with endogenous spatial weights This paper investigates the 2 stage instrumental variable
(2SIV) estimation of a spatial dynamic panel model under the setting of endogenous and time varying spatial
weights matrices.
This paper is organized as follows. Section 2 introduces the model and speci…es the source of endogeneity
in spatial weight matrices. Section 3 establishes the 2SIV estimation method and proves its asymptotic
properties. In Section 4, we study …nite sample perfermances of our proposed 2SIV method using the Monte
Carlo simulation. In Section 5, we apply this model to analyze the …scal interactions among the 91 non-oil
countries. Section 6 concludes the paper. Some lemmas and proofs are collected in the Appendices.
2
The Model
2.1
Model speci…cation
Following Jenish and Prucha (2009 & 2012), we consider spatial processes located on a (possibly) unevenly
spaced lattice D Rd , d 1. Asymptotic methods we employ are increasing domain asymptotics: growth
of the sample is ensured by an unbounded expansion of the sample region as in Jenish and Prucha (2012).1
Let f("0i;nt ; vi;nt ); i 2 Dn , n 2 N , t = 1; :::T g be a triangular double array of real random vectors de…ned
on a probability space ( ; F ; P ), where the index set Dn D is a …nite set.
In this paper, we consider a dynamic spatial panel data model
Ynt =
1 Wnt Ynt
+
2 Wn;t 1 Yn;t 1
+
1 Yn;t 1
+ X1nt + cn + Vnt ;
(1)
where Ynt = (y1t ; y2t ; :::ynt )0 and Vnt = (v1;nt ; v2;nt ; :::vn;nt )0 are n dimensional column vectors, and vi;nt ’s
are i.i.d. across i and t with zero mean and variance 2v . The X1nt is an n k1 matrix of individually
and time varying non-stochastic regressors.
is an k1 dimensional vector of coe¢ cients, and 1 , 2 , and
1 In…ll
asymptotics have not been developed for a NED process in the literature.
2
are scalar coe¢ cients. cn is an n dimensional vector of the individual …xed e¤ects and t is a scalar of
the time …xed e¤ect. The spatial weight matrix Wnt is an n n matrix with each entry constructed by:
(Wnt )ij = wij;nt = hijt (zi;nt ; zj;nt ), where zi;nt is a p dimensional row vector. For any i = 1; :::n, zi;nt has
the model
zi;nt = zi;n;t 1 2 + x02i;t + d0i;n + "0i;nt ;
where 2 is a p dimensional constant matrix, x2i;t is a k2 dimensional vector of individually and time varying
non-stochastic regressors, is an p k2 matrix of coe¢ cients, di;n is a p dimensional constant
invariant
1
0 vector
x01;2nt
C
B
..
over time„and "i;nt is a p dimensional random variable. Denote n k2 matrix X2nt = @
A, n p
.
0
B
=@
matrices Znt ; dn , and "nt with Znt
write in matrix form that
1
d01;n
1
0
"01;nt
x0
1 n;2nt
z1;nt
.. C, d = B .. C, and " = B .. C. Then we can
@ . A
@ . A
nt
. A n
0
zn;nt
"0n;nt
dn;n
Znt = Zn;t
2.2
0
1 2
+ X2nt + dn + "nt :
(2)
Source of endogeneity
We consider n agents in an area, each endowed with a predetermined location i. Due to some competition
or spillover e¤ects, at period t, each agent i has an outcome yi;nt directly a¤ected by its neighbors’current
0
outcomes yj;nt
s; its own outcome from last period yi;n;t 1 , and its neighbors’outcomes from the last period
0
yj;n;t 1 s The spatial weight wij;nt is a measure of relative strength of linkage between agents i and j at time
t, However, this weight wij;nt is not predetermined but depends on some observable random variable Znt : We
can think of zi;nt as some economic variables at location i and time t such as GDP, consumption, economic
growth rate, etc, which in‡uence strength of links across units. We have the following assumptions.
1, is in…nitely countable. All elements in D are located at
Assumption 1 The lattice D
Rd0 , d0
distances of at least dis0 > 0 from each other, i.e., 8i; j 2 D : ij
0 , where ij is the distance between
locations i and j; w.l.o.g. we assume that 0 = 1.
Assumption 2 The error terms vi;nt and "i;nt , have a joint distribution: (vi;nt ; "0i;nt )0
v"
=
2
v
0
v"
v
"
is positive de…nite,
2
v
is a scalar variance, covariance
v"
=(
i:i:d:(0;
v" ),
0
v"1 ; ::: v"p2 )
dimensional vector, and " is a p p matrix. The supi;n;t Ejvi;nt j4+ " and supi;n;t Ejj"i;nt jj4+
some " > 0. Furthermore, E(vi;nt j"i;nt ) = "0i;nt and V ar(vi;nt j"i;nt ) = 2 .
"
where
is a p
exist for
The endogeneity of Wnt comes from the correlation between vi;nt and "i;nt . If v" is zero, the spatial
weight matrix Wnt might be treated as strictly exogenous and we can apply conventional methodology of
spatial panel data models for estimation. However, if v" is not zero, Wnt becomes an endogenous spatial
weights matrix.
From the two conditional moments assumptions in Assumption 2, we have the p dimensional column
0
1
"nt , then its mean conditional
vector = " 1 v" and the scalar 2 = 2v
v" . Denote nt = Vnt
v" "
2
on "nt is zero and its conditional variance matrix is In . In particular, nt are uncorrelated with the terms
of "nt and the variance of nt is 20 In . The outcome equation (1) becomes
Ynt =
1 Wnt Ynt
+
2 Wn;t 1 Yn;t 1
+
1 Yn;t 1
+ X1nt + cn + (Znt
3
Zn;t
1 2
X2nt
dn ) +
nt ;
(3)
with E( i;nt j"i;nt ) = 0 and V ar( i;nt j"i;nt ) = 2 ; and i;n ’s are i.i.d. across i. Our subsequent asymptotic analysis will mainly rely on the di¤erenced equation of (3), where ( Znt ; Zn;t 1 ; X2nt ) are control
variables to control the endogeneity of Wnt .
Assumption 2 is relatively general without imposing a speci…c distribution on disturbances as it is based on
only conditional moments restrictions. In the special case that (vi;nt ; "0i;nt )0 has a jointly normal distribution,
0
1
then vi;nt j"i;nt i:i:d:N ("0i;nt " 1 v" ; 2v
v" ) and nt is independent of "nt in equation (2).
v" "
3
Estimation Method
3.1
The 2SIV estimation
Due to the complication of the model, we use a two step estimation.
For any variable Ant , denote Ant = Ant An;t 1 . In the 1st step, we estimate the di¤erenced equation
of (2)
Znt = Zn;t 1 2 + X2nt + "nt ;
or in the matrix form
Zn (T ) = ( Zn (T
2
1), X2n (T ))
+
"n (T ) = Kn
2
+
"n (T );
where
0
B
B
Zn (T ) = B
@
Zn;3
Zn;4
..
.
Zn;T
0
1
B
C
B
C
C , Zn (T 1) = B
@
A
Zn;T
A consistent estimator is derived from
b2
b
0
= [Kn0 q1n (q1n
q1n )
0
1
Zn;2
Zn;3
..
.
1
B
C
B
C
C ; X2n (T ) = B
@
A
B
C
B
C
C ; "n (T ) = B
@
A
X2n;T
1 0
0
0
q1n Kn ] 1 Kn0 q1n (q1n
q1n ) 1 q1n
0
1
X2n;3
X2n;4
..
.
However,
= [ (W Y )n (T ); (W Y )n (T
= Tn +
n (T ):
1); Yn (T
1); X1n (T ); "n (T )](
1;
=(
2; 1;
1;
0
Zn (T )
Zn (T
1)b2
i.e., the regression residual from the 1st step as an approximation of
0
b = [Tbn0 q2n (q2n
q2n )
X2n (T ) b
"n (T ) and estimate
1 0 b
0
0
q2n Tn ] 1 Tbn0 q2n (q2n
q2n ) 1 q2n
4
Yn (T );
2; 1;
; )0 +
"n (T ) in Tn is unknown. To solve this, we use
b
"n (T ) =
"n;T
1
C
C
C:
A
Zn (T );
where q1n is a valid instrument of Kn , for example q1n = ( X2n (T 1); X2n (T )).
Denote Tn = [ (W Y )n (T ); (W Y )n (T 1); Yn (T 1); X1n (T ); "n (T )] and
Then in the 2nd step, we want to estimate
Yn (T )
"n;3
"n;4
..
.
by
0
; )0 .
n (T )
where Tbn = [ (W Y )n (T ); (W Y )n (T 1);
0
Wn3 Yn3 Wn2 Yn2
B
Wn4 Yn4 Wn3 Yn3
B
(W Y )n (T ) = B
..
@
.
WnT YnT
Wn;T
1 Yn;T
Yn (T
1
1
C
C
C,
A
1); X1n (T ); b
"n (T )] with
0
Wn2 Yn2 Wn1 Yn1
B
Wn3 Yn3 Wn2 Yn2
B
(W Y )n (T 1) = B
..
@
.
Wn;T 1 Yn;T 1 Wn;T 2 Yn;T
and q2n is a valid instrument of Tbn , for example q2n = [ (W X1 )n (T ); (W X1 )n (T
1); X1n (T ); b
"n (T )].
3.2
1
2
C
C
C
A
1); X1n (T
Asymptotic Property
To analyze the asymptotic properties of our 2SIV estimator of the dynamic spatial panel data model, we
need the following assumptions.
Assumption 3 3.1). For any i, j, n, and t, the spatial weight wij;nt = hijt (zi;nt ; zj;nt ) for i 6= j, where
hijt ( )’s are non-negative, uniformly bounded functions of some observable variable Znt . Furthermore,
wii;nt = 0, supn;t jjWnt jj1 = cw < 1, and supn;t jjWnt jj1 = cu < 1.
3.2). The parameter = ( 1 ; 2 ; 1 ; 0 ; vec( 2 )0 ; vec( )0 ; 2 ; 0 )0 is in a compact set
in the Euclidean
space Rk , where is a vector of distinct parameters in
and k = k1 +4+k2 p+p2 +p; k1 is the dimension
of , p is the dimension of , k2 p is the number of parameters in ; and p2 is the number of parameters in
2
> 0 and " is positive de…nite. The true parameter 0 is contained in the interior of .
2 . In this set,
Furthermore, 10 , 20 , and 0 satis…y that j 10 jcw < 1.
3.3). The matrix Snt ( 1 ) is nonsingular for all 1 , and for any n and t.
3.3). Let the k n matrix Xnt collect all distinct column vectors in X1nt and X2nt : All elements in Xnt
0
are deterministic and bounded in absolute value. n1 Xnt
Xnt is nonsingular.
For the initial value, we adopt a similar but more generalized framework than Su and Yang (2013):
(i) data collection starts from the 1st period; the process starts from the t0 th period, i.e., t0 + 1 periods
(0 t0 L; L < 1) before the start of data collection, and then evolves according to the model speci…ed
by (1) and (2);
(ii) the starting position of the process Yn; t0 can treated as exogenous or endogenous. If Yn; t0 is
exogenous, so is Wn; t0 ("n; t0 ), then (Yn; t0 , "n; t0 ) are considered as random and inferences proceed by
conditioning on them;
(iii) if Yn; t0 is endogenous, it has a cross-sectional spatial model speci…cation as
Yn;
t0
=
10 Wn; t0 Yn; t0
+ X1n;
t0
0
+ con0 + Vn;
t0
with Zn;
t0
= X2n;
t0
+ don0 + "n;
t0 ;
where con0 (don0 ) is the individual speci…c intercept in the initial period, which can be di¤erent from the
individual …xed e¤ect cn0 (dn0 ) in later periods.
