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2SIV Estimation of A Dynamic Spatial Panel Data Model withEndogenous Spatial Weight Matrices
Xi QuAntai College of Economics and Management
Shanghai Jiao Tong [email protected]
Lung-fei LeeDepartment of EconomicsThe Ohio State University
Xiaoliang WangAntai 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 correlationand dynamic dependences among economic units. Conventional estimation methods rely on the keyassumption that the spatial weight matrix is strictly exogenous, which would likely be violated in someempirical applications where spatial weights are determined by economic factors. This paper studiesthe estimation method of a dynamic spatial panel model with individual �xed e¤ects when the timedimension is short. The spatial weight matrices are constructed by some economic variables and can beendogenous and time varying. We establish the consistency and asymptotic normality of the two-stageinstrumental variable (2SIV) estimator and investigate their �nite sample properties by a Monte Carlostudy. This model is applied to study the �scal interactions among the 91 non-oil countries.
JEL classi�cation: C31; C51Keywords: 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 de-pendences 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 workson 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 (Eggeret al., 2005), consumer demand (Baltagi & Li 2006), to name a few. Dynamic spatial panel data modelcan 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¤ectsand �xed e¤ects spatial panel models are most commonly used. For the random e¤ects model, Baltagi etal. (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. Mutland Pfa¤ermayr (2010) and Lee and Yu (2010b) consider the estimation of spatial panel data models withboth �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 contiguityor geographic distances among regions. But there are plenty of cases that spatial weights are constructedwith economic/socioeconomic distances. For example, Aiello and Cardamone (2008) construct their spatialweights by a variable that re�ects �rms technological similarity and geographical proximity to study anR&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 ofregions (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 theendogeneity 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 theweights are still exogenous. One may wonder whether ignoring the endogeneity in spatial weights matriceswould have severe consequences in panel setting, and whether spatial panel models with endogenous timevarying spatial weights can be easily handled and estimated. These motivate our investigation on the spatialpanel 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 spatialweights 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 asymptoticproperties. In Section 4, we study �nite sample perfermances of our proposed 2SIV method using the MonteCarlo simulation. In Section 5, we apply this model to analyze the �scal interactions among the 91 non-oilcountries. 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) unevenlyspaced lattice D � Rd, d � 1. Asymptotic methods we employ are increasing domain asymptotics: growthof 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; :::Tg be a triangular double array of real random vectors de�nedon 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 = �1WntYnt + �2Wn;t�1Yn;t�1 + �1Yn;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�sare i.i.d. across i and t with zero mean and variance �2v. The X1nt is an n � k1 matrix of individuallyand 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 ofthe 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 hasthe model
zi;nt = zi;n;t�1�2 + x02i;t� + d
0i;n + "
0i;nt;
where �2 is a p dimensional constant matrix, x2i;t is a k2 dimensional vector of individually and time varyingnon-stochastic regressors, � is an p�k2 matrix of coe¢ cients, di;n is a p dimensional constant vector invariant
over time�and "i;nt is a p dimensional random variable. Denote n � k2 matrix X2nt =
0B@ x01;2nt...
x0n;2nt
1CA, n � pmatrices Znt; dn, and "nt with Znt =
0B@ z1;nt...
zn;nt
1CA, dn =0B@ d01;n
...d0n;n
1CA, and "nt =0B@ "01;nt
..."0n;nt
1CA. Then we canwrite in matrix form that
Znt = Zn;t�1�2 +X2nt� + dn + "nt: (2)
2.2 Source of endogeneity
We consider n agents in an area, each endowed with a predetermined location i. Due to some competitionor spillover e¤ects, at period t, each agent i has an outcome yi;nt directly a¤ected by its neighbors�currentoutcomes y0j;nts; its own outcome from last period yi;n;t�1, and its neighbors�outcomes from the last periody0j;n;t�1s The spatial weight wij;nt is a measure of relative strength of linkage between agents i and j at timet, However, this weight wij;nt is not predetermined but depends on some observable random variable Znt:Wecan think of zi;nt as some economic variables at location i and time t such as GDP, consumption, economicgrowth rate, etc, which in�uence strength of links across units. We have the following assumptions.
Assumption 1 The lattice D � Rd0 , d0 � 1, is in�nitely countable. All elements in D are located atdistances of at least dis0 > 0 from each other, i.e., 8i; j 2 D : �ij � �0, where �ij is the distance betweenlocations 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 � i:i:d:(0;�v"), where
�v" =
��2v �0v"�v� �"
�is positive de�nite, �2v is a scalar variance, covariance �v" = (�v"1 ; :::�v"p2 )
0 is a p
dimensional vector, and �" is a p � p matrix. The supi;n;tEjvi;ntj4+�" and supi;n;tEjj"i;ntjj4+�" exist forsome �" > 0. Furthermore, E(vi;ntj"i;nt) = "0i;nt� and V ar(vi;ntj"i;nt) = �2�.
The endogeneity of Wnt comes from the correlation between vi;nt and "i;nt. If �v" is zero, the spatialweight matrix Wnt might be treated as strictly exogenous and we can apply conventional methodology ofspatial panel data models for estimation. However, if �v" is not zero, Wnt becomes an endogenous spatialweights matrix.From the two conditional moments assumptions in Assumption 2, we have the p dimensional column
vector � = ��1" �v" and the scalar �2� = �2v � �0v"��1" �v". Denote �nt = Vnt� "nt�, then its mean conditional
on "nt is zero and its conditional variance matrix is �2�In. In particular, �nt are uncorrelated with the termsof "nt and the variance of �nt is �
2�0In. The outcome equation (1) becomes
Ynt = �1WntYnt + �2Wn;t�1Yn;t�1 + �1Yn;t�1 +X1nt� + cn + (Znt � Zn;t�1�2 �X2nt�� dn)� + �nt; (3)
3
with E(�i;ntj"i;nt) = 0 and V ar(�i;ntj"i;nt) = �2� ; and �i;n�s are i.i.d. across i. Our subsequent asymp-totic analysis will mainly rely on the di¤erenced equation of (3), where (�Znt;�Zn;t�1;�X2nt) are controlvariables 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,
then vi;ntj"i;nt � i:i:d:N("0i;nt��1" �v"; �2v � �0v"��1" �v") and �nt is independent of "nt in equation (2).