As the time dimension T is small and the initial period is …nitely far away, the asymptotic analysis is
based on the near-epoch dependence (NED) of individual i. We need further assumptions on the structure
of Wnt .
Assumption 4 The spatial weights wij;nt = hijt (zi;nt ; zj;nt ) are bounded by some exogenous bounds. Either
of the following two cases is satis…ed.
5
4.1) 0 wij;nt
c1 ij c3 d0 for some 0 c1 and c3 > 32 . Furthermore, in any period t, there exist at
most K (K 1) columns of Wnt that the column sum exceeds cw , where K is a …xed number that does not
depend n.
4.2) The spatial weight wij;nt = 0 if ij > c , i.e., there exists a threshold c > 1 and if the geographic distancePexceeds c , then the weight is zero. For i 6= j, wij;nt = hijt (zi;nt ; zj;nt ) or wij;nt =
hijt (zi;nt ; zj;nt )=
hikt (zi;nt ; zk;nt ), where hijt ( )’s are non-negative, uniformly bounded functions.
ik
c
This assumption is adopted in Qu and Lee (2015) to show the consistency and asymptotic normality of
the QMLE of a cross-sectional SAR model with an endogenous spatial weight matrix. To be more speci…c,
Assumption 4 constrains the magnitudes of the endogenous spatial weights so that the estimator has the
spatial NED property.
The following notion of NED for random …elds is from Jenish and Prucha (2012).
De…nition 1 For any random vector Y; jjY jjp = [EjY jp ]1=p denotes its Lp -norm where jY j is the Euclidean norm of Y: Denote Fi;n (s) as a -…eld generated by the random vectors & j;n ’s located within the ball
Bi (s), which is a ball centered at the location i with a radius s in a d0 -dimensional Euclidean space D.
De…nition 2 (NED) Let H = fHi;n ; i 2 Dn ; n
1g and & = f& i;n ; i 2 Dn ; n
1g be random …elds with
jjHi;n jjp < 1; p 1, where Dn D and jDn j ! 1 as n ! 1, and let d = fdi;n ; i 2 Dn ; n 1g be an array
of …nite positive constants. Then the random …eld H is said to be Lp -near-epoch dependent on the random
…eld & if jjHi;n E(Hi;n jFi;n (s))jjp di;n '(s) for some sequence '(s) 0 such that lims!1 '(s) = 0. The
'(s), which is, without loss of generality, assumed to be non-increasing, is called the NED coe¢ cient, and
the di;n ’s are called NED scaling factors. H is said to be Lp -NED on & of size
if '(s) = O(s ) for
some > > 0. Furthermore, if supn supi2Dn di;n < 1, then H is said to be uniformly Lp -NED on &. If
'(s) = O( s ); where 0 < < 1; then H is called geometrically Lp -NED on &.
In our spatial panel setting, when we analyze the NED property, the base of Fi;n (s) is
& i;n = ("in;
t0 ; :::; "i;nT ; in; t0 ; ::: i;nT );
i.e., disturbances in location i of all periods from t0 to T:
0
0
0
0
Denote P1n = q1n (q1n
q1n ) 1 q1n
and P2n = q2n (q2n
q2n ) 1 q2n
. Then, we have
b
0
= (Tbn0 P2n Tbn )
1
Tbn0 P2n
n (T )
+ (Tbn0 P2n Tbn )
1
Tbn0 P2n Kn (Kn0 P1n Kn )
1
Kn0 P1n "n (T ) 0 :
Details can be found from Appendix. For the consistency, it is su¢ cient to show that
1 0
1 0
1 0
1 0
1 0
n (T ) = op (1); and n q1n "n (T ) = op (1):
n q2n Kn , n q1n Kn , n q1n q12n are Op (1), n q2n
We can express the reduced form of terms in Kn and Tbn . For example,
(W Y )nt
= Gnt
t+t
X0
(h)
Bnt (X1n;t
h 0
+ cn0 + "n;t
h 0
+
n;t h )
(t+t0 )
+ Gnt1 Bnt
(con0
1 0
1 0 b
n q2n Tn , n q2n q2n ,
cn0 )
h=0
Gn;t
1
t X
1+t0
(h)
Bn;t
1 (X1n;t 1 h 0
+ cn0 + "n;t
1 h 0
+
n;t 1 h )
Gn;t
(t 1+t0 ) o
(cn0
1 Bn;t 1
h=0
2 As
c0
ij
decreases faster than
c3 d 0
ij
, all the results hold for the case of 0
6
d
wij;n
c1 c0
ij
with some c1
0 and c0 > 1.
cn0 );
with
(h)
Bnt = ( 0 Sn;t1
1+
20 Gn;t
1
1 )( 0 Sn;t
2+
20 Gn;t
( 0 Sn;t1
2)
h+
20 Gn;t
h) =
h
Y
( 0 Sn;t1
k + 20 Gn;t k )
k=1
(0)
1
and Bnt = In ; where Snt = In
1 Wnt and Gnt = Wnt Snt . Therefore, in general, we want to show terms
like
T
T
1 XX 0
p
[U
Uns;H2 E(U0nt;H1 Uns;H2 )] ! 0;
n t=1 s=1 nt;H1
PH
(h)
where Unt;H = Ant h=0 Bnt & n;t h b; where & i;nt = fi (& i;nt ; Xnt ) with & nt = ("nt ; nt ) being a vector-valued
function, Ant can be either Gnt or Snt1 , and b is any conformable vector of constants.
To show the consistency and asymptotic normality of the 2SIV estimator b, in addition to the convergence
of each separated term, we need some rank conditions on relevant limiting matrices.
0
0
q1n ) and limn!1 n1 E(q2n
q2n ) exists and are nonsingular.
Assumption 5 5.1) limn!1 n1 E(q1n
1
1
0
5.2) limn!1 n E(Kn q1n ) and limn!1 n E(Tn0 q2n ) have full column rank.
Denote
0
0
2
1
0
B
B
B
=B
B
@ 0
0
2
1
0
0
1
2
..
.
..
.
1
0
2
C
C
C
C
C
A
(T
:
2) (T
2)
Theorem 1 Under Assumptions 1-5, the 2SIV estimator b is a consistent estimator of
p
d
n (b
0 ) ! N (0;
0 ), where
0
n
0.
1
plim (Tn0 P2n Tn ) 1 Tn0 P2n n P2n Tn (Tn0 P2n Tn ) 1 ; with
n
n!1
2
=
In + Kn (Kn0 P1n Kn ) 1 Kn0 P1n ( 00 "0 0 0 In )P1n Kn (Kn0 P1n Kn )
0 0
Furthermore,
=
The estimated variance-covariance matrix is
1 b0
b
=
(T P2n Tbn ) 1 Tbn0 P2n b n P2n Tbn (Tbn0 P2n Tbn ) 1 ; with
n n
b n = Kn (K 0 P1n Kn ) 1 K 0 P1n (b0 b "b 0 In )P1n Kn (K 0 P1n Kn )
n
n
n
1
Kn0 + (b2
0
1
Kn0 :
In );
where
1
1
b
"n (T )0 b
"n (T ); b2 =
2n(T 2)
2n(T
Its consistency can be found in the following claim.
b" =
Claim 3.2.1 Suppose
sistent estimator of
b2
b
"0 ,
2
b
"n (T ) =
b (T )0 b (T ):
n
n
, then b " =
1
2n(T 2)
b
"n (T )0 b
"n (T ) is a con0
b2
b
b
Zn (T ) Kn b : Similarly, if b = ( 1 ; 2 ; b1 ; b ; b)0 is a conb0 (T ) b (T ) is a consistent estimator of 2 , where b =
n
n
n
0
is a consistent estimator of
where
2)
0
0
sistent estimator of 0 , then b = 2n(T1 2)
Yn (T ) Tbn b. Furthurmore, if we replace 0 with a consistent estimator b; "n (T ) with
p
b " , and 2 with b2 , to obtain the empirical estimate of b , then b !
0.
0
7
b
"n (T );
"0
with
4
Monte Carlo simulation
4.1
Data generating process
In this section, we evaluate the 2SIV estimation method of a dynamic spatial panel data model with endogenous spatial weight matrices. The data generating process (DGP) is
Ynt = (In
1 Wnt )
1
(
2 Wn;t 1 Yn;t 1
+
1 Yn;t 1
+ X1nt + cn + Vnt );
with the initial value
Yn;
t0
=
10 Wn; t0 Yn; t0
+ X1n;
t0
0
+ con0 + Vn;
t0 ;
where t0 = 8, i.e., 8 periods before the zero period, T = 10; i.e., the total observed periods are 10, X1n
N (0; 1), 1 = 0:5; 2 = 0:1, 1 = 0:2, = 1; and cn
U (0; 1). The endogenous, row-normalized Wnt =
(wij;nt ) is constructed as follows:
1. Generate bivariate normal random variables (vi;n ; "i;n ) from i:i:d N
0;
1
1
as disturbances in
the outcome equation and the spatial weight equation.
d
e
, i.e., wij;nt = wij;n
2. Construct the spatial weight matrix as the Hadamard product Wnt = Wnd Wnt
e
d
d
wij;nt , where Wn is a predetermined matrix based on geographic distance (not time-varying): wij;n = 1
e
is a time-varying matrix based on economic
if the two locations are neighbors and otherwise 0; Wnt
e
e
similarity: wij;nt = 1=jzi;nt zj;nt j if i 6= j and wii;nt = 0; where elements of zi;nt is generated by
zi;nt = zi;n;t 1 2 + x2i;t + di;n + "i;nt with the initial value zi;n; t0 = x2i; t0 + di;n + "n; t0 , where
X2n N (0; 1), di;n U (0; 1), 2 = 0:7; and = 0:8.
3. Row-normalize Wn .
For the predetermined Wnd , we use four examples. First, the U.S. states spatial weight matrix W S(49 49);
based on the contiguity of the 48 contiguous states and D.C.; second, the Toledo spatial weight matrix
W O(98 98); based on the 5 nearest neighbors of 98 census tracts in Toledo, Ohio; third, the Iowa “County"
spatial weight matrix W C(361 361), based on whether the school districts are in the same county in Iowa
in 2009; and lastly, the Iowa “Adjacency" spatial weight matrix W A(361 361), based on the adjacency of
361 school districts in Iowa in 2009.
In the simulation, we compare our 2SIV method with the conventional IV method which refers to the
case of treating Wnt as exogenous. Here the conventional IV only estimates the outcome equation (Znt
equation is not estimated) since it treats Wnt as exogenous. For the simulation, we choose (W X)nt to
instrument (W Y )nt ; Xn;t 1 to instrument Yn;t 1 . The 2SIV estimates both equations. In the 1st step,
we use Xn;t 1 as the instrument of Zn;t 1 to estimate the Znt equation, and in the 2nd step, (W X)nt ,
(W X)n;t 1 , and Xn;t 1 are used as instruments of (W Y )nt , (W Y )n;t 1 , and Yn;t 1 to estimate
the Ynt equation. Of particular interest, we want to see how large the bias is for this conventional estimation
method when Wnt is indeed endogenous. To generate di¤erent degrees of endogeneity, we choose correlation
coe¢ cients = 0:2, 0:5, and 0:8. We also let the spatial correlation to be = 0:2 and 0:4 to investigate how
the spatial correlation parameter a¤ects estimates. 1000 replications are carried out for each setting3 .
8
Table 1: Estimates from the U.S. states spatial weight matrix
= 0:2
IV
2SIV
= 0:5
IV
2SIV
= 0:8
IV
2SIV
Note: Observations
WS(49)
= 0:5
=
0:2
=1
= 0:8
= 0:2
2 = 0:1
1
2 = 0:7
0:5069
0:0844
0:2048
0:9988
(0:0811) (0:1025) (0:0784) (0:0692)
[0:0835]
[0:1067] [0:0804] [0:0688]
0:4954
0:0958
0:2043
1:0027
0:7130
0:8062
0:1989
(0:0802) (0:1007) (0:0782) (0:0693) (0:1518) (0:0779) (0:0627)
[0:0827]
[0:1050] [0:0801] [0:0684] [0:1546] [0:0795] [0:0617]
=1
= 0:8
= 0:5
1 = 0:5
2 = 0:1
1 = 0:2
2 = 0:7
0:5251
0:0685
0:2052
0:9922
(0:0813) (0:1052) (0:0795) (0:0696)
[0:0834]
[0:1054] [0:0811] [0:0697]
0:4976
0:0999
0:2031
1:0027
0:7121
0:8060
0:4970
(0:0722) (0:0919) (0:0777) (0:0689) (0:1517) (0:0779) (0:0644)
[0:0746]
[0:0928] [0:0787] [0:0682] [0:1545] [0:0782] [0:0628]
=
0:5
=1
= 0:8
= 0:8
1
2 = 0:1
1 = 0:2
2 = 0:7
0:5432
0:0490
0:2064
0:9845
(0:0817) (0:1078) (0:0808) (0:0701)
[0:0801]
[0:1094] [0:0835] [0:0700]
0:4994
0:0999
0:2022
1:0024
0:7110
0:8058
0:7950
(0:0522) (0:0670) (0:0747) (0:0670) (0:1514) (0:0778) (0:0579)
[0:0533]
[0:0695] [0:0758] [0:0653] [0:1543] [0:0769] [0:0570]
n = 49: Estimated standard error based on an asymptotic variance-covariance matrix is
in parentheses; and empirical standard deviation is in brackets.
1
9
Table 2: Estimates from the Ohio State spatial weight matrix
= 0:2
IV
2SIV
= 0:5
IV
2SIV
= 0:8
IV
2SIV
Note: Observations
WO(98)
= 0:5
=
0:2
=1
= 0:8
= 0:2
2 = 0:1
1
2 = 0:7
0:5130
0:0812
0:2022
0:9950
(0:0615) (0:0764) (0:0549) (0:0483)
[0:0609]
[0:0747] [0:0528] [0:0476]
0:4974
0:0939
0:2020
0:9994
0:7003
0:8027
0:2006
(0:0610) (0:0751) (0:0548) (0:0484) (0:1045) (0:0539) (0:0445)
[0:0605]
[0:0740] [0:0526] [0:0475] [0:1059] [0:0546] [0:0454]
=1
= 0:8
= 0:5
1 = 0:5
2 = 0:1
1 = 0:2
2 = 0:7
0:5360
0:0602
0:2029
0:9874
(0:0623) (0:0792) (0:0559) (0:0488)
[0:0621]
[0:0764] [0:0544] [0:0483]
0:4969
0:0935
0:2020
0:9999
0:7002
0:8020
0:5020
(0:0556) (0:0692) (0:0546) (0:0484) (0:1046) (0:0540) (0:0457)
[0:0559
[0:0689] [0:0526] [0:0476] [0:1038] [0:0529] [0:0467]
=
0:5
=1
= 0:8
= 0:8
1
2 = 0:1
1 = 0:2
2 = 0:7
0:5632
0:0373
0:2039
0:9767
(0:0627) (0:0820) (0:0571) (0:0492)
[0:0631]
[0:0813] [0:0563] [0:0491]
0:4988
0:0959
0:2015
0:9999
0:7008
0:8014
0:8013
(0:0403) (0:0505) (0:0526) (0:0472) (0:1049) (0:0540) (0:0409)
[0:0408]
[0:0511] [0:0508] [0:0467] [0:1013] [0:0514] [0:0402]
n = 98: Estimated standard error based on an asymptotic variance-covariance matrix is
in parentheses; and empirical standard deviation is in brackets.
1
10
Table 3: Estimates from the Iowa State county spatial weight matrix
WC(361)
= 0:5
=
0:2
=1
= 0:8
= 0:2
2 = 0:1
1
2 = 0:7
0:5054
0:0937
0:1996
0:9952
(0:0241) (0:0324) (0:0284) (0:0251)
[0:0244]
[0:0311] [0:0284] [0:0245]
2SIV
0:4983
0:1007
0:1996
0:9983
0:6991
0:8001
0:1997
(0:0238) (0:0318) (0:0283) (0:0251) (0:0541) (0:0279) (0:0228)
[0:0240]
[0:0308] [0:0283] [0:0244] [0:0546] [0:0275] [0:0230]
= 0:5
=1
= 0:8
= 0:5
1 = 0:5
2 = 0:1
1 = 0:2
2 = 0:7
IV
0:5153
0:0827
0:1998
0:9906
(0:0242) (0:0329) (0:0286) (0:0252)
[0:0248]
[0:0314] [0:0286] [0:0248]
2SIV
0:4978
0:1003
0:1998
0:9985
0:6987
0:7993
0:5001
(0:0214) (0:0289) (0:0281) (0:0250) (0:0541) (0:0279) (0:0232)
[0:0218
[0:0283] [0:0280] [0:0242] [0:0544] [0:0273] [0:0227]
= 0:8
=
0:5
=1
= 0:8
= 0:8
1
2 = 0:1
1 = 0:2
2 = 0:7
IV
0:5249
0:0699
0:2007
0:9856
(0:0243) (0:0335) (0:0288) (0:0253)
[0:0251]
[0:0318] [0:0291] [0:0251]
2SIV
0:4983
0:0991
0:2000
0:9984
0:6986
0:7985
0:8004
(0:0153) (0:0211) (0:0272) (0:0243) (0:0541) (0:0278) (0:0206)
[0:0152]
[0:0210] [0:0274] [0:0236] [0:0543] [0:0271] [0:0200]
Note: Observations n = 361: Estimated standard error based on an asymptotic variance-covariance matrix
is in parentheses; and empirical standard deviation is in brackets.
= 0:2
IV
1
11
Table 4: Estimates from the Iowa State adjacency spatial weight matrix
WA(361)
= 0:5
=
0:2
=1
= 0:8
= 0:2
2 = 0:1
1
2 = 0:7
0:5138
0:0886
0:1997
0:9933
(0:0313) (0:0391) (0:0286) (0:0252)
[0:0307]
[0:0376] [0:0283] [0:0242]
2SIV
0:4981
0:1014
0:1994
0:9983
0:6991
0:8001
0:1998
(0:0310) (0:0385) (0:0285) (0:0253) (0:0541) (0:0279) (0:0230)
[0:0305]
[0:0370] [0:0282] [0:0241] [0:0546] [0:0275] [0:0235]
= 0:5
=1
= 0:8
= 0:5
1 = 0:5
2 = 0:1
1 = 0:2
2 = 0:7
IV
0:5371
0:0678
0:2006
0:9846
(0:0316) (0:0405) (0:0292) (0:0255)
[0:0313]
[0:0393] [0:0289] [0:0251]
2SIV
0:4980
0:1008
0:1996
0:9984
0:6987
0:7993
0:5001
(0:0282) (0:0353) (0:0284) (0:0252) (0:0541) (0:0279) (0:0237)
[0:0284]
[0:0349] [0:0283] [0:0247] [0:0544] [0:0273] [0:0235]
= 0:8
=
0:5
=1
= 0:8
= 0:8
1
2 = 0:1
1 = 0:2
2 = 0:7
IV
0:5610
0:0454
0:2019
0:9741
(0:0317) (0:0418) (0:0298) (0:0257)
[0:0309]
[0:0407] [0:0301] [0:0252]
2SIV
0:4981
0:0994
0:1999
0:9984
0:6986
0:7985
0:8006
(0:0205) (0:0258) (0:0272) (0:0245) (0:0541) (0:0278) (0:0213)
[0:0203]
[0:0260] [0:0275] [0:0235] [0:0543] [0:0271] [0:0207]
Note: Observations n = 361: Estimated standard error based on an asymptotic variance-covariance matrix
is in parentheses; and empirical standard deviation is in brackets.
= 0:2
IV
1
12
4.2
Monte Carlo results
In this section, we examine the …nite sample performance of our 2SIV estimator.
Tables 1-4 list the empirical mean of each estimator, the empirical mean of its estimated standard
error based on the corresponding asymptotic variance-covariance matrix (in parentheses), and the empirical
standard deviation of the estimator (in brackets) based on 1000 replications using W S; W O; W C; or W A
as the predetermined spatial weight matrices. To see how the di¤erent estimation methods react to the
magnitude of endogeneity, we have three sets of results. In each table, the upper panel shows the results
with = 0:2; the middle panel is for = 0:5, and the lower panel is for = 0:8, so the endogeneity is
increasing.
There are several patterns in the estimation results:
1. For the accuracy of estimation, we consider the empirical standard deviation based on 1000 replications
and the estimated standard error based on the asymptotic variance-covariance matrix. These two values
are very close in all cases. Compared across di¤erent estimation methods, accuracy of IV is very close
to 2SIV. In the case of W S, when the endogeneity is high, 2SIV estimation has a slightly smaller
variance. All the other cases, variances of the two estimation methods are almost the same.
2. For the bias, the more serious the endogeneity is, i.e., the larger the correlation coe¢ cient is, the
more bias IV method creates. Most bias comes from the estimator of the spatial correlation coe¢ cients
and 1 are much less biased. b2 from IV su¤ers
1 and 2 , especially 2 . In general, estimators for
severe downward bias when achieves 0:5, in some cases exceeding 100%. Correspondingly, b1 from IV
has the upword bias, but the bias is not as big as b2 . Although biases of and 1 in the conventional
IV method can be neglectable, interpetation of these coe¢ cients signi…cantly depends on ’s. Also the
existense of spatial correlation indicates the multiplier e¤ects. In all cases, 2SIV has estimates very
close to the true value. From this, we can conclude that with the endogeneous spatial weight matrices,
our 2SIV estimation method is preferred.
3. From Table 1 to Table 4, as sample size increases while the number of neighbors for each agent grows
in a slower rate, we observe that the bias and standard error of estimators decrease. When the sample
size is 361, such as in the case of W A and W C, the estimates appear quite precise and close to the
true parameters.
5
An empirical application
In this section, we provide an illustration of the use of our method. We employ the dynamic spatial panel data
model to investigate the …scal interactions among the 91 non-oil countries used in Mankiw et al. (1992) and
Ertur and Koch (2007). Now, we will give some explanations why we use such a model to estimate the …scal
reaction function. Suggested by Michela (2007), we believe that state’s …scal policies are dependent on their
neighbour’s policies by the following reasons. First, there exist expenditure externalities among jurisdiction,
like the free riding behavior. Second, the citizens can make their voting decision by comparing the same
policy choices taken by the neighboring countries’policy makers, which is kind of the so-called “yardstick”
competition. Third, the states compete with other states to attract more investments and therefore tax
bases. Moreover, policy makers among the di¤erent countries may follow a “common intellectual trend”,
proposed by Manski (1993), by sharing views at some international meetings or taking the advice from
3 We
try the DGP of some other values of
1,
2,
1,
2,
, and . The results are similar.
13
the international organizations, like IMF and World Bank. So, it is necessary to take the inderdepedence
among countries into consideration when we try to evaluate the …scal reaction functions. At the same time,
we include the lagged variable based on the idea that …scal policies are serially correlated, because of the
adjusted cost caused by the abrupt change in …scal system or objection from the interest group at the political
level. Similarly, we also want to know whether the serially correlated e¤ect of other countries will a¤ect the
policy makers’ decision. So, that is why we incorporate the last lagged period spatial term. In sum, it is
reasonable to construct the dynamic spatial panel data model in (1) to estimate the …scal reaction function.
In this empirical application, we will evaluate both taxes and public expenditures by using an dynamic
spatial panel data model . Following the theoretical part, we use a two step estimation and construct the
weight matrix based on both “physical” and “economic” distance: inverse of the product of geographic
distance and distance between GDP per capita. In the …rst step, we estimate the growth equation by using
dynamic panel data model. Then, by noting equation (3), so we can study the …scal interdependence among
states by using the estimated result in the …rst step.
In the …rst step, like Islam (1995) and Caselli et al. (1996) who study the empirical growth by using the
dynamic panel data model, we consider the following regression model.
ln Git =
i
+ ln Gi;t
1
+
>
x1it + "it ;
t = 1; 2; :::; 10;
where Git denotes the real GDP per capita in country i in the tth …nite …ve-year period, e.g., 1965-2010
and the initial period is 1960. x1it is a list of country i’s characteristics at time period t including variables
measuring real income, investment and population, suggested by Ertur and Koch (2007). We list the speci…c
variables of x1it in Table (5). By using the estimated result b
"it from this step and noting equation (3), we
can estimate the regression model in the second step.
Next, in the second step, we consider the following dynamic spatial panel data model:
X
X
Eit = ci + Ei;t 1 + 1
wijt Ejt + 2
wij;t 1 Ej;t 1 + > x2it + vit ; t = 1; 2; :::; 10;
j6=i
j6=i
where Eit can be either the aggregated level of public expenditure or total tax revenue, ci is the unobservable
…xed e¤ect, Ei;t 1 is the one period lagged term and wijt , wij;t 1 is the (i; j)th entry of the weight matrix
Wnt and Wn;t 1 respectively. x2it is a set of variables which are conventionally assumed to a¤ect the
determination of the …scal choices. We use the speci…c variables in Michena (2007) and also list them in
Table (5). Indeed, this framework is general enough to include many empirical models, like that in Baicker
(2005), Ertur and Koch (2007) and Michela (2007). Additionally, it is worthy to point out that we can
rewrite this equation in matrix format as that in (1).
For the speci…c form of the spatial weight matrices, we consider the following two possible choices
8
1
< PjGit Gjt jdij ; if j 6= i
1
j jGit Gjt jdij
;
wijt =
:
0;
Otherwise
where dij denotes the geographic distance between county i and country j. Alternatively, we can construct
the weight matrix as follows
8
1
< PjGit Gjt j ; if j 6= i and i, j country share a border
1
j jGit Gjt j
:
wijt =
:
0
Otherwise
14
In those two weight matrices, we take both of the exogenous “physical”distance and endogenous “economic”
distance into consideration. Virtually, the essence of those two distances is similar, and we can let dij = 1 if
Country i and j share border, otherwise dij = 0. So the later one is a special case of the former one.
As mentioned before, we consider a sample of 91 non-oil countries in Mankiw et al. (1992) over period
1965-2010, with the initial time 1960, which is used by Ertur and Koch (2007). We will list the used countries
in the supplemental materials. And all of the data we use are collected from World Bank-World Development
Indicator.
6
Conclusion
In this paper, we consider the speci…cation and estimation of a dynamic spatial panel data model with
endogenous spatial weight matrices. First, we specify two sets of equations: one is for the dynamic outcome
involving spatial correlation, and the other is for entries of the spatial weight matrices. The source of endogeneity is the correlation between the disturbances in the dynamic spatial outcome equation and the errors
in the spatial weight entry equation. Second, under the conditional moment assumptions on disturbances,
we propose the 2SIV estimation method and prove its consistency and asymptotic normality by employing
the theory of asymptotic inference under near-epoch dependence. We consider two types of spatial weight
matrices: one is sparse and another one has its entries decreasing su¢ ciently fast as the physical distance
increases. To examine the behavior of our proposed estimator in …nite samples, we conduct a Monte Carlo
simulation study. The simulation results indicate that the commonly used estimates under exogenous weight
matrix su¤er serious downward bias when the true weight matrix is endogenous. On the other hand, our
2SIV estimates have good …nite sample properties. As the sample size increases and the number of neighbors
grows more slowly, our estimates quickly converge to true parameters.
This paper focuses on estimating a dynamic spatial panel data model with a speci…ed source of endogeneity for the time-varying spatial weight matrices when the time period T is short. In future research,
we may consider the dynimic spatial panel data model with endogenous weights when the time period T is
long. This could be a technical challenging issue as the near-epoch assumption needs to extend to the space
and time dimension.
Appendix
The model
Based on the conditional moment conditions in Assumption 2, our model can be written as
Ynt =
where "nt = Znt
Zn;t
Ynt =
1 Wnt Ynt
1 2
1
+
X2nt
(W Y )nt +
where "nt = Znt
Zn;t 1
In the 1st step, we estimate
2
Zn (T ) = ( Zn (T
2 Wn;t 1 Yn;t 1
+
1 Yn;t 1
+ X1nt + cn + "nt +
nt ;
(4)
dn : Taking di¤erence to eliminate the individual …xed e¤ects, we have
2
(W Y )n;t
1
+
1
Yn;t
1
+
X1nt + ( "nt ) +
X2nt :
2
1), X2n (T ))
15
+
"n (T ) = Kn
2
+
"n (T )
nt ;
(5)
by
b2
b
= (Kn0 P1n Kn )
1
0
Kn0 P1n Zn (T ) =
0
+ (Kn0 P1n Kn )
1
Kn0 P1n "n (T );
0
0
where P1n = q1n (q1n
q1n ) 1 q1n
with q1n being a valid instrument, for example q1n = ( X2n (T 1); X2n (T )).
In the 2nd step, we treat
b
"n (T )
as an approximate of
Yn (T )
where
and
=
=
Zn (T )
"n (T )
Zn (T 1)b2
X2n (T ) b
0
0
0
0
Kn [Kn q1n (q1n q1n ) 1 q1n
Kn ] 1 Kn0 q1n (q1n
q1n )
1 0
q1n
"n (T )
"n (T ) and estimate the equation
=
[ (W Y )n (T ); (W Y )n (T 1); Yn (T
0
b
= Tn +
n (T ) = Tn + Kn (Kn P1n Kn )
1); X1n (T ); "n (T )](
1
Kn0 P1n
"n (T )
0
+
1;
2; 1;
1); Yn (T
1); X1n (T ); "n (T )]
Tbn = [ (W Y )n (T ); (W Y )n (T
1); Yn (T
1); X1n (T ); b
"n (T )]:
Some useful lemmas
; )0 +
n (T )
n (T );
Tn = [ (W Y )n (T ); (W Y )n (T
0
If q2n is choosen to be the instrument variable and denote P2n = q2n (q2n
q2n )
0
1 0
q2n ,
then b = (Tbn0 P2n Tbn )
1
Tbn0 P2n Yn (T ).
Let & i;n = fi ("i;n ; i;n ; Xn ; 0 ). As Xn is deterministic, & i;n is purely determined by the location i, independent of error terms associated with any other places.
Claim 6.1 For any distance , there are at most c5 d0 points in Bi ( ) and at most c4
space Bi ( + 1)nBi ( ), where c4 and c5 are positive constants.
d0 1
points in the
From Jenish and Prucha (2012), we have the following two Claims for LLN and CLT under NED.
Claim 6.2 Under Assumption 1, if the random …eld fTi;n ; i 2 Dn ; n
i.i.d., and fTi;n g’s are uniformly Lp bounded for some p > 1, then
1
n
1g is L1 -NED, the base f& i;n g’s are
Pn
L
ETi;n ) !1 0:
i=1 (Ti;n
Claim 6.3 Let fTi;n ; i 2 Dn ; n
1g be a random …eld that is L2 -NED on an i.i.d. random …eld &. If
Assumption 1 and the following conditions are met:
(1) fTi;n ; i 2 Dn ; n 1g is uniformly
PnL2+ -bounded for some > 0;
(2) inf n n1 2n > 0 where 2n =PV ar( i=1 Ti;n );
1
(3) NED coe¢ cients satisfy r=1 rd0 1 '(r) < 1;
(4) NED scaling factors satisfy supn;i2D di;n < 1,
Pn
d
then n 1 i=1 (Ti;n ETi;n ) ! N (0; 1).
Claim 6.0.1 If for any t; r = 1; :::T , jjt1i;nt E(t1i;nt jFi;n (s))jj4 C1 '1 (s) and jjt2i;nr E(t2i;nr jFi;n (s))jj4
C2 '2 (s); with maxi;n;t (jjt1i;nt jj4 ; jjt2i;nt jj4 )
C, then jjt1i;nt t2i;nr
E(t1i;nt t2i;nr jFi;n (s))jj2
2C(C1 +
C2 )'(s); where '(s) = max('1 (s); '2 (s)):
16
Consider the product of t1i;nt t2i;nr ,
jjt1i;nt t2i;nr E(t1i;nt t2i;nr jFi;n (s))jj2 jjt1i;nt t2i;nr E(t1i;nt jFi;n (s))E(t2i;nr jFi;n (s))jj2
jjt2i;nr [t1i;nt E(t1i;nt jFi;n (s))]jj2 + jjE(t1i;nt jFi;n (s))[t2i;nr E(t2i;nr jFi;n (s))]jj2
jjt2i;nr jj4 jjt1i;nt E(t1i;nt jFi;n (s))jj4 + jjt1i;nt jj4 jjt2i;nr E(t2i;nr jFi;n (s))jj4
C(C1 '1 (s) + C2 '2 (s)) C(C1 + C2 ) max('1 (s); '2 (s)):
NED properties in the case under Assumption 4.1)
Claim 6.0.2 Under Assumptions 1, 3.1), and 4.1), for any n and positive integer q, jj
1)cu (L + T )Kcqw
+ cu cqw
1
l=1
Wn;tl jj1
(q
qcu (L + T )Kcqw 1 , where cu = supn;t jjWnt jj1 and cw = supn;t jjWnt jj1 .
Pn
Proof of Claim 6.0.2. Denote an index set Vn with at least one period t satis…es cw
j=1 wji;nt < cu
Pn
if i 2 Vn and j=1 wji;nt < cw for all t if i 2
= Vn . Then Assumption 6.1) constrains that jVn j (L + T )K
q
Q
for any n, where (L + T ) is the total number of periods. Consider the kth column sum of
Wn;tl , i.e.,
e0n
1
q
Q
l=1
q
Q
0
Wn;tl ek;n , where en = (1; :::; 1) and ek;n is the unit column vector with one in its kth entry and zeros
Pn
in its other entries. As In = i=1 ei;n e0i;n ,
l=1
e0n
q
Y
Wn;tl ek;n
=
n
X
e0n Wn;t1 ei;n e0i;n
i=1
l=1
X
=
q
Y
Wn;tl ek;n
l=2
q
Y
e0n Wn;t1 ei;n e0i;n
i2Vn
Wn;tl ek;n +
l=2
(L + T )K
max e0n Wn;t1 ei;n
i2Vn
(L + T )Kcu jj
q
Y
l=2
X
i2V
= n
max e0i;n
i2Vn
Wn;tl jj1 + cw jj
q
Y
l=2
q
Q
l=1
Wn;tl jj1
(q
q
Y
q
Q
l=1
1)cu (L + T )Kcqw
Wn;tl ek;n
l=2
Wn;tl jj1
Since this inequality holds for any k = 1; :::; n, we have jj
By deduction, we have jj
e0n Wn;t1 ei;n e0i;n
Wn;tl ek;n
l=2
+ max e0n Wn;t1 ei;n
i2V
= n
(L + T )Kcu cqw
+ cu cqw
1
+ cw jj
cu (L + T )Kcqw
Wn;tl jj1
1
!
q
Y
1
1
q
Y
l=2
i2V
= n
e0i;n
q
Y
l=2
Wn;tl jj1 :
+ cw jj
q 1
qcu (L + T )Kcw
.
X
q
Q
l=2
Wn;tl jj1 :
Claim 6.0.3 Suppose Wn;l is an n n square matrix which can be decomposed into the sum of two n n
matrices such that Wn;tl = An;tl + Bn;tl . Denote jAjmax = maxfjaij;nl j : i; j = 1; :::; ng: Then for any positive
integer k and any i; j = 1; :::; n,
k
Y
( Wn;tl
l=1
k
Y
l=1
Bn;tl )ij
jAjmax
k
X1
m=0
17
jj
m
Y
l1 =1
Bn;tl1 jj1 jj
k
Y
l2 =m+2
Wn;tl2 jj1 :
Wn;tl ek;n
Proof of Claim 6.0.3. By expansion,
k
Y
Wn;tl
l=1
k
Y
Bn;l
=
(An;t1 + Bn;t1 )
l=1
k
Y
k
Y
Wn;tl = An;t1
l=2
=
k
X1
m=0
m
Y
!
Bn;tl1
l1 =1
Wn;tl + Bn;t1 An;t2
l=2
k
Y
An;tm+1
Wn;tl2
l2 =m+2
!
k
Y
Wn;tl + ::: +
l=3
kY1
Bn;tl An;tk
l=1
:
k
k
Q
Q
Denote ei;n = (0; :::0; 1; 0; :::; 0)0 , which is the ith unit vector of order n; then ( Wn;tl
Bn;l )ij =
l=1
l=1
!
!
m
k
Q
Q
Pk 1 0
Bn;tl1 An;tm+1
Wn;tl2 ej;n . For any matrix M and vector e of dimension n, it is
m=0 ei;n
l1 =1
l2 =m+2
easy to see that jjM ejj1 jM jmax jjejj1 : Thus, for any integer m = 0; :::; k 1,
!
!
!
m
k
m
Y
Y
Y
0
0
ei;n
Bn;tl1 An;tm+1
Wn;tl2 ej;n jjei;n
Bn;tl1 jj1 jjAn;tm+1
l1 =1
l2 =m+2
jj
m
Y
l1 =1
Bn;tl1 jj1 jAn;tm+1
l1 =1
k
Y
l2 =m+2
Wn;tl2 jmax
jAjmax jj
m
Y
l1 =1
Bn;tl1 jj1 jj
k
Y
Wn;tl2
l2 =m+2
k
Y
l2 =m+2
!
ej;n jj1
Wn;tl2 jj1
Together, we have the result.
Ym
Claim 6.0.4 Let ti;nt (m) be the ith element of the vector
Wn;tl & n;tm a, where & i;nt = fi (& i;nt ; Xnt ) with
l=1
& nt = ("nt ; nt ) is a vector-valued function and a is any conformable vector of constants. Under Assumptions
1, 3.1), and 4.1), suppose supi;n;t jj& i;nt jjp < 1, then supi;n;t jjti;nt (m)jjp mc3 d0 +2 cm
w Cap and
3+c3 d0 (2 c3 )d0
s
with
C
being
a …nite constant.
m
supi;n;t jjti;nt (m) E(ti;nt (m)jFi;n;T (s))jjp Cap cm
ap
w
Proof. Note that for any integer 1
k
Y
( Wn;tl )iik
k
m and ik 2
= Bi (s), we can show that
C0 mc3 d0 +2 ckw iikc3 d0
and
l=1
X
k
Y
( Wn;tl )iik
C1 ckw mc3 d0 +2 s(1
c3 )d0
(6)
ik 2B
= i (s) l=1
with C0 and C1 being positive constants, not depend on k. To show this, we construct two matrices An;tl and
Bn;tl as follows: aij;ntl = wij;ntl I(wij;ntl c1 ( iik =m) c3 d0 ) and bij;ntl = wij;ntl I(wij;ntl > c1 ( iik =m) c3 d0 ),
then Wn;tl = An;tl + Bn;tl and aij;ntl bij;ntl = 0: As
e0i;n
k
Y
l=1
Wn;tl eik ;n =
n X
n
X
i1 =1 i2 =1
n
X
wii1 ;nt1 wi1 i2 ;nt2
wik
1 ik ;ntk
;
ik =1
if ik 2
= Bi (s), at least one of the items wii1 ;nt1 , wi1 i2 ;nt2 ;
; wik 1 ik ;ntk , say wiq 1 iq ;ntq , satis…es that
wiq 1 iq ;ntq
c1 ( iik =k) c3 d0
c1 ( iik =m) c3 d0 , because there exist at least two neighboring nodes in
Yk
the chain i ! i1 !
! ik such that their distance is at least iik =k. Hence, (
Bn;tl )iik =
l=1
18
P
P
i1
ik
1
wii1 ;nt1 wi1 i2 ;nt2
k
Y
( Wn;tl )iik
wik
1 ik ;ntk
k
Y
( Wn;tl
=
l=1
k
Y
l=1
c1 (
iik
m
)
I(all w’s> c1 (
Bn;tl )iik
sup jAn;tl jmax
n;tl
l=1
c3 d0
k
X1
cqw (k
c3 d0
iik =m)
1
k
X1
q=0
) = 0, and we have
jj
k
q)K(L + T )cu cw
q
Y
l1 =1
Bn;l1 jj1 jj
k
Y
l2 =q+2
Wn;l2 jj1
q 2
q=0
K(L + T )cu c1 (
iik =m)
c3 d0 2 k 2
k cw
C0 mc3 d0 +2 ckw
c3 d0
;
iik
where the …rst inequality is from Claim 6.0.3; the second one is from Claim 6.0.2 and all elements in An;tl
are c1 ( iik =m) c3 d0 . And hence, for any i;
X
k
Y
( Wn;tl )iik
C0 mc3 d0 +2 ckw
ik 2B
= i (s) l=1
X
c3 d0
iik
C0 mc3 d0 +2 ckw c4
ik 2B
= i (s)
1
X
(1 c3 )d0 1
C1 ckw mc3 d0 +2 s(1
c3 )d0
:
=[s]
Ym
Any element of e0i;n
Wn;tl & n;tm a has a chain of w’s starting from i and ending in im with m steps.
l=1
We can divide these chains into two sets: one set has all its paths staying within Bi (s), and the other set
has some paths falling outside of Bi (s). For the …rst set with all nodes i’s in Bi (s), obviously ti;nt (m)
E(ti;nt (m)jFi;n (s)) = 0. For the second set, divide it into m mutually exclusive subsets:
Ym
P
(i) im 2B
Wn;tl )iim & im ;ntm a;
= i (s) (
l=1
Ym 1
P
P
(ii) im 1 2B
(
Wn;tl )iim 1 wim 1 im ;ntm & im ;ntm a; etc.
= i (s)
im 2Bi (s)
l=1
Such subset can be written as
im
X
X
= i (s) im
k 2B
k+1 2Bi (s)
m
Yk
X
(
Wn;tl )iim
k
wim
k im
k+1;ntm
wim
k+1
1 im ;ntm
& im ;ntm a
im 2Bi (s) l=1
for k = 0; :::m 1.
Consider (i): As
j
X
m
Y
( Wn;tl )iim & im ;ntm aj
im 2B
= i (s) l=1
X
m
Y
( Wn;tl )iim j& im ;ntm aj
im 2B
= i (s) l=1
X
im 2B
= i (s)
j& im ;n ajC0 mc3 d0 +2 cm
w
c3 d0
iim ;
we have
jj
X
(Wnm )iim & im ;ntm ajjp
cap C0 mc3 d0 +2 cm
w
im 2B
= i (s)
1
X
c4
(1 c3 )d0 1
(1
C2 cap mc3 d0 +2 cm
ws
c3 )d0
;
(7)
=[s]
where cap = supi;n;t (Ej& i;nt ajp )1=p . For (ii),
j
im
X
X
m
Y1
(
= i (s) im 2Bi (s) l=1
1 2B
Wn;tl )iim 1 wim
j
1 im ;ntm
jjWn;tm jj1
19
im
X
m
Y1
(
= i (s) l=1
1 2B
Wn;tl )iim
1
c3 d0 +2 (1
cm
s
wm
c3 )d0
C1 ;
and hence,
jj
im
X
X
m
Y1
(
= i (s) im 2Bi (s)
1 2B
Wn;tl )iim 1 wim
jj
= i (s) im
k 2B
m
Y
l=m k+1
X
im
c3 )d0
k
X
X
cap
c5 sd0 cap
c3 d0 +2 (2
C3 cm
s
wm
2
= Bi (s) where 1
m
Yk
(
Wn;tl )iim
k
k
wim
m
k im
c3 )d0
cap by (6).
1; we have
k+1;ntm
wim
k+1
1 im ;ntm
im 2Bi (s) l=1
k+1 2Bi (s)
Wn;tl jj1
c3 )d0
im 2Bi (s)
And similarly, for subset (k) with im
im
c3 d0 +2 (1
C1 cm
s
wm
& im ;ntm ajjp
l=1
c3 d0 +2 (1
C1 cm
s
wm
X
1 im ;ntm
X
m
Yk
(
= i (s)
k 2B
Wn;tl )iim
k
c3 d0 +2 (1
cm
s
wm
c3 )d0
l=1
C1 as jj
m
Y
l=m k+1
Wn;tl jj1
ckw :
Hence,
jj
im
X
= i (s) im
k 2B
X
X
c3 )d0
Wn;tl )iim
k
wim
k im
k+1;ntm
k+1
wim
1 im ;ntm
& im ;ntm ajjp
im 2Bi (s) l=1
k+1 2Bi (s)
c3 d0 +2 (2
C3 cm
s
wm
m
Yk
(
cap :
These together imply
jjti;nt (m)
E(ti;nt (m)jFi;n (s))jjp
(1
2 C2 mc3 d0 +2 cm
ws
2jjpaths in ti;nt (m) with at least one node im k 2
= Bi (s)jjp
!
m
1
X
c3 )d0
c3 d0 +2 (2 c3 )d0
c3 d0 +3 (2 c3 )d0
cap + C3
cm
s
cap
cm
s
Cap :
wm
wm
k=1
3+c3 d0
'(s) with '(s) = s(2 c3 )d0 .
To conclude, for any integer m, jjti;nt (m) E(ti;nt (m)jFi;n (s))jjp Cap cm
wm
c3 d0 +2 m
Now we show jjti;nt (m)jjp m
cw Cap . Divide the whole space D into exclusive subsets Bi (1) and
Bi ( + 1)nBi ( ), = 1; 2;
: Consider the case with im 2 Bi ( + 1)nBi ( ). For each
1, from equation
(7), we have
m
X
Y
(1 c3 )d0
jj
( Wn;tl )iim & im ;ntm ajjp C2 cap mc3 d0 +2 cm
:
w
im 2Bi ( +1)nBi ( ) l=1
Ym
For Bi (1); there are two cases: im = i and im 6= i: For the case im = i, we have jj(
Wn;tl )ii & i;n ajjp
l=1
Ym
P
m
cap cw : For the case im 6= i, it must be iim = 1 from Assumption 1. Hence, jj im ; ii =1 (
Wn;tl )iim & im ;n ajjp
m
l=1
m
cap c5 cw : Since
ti;nt (m) = e0i;n
m
Y
l=1
Wn;tl & ntm a =
1
X
X
m
Y
( Wn;tl )iim & im ;ntm a +
=1 im 2Bi ( +1)nBi ( ) l=1
20
X
m
Y
( Wn;tl )iim & im ;ntm a;
im 2Bi (1) l=1
we have
1
X
jjti;nt (m)jjp
=1
jj
X
m
Y
( Wn;tl )iim & im ;ntm ajjp + jj
im 2Bi ( +1)nBi ( ) l=1
C2 cap mc3 d0 +2 cm
w
1
X
(1 c3 )d0
m
+ cap c5 cm
w + cw cap
X
m
Y
( Wn;tl )iim & im ;ntm ajjp
im 2Bi (1) l=1
mc3 d0 +2 cm
w Cap :
=1
Ym
Claim 6.0.5 Let gi;nt (m) = e0i;n
Gn;tl ( 1 )& n;tm a, where & nt and a are the same as Claim 6.0.4.
l=1
Under Assumptions 1, 3.1), and 4.1), suppose supi;n;t jj& i;nt jjp < 1, then supi;n;t jjgi;nt (m)jjp < 1 and
supi;n;t jjgi;nt (m) E(gi;nt (m)jFi;n (s))jjp Capm s(2 c3 )d0 with Capm being a …nite constant.
As
m
Y
Gn;tl (
l=1
1)
1 X
1
X
=
1
X
l1 =0 l2 =0
where q = l1 + l2 +
l1 +l2 +
1
+lm
l1 +1
l2 +1
Wn;t
Wn;t
1
2
lm +1
Wn;t
m
lm =0
1
X
q+m
=
m
q=0
1
q
1
q+m
Y
Wn;tl ;
l=1
+ lm , by using the results for ti;nt (q + m) in Claim 6.0.4, for any …nite m, we have
jjgi;nt (m)jjp
1
X
q=0
qm j
q
1j
cm
w Cap
jjti;nt (q + m)jjp
1
X
q=0
(q + m)m+c3 d0 +2 j
q
1 cw j
<1
and
jjgi;nt (m)
E(gi;nt (m)jFi;n (s))jjp
1
X
q=0
Cap
qm j
1
X
q=0
q
1 j jjti;nt (q
j
+ m)
q q+m
(q
1 j cw
E(ti;nt (q + m)jFi;n (s))jjp
+ m)m+3+c3 d0 s(2
c3 )d0
Capm s(2
c3 )d0
:
PH
(h)
Claim 6.0.6 Denote Unt;H = Gnt h=0 Bnt & 1n;t h . Under Assumptions 1, 2.1), 3.1), and 4.1), we have
P
(h)
H
the NED property of ui;nt , i.e., jje0i;n Gnt h=0 Bnt & 1n;t h jjp < Cu1 and
jje0i;n Gnt
H
X
(h)
Bnt & 1n;t h
E(e0i;n Gnt
h=0
H
X
(h)
Bnt & 1n;t
h=0
h jFi;n (s))jjp
d'(s)
such that lims!1 '(s) = 0.
(h)
(h)
Proof. First we show jje0i;n Gnt Bnt & 1n;t h E(e0i;n Gnt Bnt & 1n;t
Yhnl
Yl 1
Yh
0; :::H. Denote
An;k =
An;k
An;k ,
k=1
k=1
k=l+1
21
h jFi;n (s))jjp
d1 '(s) holds for any h =
Yhnl1 ;l2
k=1
(h)
Bnt
An;k =
=
h
Y
Yl1
1
k=1
( 0 Sn;t1
An;k
k +
Yl2
1
k=l1 +1
20 Gn;t
k) =
k=1
=
(
Yl2 +1
An;k
k=l1 +1
h
Y
An;k
[ 0 In + (
Yh
k=l3 +1
0 10
20 )Gn;t k ]
k=1
0 10
h
+
20 )
h
Y
Gn;t
k
+(
0 10
h 1
+
20 )
0
k=1
+(
+
An;k , and so on. As
0 10
+
hnl
h Y
X
Gn;t
k
l=1 k=1
h X hnl
1 ;l2
X
Y
h 2 2
20 )
0
Gn;t
k
+
+(
0 10
+
h 1
20 ) 0
l1 =1 l2 6=l1 k=1
h
X
Gn;t
k
+
h
0 In ;
k=1
we have
(h)
jje0i;n Gnt Bnt & 1n;t
j
0 10
+
h
20 j
(h)
E(e0i;n Gnt Bnt & 1n;t
h
(2 c3 )d0
Cap(h+1) s
+ hj
0 20
h jFi;n (s))jjp
+
h 1
j 0 jCaph s(2 c3 )d0
20 j
h
j 0 20 + 20 jh 2 j 0 j2 Cap(h 1) s(2 c3 )d0 +
2
h
+
j 0 20 + 20 j j 0 jh 1 Cap(2) s(2 c3 )d0 + j 0 jh Cap(1) s(2
h 1
+
h
X
h
j 0 jq j
q
q=0
=
(j 0 j + j
0 10
0 20
+
+
c3 )d0
h q
CapH s(2 c3 )d0
20 j
h
(2 c3 )d0
;
20 j) CapH s
(8)
where CapH = max0 h H Caph < 1 is a …nite constant not depending on n: The …rst inequality is from
Ym
Ym
Claim 6.0.5 that jje0i;n
Gn;tl ( 1 )& n;tm a E(e0i;n
Gn;tl ( 1 )& n;tm ajFi;n;T (s))jjp Capm s(2 c3 )d0 : As
l=1
l=1
8 holds for any h = 0; :::H, we have
jje0i;n Gnt
H
X
h=0
H
X
(h)
Bnt & 1n;t
E(e0i;n Gnt
h=0
H
X
(h)
Bnt & 1n;t
h=0
(h)
jje0i;n Gnt Bnt & 1n;t
CapH s(2
h
c3 )d0
H
X
h=0
(h)
h
(j 0 j + j
E(e0i;n Gnt Bnt & 1n;t
0 20
+
h
20 j)
h jFi;n (s))jjp
h jFi;n (s))jjp
Cp;H s(2
c3 )d0
:
Similarly,
jje0i;n Gnt
H
X
h=0
(h)
Bnt & 1n;t h jjp
H
X
h=0
(h)
jje0i;n Gnt Bnt & 1n;t h jjp
NED properties in the case under Assumption 4.2)
22
CapH
H
X
h=0
(j 0 j + j
0 20
+
h
20 j)
Cp;H :
Ym
Claim 6.0.7 If the i; jth element of
l=1
Wn;tl is not zero, then
m c.
ij
Ym
P P
P
Proof of Claim 6.0.7. The i; jth element of
Wn;tl is i1 i2
wim 1 j;ntm :
im 1 wii1 ;nt1 wi1 i2 ;nt2
l=1
If it is not zero, then there exists at least one path i ! i1 !
im 1 ! j such that all wii1 ;nt1 , wi1 i2 ;nt2 ;
;
wim 1 j;ntm are positive. As wij;nt = 0 if j 2
= Bi ( c ) holds for any period t, it must be i1 2 Bi ( c ),
m c.
i2 2 Bi1 ( c ),..., j 2 Bim 1 ( c ). Therefore, ij
ii1 + i1 i2 + ::: + im 1 j
Claim 6.0.8 Under Assumptions 1, 3.1), and 4.2), for any positive integer q, supn;t jj
Proof of Claim 6.0.8. Consider the kth column sum of
q
Q
q
Q
l=1
Wn;tl jj1
cqw c5 (q c )d0 .
Wn;tl , as all elements in Wnt are non-negative,
l=1
e0n
q
Y
Wn;tl ek;n
n
X
=
l=1
i2Bk (q
sup
i;n
e0i;n
e0i;n
j=1
qY1
n
X
Wn;tl ek;n =
l=1
c)
n
X
q
Y
i2Bk (q
Wn;tl ej;n
l=1
sup
j;k;n;tq
c)
n
X
e0i;n
j=1
n
X
i2Bk (q
qY1
Wn;tl ej;n e0j;n Wn;tq ek;n
l=1
e0j;n Wn;tq ek;n
cqw c5 (q c )d0 :
c)
The …rst “=” holds because of Claim 6.0.7. The last “ ” holds because supn;t jj
supj;k;n;tq e0j;n Wn;tq ek;n < cw .
m
Q
l=1
Wn;tl jj1
cm
w and
Claim 6.0.9 For any
p=( ln q) + 1, then there exists a …nite
P positive integer p and 0 < q < 1, if s
constant c such that l=[s] lp q l < csp q s , where [s] denotes the largest integer less than or equal to s.
This is from Qu and Lee (2015).
Ym
Claim 6.0.10 Let ti;nt (m) = e0i;n
Wn;tl & n;tm a; where & nt and a are the same as Claim 6.0.4. Under
l=1
Assumptions 1, 3.1) and 4.2), suppose supi;n;t jj& i;nt jjp < 1, then supi;n;t jjti;nt (m)jjp
Cap md0 cm
w and
supi;n;t jjti;n (m) E(ti;n (m)jFi;n (s))jjp Cap1 '(s) with Cap and Cap1 being positive constants; '(s) = 1 if
s m c and '(s) = 0 if s > m c .
Proof of Claim 6.0.10. As
jti;nt (m)j = j
X
k
e0i;n
m
Y
l=1
max je0i;n
k;n;t
m
Y
l=1
we have
jjti;nt (m)jjp
Wn;tl ek;n e0k;n & n;tm aj = j
cm
w
Wn;tl ek;n j
X
k2Bi (m
c)
X
k2Bi (m
c)
jj& k;ntm ajjp
where cap = supi;n;t jj& i;nt ajjp and Cap = cap c5
d0
c .
23
X
k2Bi (m
e0i;n
c)
m
Y
Wn;tl ek;n e0k;n & n;tm aj
l=1
j& k;ntm aj;
d0
m d0
cm
w c5 (m c ) cap = Cap cw m ;
Next, we show the NED property. For the spatial weight matrix without row-normalization, wij;nt is a
function of & i;nt and & j;nt , and wij;nt = 0 if j 2
= Bi ( c ). For the row-normalized case, wij;nt may be related
to all points in Bi ( c ) and in general is a function of &’s at those locations. In both cases, all the locations
Ym
of nodes in the chains of e0i;n
Wn;tl & n;tm a related to ti;nt (m) are within the ball Bi (m c ). Hence, when
l=1
s > m c , ti;nt (m) E(ti;nt (m)jFi;n (s)) = 0. With s m c ;
jjti;nt (m)
E(ti;nt (m)jFi;n (s))jjp
d0
2Cap cm
wm :
2jjti;nt (m)jjp
Therefore, the NED property follows if we choose '(s) = 1 for s
m
c
and '(s) = 0 for s > m c .
Ym
Claim 6.0.11 Denote gi;nt (m) = e0i;n
Gn;tl ( 1 )& n;tm a, where & nt and a are the same as Claim 6.0.4.
l=1
Under Assumptions 1, 3.1), and 4.2), suppose supi;n;t jj& i;nt jjp < 1, then supi;n;t jjgi;nt (m)jjp < 1 and
supi;n;t jjgi;nt (m) E(gi;nt (m)jFi;n (s))jjp Capm '(s) with Capm being a …nite constant; '(s) = 1 if s m c
and '(s) = sd0 +m j cw js= c if s > m c .
Proof of Claim 6.0.11. From the proof of Claim 6.0.5,
m
Y
Gn;tl (
1)
=
l=1
1
X
q+m
m
q=0
1
q
1
q+m
Y
Wn;tl ;
l=1
where q = l1 + l2 +
+ lm . If 1 = 0, then gi;nt (m) = ti;nt (m) and the Claim follows directly from Claim
(m + q)d0 and
6.0.10. For 1 6= 0, by Claim 6.0.10, for any i; n, and t; jjti;nt (m + q)jjp Cap cm+q
w
jjgi;nt (m)jjp
cm
w Cap
1
X
q
1 cw j (q
j
q=0
+ m)d0 +m
Cm ;
where Cm is …nite. Thus, jjgi;nt (m) E(gi;nt (m)jFi;n (s))jjp
2jjgi;nt (m)jjp
2Cm . Now consider the
case when s > m c . Given such an s, from Claim 6.0.10, ti;nt (m + q) E(ti;nt (m + q)jFi;n (s)) = 0 for
any nonnegative integer q such that s > (m + q) c . Such a set of q will be determined by q < ( s
m).
c
Therefore, when s > m c ,
jjgi;nt (m) E(gi;nt (m)jFi;n (s))jjp
1
X
2
(q + m)m j 1 jq jjti;nt (q + m)jjp
q=[
s
c
m]
2Cap cm
w
1
X
q=[
s
c
m]
j
q
1 cw j (q
+ m)m+d0 ;
where the last inequality follows from Claim 6.0.10. By the inequality in Claim 6.0.9, as s=
1
X
q=[
s
c
m]
j
q+m
(q + m)m+d0 =j
1 cw j
m
1j =
The Claim would follow if we set '(s) = 1 if s
1
X
q=[
m
c
s
c
]
j
q m+d0
=j 1 jm
1 cw j q
and '(s) = sd0 +m j
24
= O(sm+d0 j
s=
1 cw j
c
c
> m, we have
s=
1 cw j
if s > m c .
c
):
PH
(h)
Claim 6.0.12 Denote Unt;H = Gnt h=0 Bnt & 1n;t h . Under Assumptions 1, 2.1), 3.1), and 4.2), we have
P
(h)
H
the NED property of ui;nt , i.e., jje0i;n Gnt h=0 Bnt & 1n;t h jjp < Cu1 and
jje0i;n Gnt
such that '(s) = 1 if s
H
X
(h)
Bnt & 1n;t
H
X
E(e0i;n Gnt
h
h=0
(h)
Bnt & 1n;t
h=0
(H + 1)
c
and '(s) = sd0 +H j
s=
1 cw j
c
h jFi;n (s))jjp
d'(s)
if s > (H + 1) c .
Proof. Similarly to the proof of Claim 6.0.6, From Claim 6.0.11,
jje0i;n Gnt
If s
H
X
h=0
(h + 1)
(h)
Bnt & 1n;t h jjp
H
X
h=0
(h)
jje0i;n Gnt Bnt & 1n;t h jjp
C(H+1)
H
X
h=0
(j 0 j + j
0 20
h
20 j)
+
Cp;H+1 :
(H + 1) c ;
c
(h)
(h)
jje0i;n Gnt Bnt & 1n;t
Cj
+
0 10
h
20 j
h 2
j 0 j2 C
20 j
+
h
X
h
j 0 jq j
q
q=0
E(e0i;n Gnt Bnt & 1n;t
h
+ hCj
+
0 10
+
0 10
h 1
j 0j
20 j
+
h
h
h q
C
20 j
+
j
1
0 10
h jFi;n;T (s))jjp
h
j
2
+
+
20 j
= (j 0 j + j
0 20
j 0 jh
0 10
+
1
C + j 0 jh C
h
20 j) C
(9)
and
jje0i;n Gnt
H
X
(h)
Bnt & 1n;t
E(e0i;n Gnt
h
h=0
H
X
(h)
Bnt & 1n;t
h=0
H
X
h jFi;n (s))jjp
h=0
(j 0 j + j
0 10
h
20 j) C
+
= CH :
If s > (H + 1) c ;
(h)
jje0i;n Gnt Bnt & 1n;t
j
0 10
+
h
20 j
(h)
E(e0i;n Gnt Bnt & 1n;t
h
d0 +h
Cs
j
s=
1 cw j
c
+ hj
0 10
h jFi;n (s))jjp
+
h 1
j 0 jCsd0 +h 1 j 1 cw js=
20 j
h
j 0 10 + 20 jh 2 j 0 j2 Csd0 +h 2 j 1 cw js= c +
2
h
+
j 0 10 + 20 j j 0 jh 1 Csd0 +1 j 1 cw js= c + j 0 jh Csd0 j
h 1
c
+
(j 0 j + j
0 10
+
h
d0 +h
j 1 cw js=
20 j) Cs
H
X
Bnt & 1n;t
s=
1 cw j
c
(10)
c
and
jje0i;n Gnt
H
X
h=0
(h)
h
E(e0i;n Gnt
h=0
(j 0 j + j
H
X
(h)
Bnt & 1n;t
h=0
0 10
+
h
d0 +h
j 1 cw js=
20 j) Cs
25
c
h jFi;n (s))jjp
CH sd0 +H j
s=
1 cw j
c
:
PH
PH
(h)
(h)
Lemma 1 Denote U1nt;H = A1nt h=0 Bnt & 1n;t h and U2nt;H = A2nt h=0 Bnt & 1n;t h . Under Assump1
0
0
tions 1, 2.1), 3.1), and 4), n [ U1nt;H1 U2nr;H2 E( U1nt;H1 U2nr;H2 )] = op (1), where H1 and H2 can be
any integer from 0 to T + L:
Pn
0
0
Unr;H2 E( Unt;H
Unr;H2 )] = n1 i=1 qi;n with
Proof. Denote n1 [ Unt;H
1
1
qi;n = u1i;nt;H1 u2i;nr;H2
u1i;n;t
1;H1 u2i;nr;H2
u1i;nt;H1 u2i;n;r
1;H2
+ u1i;n;t
1;H1 u2i;n;r 1;H2 :
Under Assumption 4.1), we have supi;n;t;H jju1i;nt;H
E(u1i;nt;H jFi;n (s))jj4
C4;T +L s(2 c3 )d0 and
(2 c3 )d0
supi;n;t;H jju2i;nt;H E(u2i;nt;H jFi;n (s))jj4 C4;T +L s
. Together with maxi;n;t;H (jju1i;nt;H jj4 ; jju2i;nt;H jj4 )
C1 from Claim 6.0.6, we have jjqi;n E(qi;n jFi;n;T (s))jj2 8C1 C4;T +L s(2 c3 )d0 from Claim 6.0.1.
Under Assumption 4.2), we have supi;n;t;H jju1i;nt;H E(u1i;nt;H jFi;n (s))jj4 C'(s) and supi;n;t;H jju2i;nt;H
E(u2i;nt;H jFi;n (s))jj4 C'(s) with '(s) = 1 if s (T +L) c and '(s) = sd0 +T +L j 1 cw js= c if s > (T +L) c .
Pn
L
And maxi;n;t;H (jju1i;nt;H jj4 ; jju2i;nt;H jj4 ) Cp;T +L from Claim 6.0.12. Therefore, n1 i=1 [qi;n E(qi;n )] !1 0
as conditions of the LLNs under the spatial NED in Jenish and Prucha (2012) hold.
Proof of the main results
Proof of Theorem 1.
b
0
(Tbn0 P2n Tbn )
= (Tbn0 P2n Tbn )
Tbn0 P2n Yn (T )
0
1 b0
b
Tn P2n n (T ) + (Tn0 P2n Tbn )
1
=
1
Tbn0 P2n Kn (Kn0 P1n Kn )
1
Kn0 P1n "n (T ) 0 :
0 b
0
0
0
0
0
Tn , n1 q2n
q2n , n1 q2n
Kn , n1 q1n
Kn , n1 q1n
q12n are Op (1), n1 q2n
For the consistency, we need to show that n1 q2n
1 0
op (1); and n q1n "n (T ) = op (1):
Since
Tbn = [ (W Y )n (T ); (W Y )n (T 1); Yn (T 1); X1n (T ); b
"n (T )] with
b
"n (T ) =
0
Kn [Kn0 q1n (q1n
q1n )
"n (T )
if we choose q2n = [ (W X)n (T ); (W X)n (T
cated term we need to show is
1 0
0
0
q1n Kn ] 1 Kn0 q1n (q1n
q1n ) 1 q1n
T
Wn;t
"n (T );
1); X1n (T ); b
"n (T )], then the most compli-
1); X1n (T
1
1X
(Wnt Xnt
(W X)n (T )0 (W Y )n (T ) =
n
n t=2
n (T )
0
1 Xn;t 1 )
(Wnt Ynt
Wn;t
1 Yn;t 1 )
= Op (1):
In our setting, the reduced form is
Znt
=
=
t+t
0 1
X
h=0
t+t
X0
h
20 (X2n;t h 0
h
20 (X2n;t h 0
+ dn0 + "n;t
+ dn0 + "n;t
h)
h)
+
+
t+t0
20 Zn; t0
t+t0
o
20 (dn0
dn0 );
h=0
Ynt
= Snt1
= Snt1
t+t
0 1
X
h=0
t+t
X0
(h)
Bnt (X1n;t
h 0
(h)
Bnt (X1n;t
h 0
+ cn0 + "n;t
+ cn0 + "n;t
h 0
h 0
+
+
h=0
26
n;t h )
n;t h )
(t+t0 1)
+ Snt1 Bnt
(t+t0 )
+ Snt1 Bnt
(con0
(
10 In
+
cn0 ):
20 Wn; t0 )Yn; t0
=
(h)
where Bnt = ( 0 Sn;t1
(0)
Bnt
20 Gn;t k ) with
con0 + "n; t0 0 +
Therefore,
Znt
n;
t+t
X0
=
1
+
1
20 Gn;t 1 )( 0 Sn;t 2
= In and Yn;
t0 ).
t0
h
20 (X2n;t h 0
+
( 0 Sn;t1
20 Gn;t 2 )
h
+
can be treated either exogenous or Yn;
+ dn0 + "n;t
h)
t+t0 o
20 (dn0
+
20 Gn;t h )
t0
= (In
=
Yh
k=1
1 Wn; t0 )
( 0 Sn;t1
1
(X1n;
k
+
t0
+
dn0 )
h=0
t+t
0 1
X
Ynt
h=0
t+t
X0
Snt1
h=0
=
Sn;t1
h
20 (X2n;t 1 h 0
(h)
1
+ dn0 + "n;t
Bnt (X1n;t
h 0
t X
1+t0
1 (X1n;t 1 h 0
(h)
Bn;t
+ cn0 + "n;t
t+t0 1 o
(dn0
20
1 h)
h 0
+
(t+t0 )
+ Snt1 Bnt
n;t h )
+ cn0 + "n;t
dn0 );
1 h 0
+
(con0
n;t 1 h )
cn0 )
Sn;t1
(t 1+t0 ) o
(cn0
1 Bn;t 1
cn0 );
h=0
(W Y )nt
= Gnt
t+t
X0
(h)
Bnt (X1n;t
h 0
+ cn0 + "n;t
+
h 0
(t+t0 )
n;t h )
+ Gnt Bnt
(con0
cn0 )
h=0
Gn;t
1
t X
1+t0
(h)
Bn;t
1 (X1n;t 1 h 0
+ cn0 + "n;t
1 h 0
+
n;t 1 h )
Gn;t
(t 1+t0 ) o
(cn0
1 Bn;t 1
h=0
1 0
1 0 b
1 0
0
to show n1 q2n
n (T ) = op (1) and n q1n "n (T ) = op (1); n q2n Tn and n qn Kn are Op (1), in general, we need
to show terms like
"
!#
t+t
t+t
X0 (h)
X0 (h)
1 0
0
0
0
&
W Gnt
Bnt & 1n;t h E & 2n;t Wnt Gnt
Bnt & 1n;t h
= op (1);
n 2n;t nt
h=0
where & can be either X or " or
sup jje0in Gnt
i;n;t
t+t
X0
h=0
or a constant. From Claim 6.0.1, it is su¢ cient to show that
(h)
Bnt & 1n;t
h
t+t
X0
E(e0in Gnt
h=0
(h)
Bnt & 1n;t
and sup jje0in Gnt
i;;n;t
For the asymptotic distribution of b, we have
p
n(b
0
=
n
=
1
plim (Tn0 P2n Tn )
n!1 n
2
0 0
h jFi;n (s))jj4
h=0
1
Tn0 P2n
0)
d
! N (0;
t+t
X0
(h)
Bnt & 1n;t
h=0
1
Kn0 P1n (
27
0
0
< 1:
0 ):
0
1
;
n P2n Tn (Tn P2n Tn )
In + Kn (Kn0 P1n Kn )
h jj4
C1 '1 (s);
"0 0
0
with
In )P1n Kn (Kn0 P1n Kn )
1
Kn0 :
cn0 );
We can write
p
n(b
0)
=
1
( Tbn0 P2n Tbn )
n
1
n
=
1
p Tbn0 P2n
n
n (T )
1
+ ( Tbn0 P2n Tbn )
n
1
T
X
1 X
p
ri;n + op (1); with ri;n =
(q1i;nt
n i=1
t=3
i;nt
1 b0
1
T P2n Kn ( Kn0 P1n Kn )
n n
n
where
bn
=
1 b0
(T P2n Tbn )
n n
b" =
1
2n(T
"0 .
p
Then we show that b2 !
0
Kn0 P1n (b b "b
p
2)
b
"n (T )0 b
"n (T ); b2 =
Zn (T )
Kn
b2
b
=
1
2n(T
2)
1
Kn0 + (b2
In );
0
b (T )0 b (T ):
n
n
"n (T ) + Kn
20
0
2)
b2
b
;
"0 :
As
Ynt
0
0 ).
; with
1
p 1
b
"n (T )0 b
"n (T ) !
"n (T )0 "n (T ) = 2(T
n
n
2
0.
n(b
In )P1n Kn (Kn0 P1n Kn )
0
Tbnt b =
(W Y )nt ( 10
nt (T ) +
b
+ X1nt ( 0
) + "nt 0
b
"nt (T )b
=
Ynt Tbnt b =
(W Y )nt ( 10
nt (T ) +
=
+ X1nt (
we have
1
1
This holds because
b
"n (T ) =
nt
Tbn0 P2n b n P2n Tbn (Tbn0 P2n Tbn )
= Kn (Kn0 P1n Kn )
p
First we show that b " !
b
1
1
p Kn0 P1n "n (T )
n
+ q2i;nt "i;nt ):
With the NED property of ri;n , we can show the asymptotic normality of
Proof of Claim 3.2.1. Estimated variance-covariance matrix
b
1
b) +
"nt
0
b
1 b
1
p
(T )0 bn (T ) ! lim E(
n!1 n
n n
Kn
0
nt (T )
b1 ) +
b1 ) +
20
0
nt (T ))
(W Y )n;t
1 ( 20
(W Y )n;t
b2
;
b
1 ( 20
=
2
0 tr(
0
b2 ) +
b2 ) +
In ) = 2(T
Yn;t
1 ( 10
Yn;t
1 ( 10
2)
b1 )
b1 )
2
0
References
Aiello, F. & Cardamone, P. (2009) R&D spillovers and …rms’performance in Italy Evidence from a ‡exible
production function, In Spatial Econometrics (pp. 143-166). Physica-Verlag HD.
Anselin, L., LeGallo, J. & Jayet, H. (2008) Spatial panel econometrics, in: L. Matyas & P. Sevestre (eds)
The Econometrics of Panel Data, Chapter 19, pp. 625 660, Berlin, Springer-Verlag.
Baicker, K. (2005) The spillover e¤ects of state spending, Journal of Public Economics, 89, 529-544.
28
0
Baltagi, B. & Li, D. (2006) Prediction in the panel data model with spatial correlation: the case of liquor,
Spatial Economic Analysis, 1, 175-185.
Baltagi, B., Song, S. H. & Koh, W. (2003) Testing panel data regression models with spatial error
correlation, Journal of Econometrics, 117, 123-150.
Baltagi, B., Egger, P. & Pfa¤ermayr, M. (2007a) A generalized spatial panel data model with random
e¤ects, Working Paper, Syracuse University.
Baltagi, B., Song, S. H., Jung, B. C. & Koh, W. (2007b) Testing for serial correlation, spatial autocorrelation and random e¤ects using panel data, Journal of Econometrics, 140, 5-51.
Brueckner, J. K. (1998) Testing for strategic interaction among local governments: the case of growth
controls, Journal of Urban Economics, 44, 438-467.
Brueckner, J. K. & Saavedra, L. A. (2001) Do local governments engage in strategic property tax competition? National Tax Journal, 54, 203-229.
Case, A., Hines, J. R. & Rosen, H. S. (1993) Budget spillovers and …scal policy interdependence: evidence
from the States, Journal of Public Economics, 52, 285-307.
Caselli, F., Esquivel, G. & Lefort, F. (1996) Reopening the convergence debate: a new look at crosscountry growth empirics, Journal of economic growth, 1(3), 363-389.
Cli¤, A. D. & Ord, J. K. (1973) Spatial Autocorrelation, London, Pion Ltd.
Crabbé, K. & Vandenbussche, H. (2008) Spatial tax competition in EU15, Manuscript, Catholic University Leuven.
Druska, V. & Horrace, W. C. (2004) Generalized moments estimation for spatial panel data: Indonesian
rice farming, American Journal of Agricultural Economics, 86, 185-198.
Egger, P., Pfa¤ermayr, M. & Winner, H. (2005) An unbalanced spatial panel data approach to US state
tax competition, Economics Letters, 88, 329-335.
Elhorst, J. P. (2005) Unconditional Maximum Likelihood Estimation of Linear and Log-Linear Dynamic
Models for Spatial Panels, Geographical Analysis, 37(1), 85-106.
Ertur, C. & Koch, W. (2007) Growth, technological interdependence and spatial externalities: theory
and evidence, Journal of Applied Econometrics, 22(6), 1033-1062.
Foote, C. L. (2007) Space and time in macroeconomic panel data: young workers and state-level unemployment revisited, Working Paper No. 07-10, Federal Reserve Bank of Boston.
Franzese, R. J. & Hays, J. C. (2007) Spatial econometric models of cross-sectional interdependence in
political science panel and time-series-cross-section data, Political Analysis, 15(2), 140-164.
Frazier, C. & Kockelman, K. M. (2004) Chaos theory and transportation systems: instructive example,
Transportation Research Record: Journal of the Transportation Research Board, 1897(1), 9-17.
Islam, N. (1995) Growth empirics: a panel data approach, The Quarterly Journal of Economics, 11271170.
Kapoor, M., Kelejian, H. H. & Prucha, I. R. (2007) Panel data models with spatially correlated error
components, Journal of Econometrics, 140, 97-130.
Kelejian, H. H. & Prucha, I. R. (1998) A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbance, Journal of Real Estate Finance and
Economics, 17, 99-121.
Kelejian, H. H. & Prucha, I. R. (2001) On the asymptotic distribution of the Moran I test statistic with
applications, Journal of Econometrics, 104, 219-257.
Keller, W. & Shiue, C. H. (2007) The origin of spatial interaction, Journal of Econometrics, 140, 304-332.
Korniotis, G. M. (2010) Estimating panel models with internal and external habit formation, Journal of
Business and Economic Statistics, 28, 145-158.
29
Lee, L. F. (2004) Asymptotic distributions of quasi-maximum likelihood estimators for spatial econometric
models, Econometrica, 72, 1899-1925.
Lee, L. F. (2007) GMM and 2SLS estimation of mixed regressive, spatial autoregressive models, Journal
of Econometrics, 137, 489-514.
Lee, L. F. & Yu, J. (2010a) Estimation of spatial autoregressive panel data models with …xed e¤ects,
Journal of Econometrics, 154, 165-185.
Lee, L. F. & Yu, J. (2010b) Estimation of spatial panels: random components vs. …xed e¤ects, Manuscript, The Ohio State University.
Lee, L. F. & Yu, J. (2010c) A spatial dynamic panel data model with both time and individual …xed
e¤ects, Econometric Theory, 26, 564-597.
Lee, L. F. & Yu, J. (2010d) Some recent developments in spatial panel data models, Regional Science
and Urban Economics, 40, 255-271.
Lee, L. F. & Yu, J. (2012) QML estimation of spatial dynamic panel data models with time varying
spatial weights matrices, Spatial Economic Analysis, 7, 31-74.
LeSage, J. P. & Pace, R. K. (2009) Introduction to Spatial Econometrics, Boca Raton, FL, Chapman
and Hall/CRC.
Mankiw, N. G., Romer, D. & Weil, D. N. (1992) A contribution to the empirics of economic growth, The
Quarterly Journal of Economics, 107, 407-437.
Manski, C. F. (1993) Identi…cation of endogenous social e¤ects: The re‡ection problem, The review of
economic studies, 60(3), 531-542.
Michela, R. (2007) Fiscal interactions among European countries: does the EU matter? Manuscript, The
University of Warwick.
Mutl, J. (2006) Dynamic panel data models with spatially correlated disturbances, PhD thesis, University
of Maryland, College Park.
Mutl, J. & Pfa¤ermayr, M. (2011) The Hausman test in a Cli¤ and Ord panel model, Econometrics
Journal 14, 48-76.
Qu, X. & Lee, L.F. (2015) Estimating a spatial autoregressive model with an endogenous spatial weight
matrix, Journal of Econometrics, 184, 209-232.
Revelli, F. (2001) Spatial patterns in local taxation: taxmimicking or errormimicking? Applied Economics, 33, 1101-1107.
Rincke, J. (2010) A commuting-based re…nement of the contiguity matrix for spatial models, and an
application to local police expenditures, Regional Science and Urban Economics, 40, 324-330.
Su, L. & Yang, Z. (2007) QML estimation of dynamic panel data models with spatial errors, Manuscript,
Singapore Management University.
Tao, J. (2005) Analysis of local school expenditures in a dynamic fame, Manuscript, Shanghai University
of Finance and Economics.
Yu, J., de Jong R. & Lee, L.F. (2007) Quasi-maximum likelihood estimators for spatial dynamic panel
data with …xed e¤ects when both n and T are large: a nonstationary case, Manuscript, The Ohio State
University.
Yu, J., de Jong, R. & Lee, L.F. (2008) Quasi-maximum likelihood estimators for spatial dynamic panel
data with …xed e¤ects when both N and T are large, Journal of Econometrics, 146, 118-134.
Yu, J., de Jong R. & Lee, L.F. (2012) Estimation for spatial dynamic panel data with …xed e¤ects: the
case of spatial cointegration, Journal of Econometrics, 167, 16-37.
30