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 � 1),�X2n(T ))��2�
�+�"n(T ) = Kn
��2�
�+�"n(T );
where
�Zn(T ) =
0BBB@�Zn;3�Zn;4...
�Zn;T
1CCCA ,�Zn(T�1) =0BBB@
�Zn;2�Zn;3...
�Zn;T�1
1CCCA ;�X2n(T ) =0BBB@�X2n;3�X2n;4...
�X2n;T
1CCCA ;�"n(T ) =0BBB@�"n;3�"n;4...
�"n;T
1CCCA :A consistent estimator is derived from�b�2b�
�= [K 0
nq1n(q01nq1n)
�1q01nKn]�1K 0
nq1n(q01nq1n)
�1q01n�Zn(T );
where q1n is a valid instrument of Kn, for example q1n = (�X2n(T � 1);�X2n(T )).Denote Tn = [�(WY )n(T );�(WY )n(T � 1);�Yn(T � 1);�X1n(T );�"n(T )] and � = (�1; �2; �1; �0; �)0.
Then in the 2nd step, we want to estimate
�Yn(T ) = [�(WY )n(T );�(WY )n(T � 1);�Yn(T � 1);�X1n(T );�"n(T )](�1; �2; �1; �0; �)0 +��n(T )= Tn�+��n(T ):
However, �"n(T ) in Tn is unknown. To solve this, we use
�b"n(T ) = �Zn(T )��Zn(T � 1)b�2 ��X2n(T )b�i.e., the regression residual from the 1st step as an approximation of �"n(T ) and estimate � by
b� = [bT 0nq2n(q02nq2n)�1q02n bTn]�1 bT 0nq2n(q02nq2n)�1q02n�Yn(T );4
where bTn = [�(WY )n(T );�(WY )n(T � 1);�Yn(T � 1);�X1n(T );�b"n(T )] with�(WY )n(T ) =
0BBB@Wn3Yn3 �Wn2Yn2Wn4Yn4 �Wn3Yn3
...WnTYnT �Wn;T�1Yn;T�1
1CCCA , �(WY )n(T � 1) =0BBB@
Wn2Yn2 �Wn1Yn1Wn3Yn3 �Wn2Yn2
...Wn;T�1Yn;T�1 �Wn;T�2Yn;T�2
1CCCAand q2n is a valid instrument of bTn, for example q2n = [�(WX1)n(T );�(WX1)n(T � 1);�X1n(T �1);�X1n(T );�b"n(T )].3.2 Asymptotic Property
To analyze the asymptotic properties of our 2SIV estimator of the dynamic spatial panel data model, weneed 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, wherehijt(�)�s are non-negative, uniformly bounded functions of some observable variable Znt. Furthermore,wii;nt = 0, supn;t jjWntjj1 = cw <1, and supn;t jjWntjj1 = 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 Euclideanspace Rk� , where � is a vector of distinct parameters in �� and k� = k1+4+k2p+p2+p; k1 is the dimensionof �, p is the dimension of �, k2p is the number of parameters in �; and p2 is the number of parameters in�2. In this set, �
2� > 0 and �" is positive de�nite. The true parameter �0 is contained in the interior of �.
Furthermore, �10, �20, and �0 satis�y that j�10jcw < 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
are deterministic and bounded in absolute value. 1nX
0ntXnt 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 �t0th 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�edby (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 byconditioning on them;(iii) if Yn;�t0 is endogenous, it has a cross-sectional spatial model speci�cation as
Yn;�t0 = �10Wn;�t0Yn;�t0 +X1n;�t0�0 + con0 + Vn;�t0 with Zn;�t0 = X2n;�t0� + d
on0 + "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 structureof Wnt.
Assumption 4 The spatial weights wij;nt = hijt(zi;nt; zj;nt) are bounded by some exogenous bounds. Eitherof the following two cases is satis�ed.
5
4.1) 0 � wij;nt � c1�ij�c3d0 for some 0 � c1 and c3 > 32 . Furthermore, in any period t, there exist atmost K (K � 1) columns of Wnt that the column sum exceeds cw, where K is a �xed number that does notdepend n.4.2) The spatial weight wij;nt = 0 if �ij > �c, i.e., there exists a threshold �c > 1 and if the ge-
ographic distance exceeds �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)=
P�ik��c hikt(zi;nt; zk;nt), where hijt(�)�s are non-negative, uniformly bounded functions.
This assumption is adopted in Qu and Lee (2015) to show the consistency and asymptotic normality ofthe 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 thespatial 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 Euclid-ean norm of Y: Denote Fi;n(s) as a �-�eld generated by the random vectors &j;n�s located within the ballBi(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 withjjHi;njjp <1; p � 1, where Dn � D and jDnj ! 1 as n!1, and let d = fdi;n; i 2 Dn; n � 1g be an arrayof �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;njFi;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, andthe di;n�s are called NED scaling factors. H is said to be Lp-NED on & of size �� if '(s) = O(s��) forsome � > � > 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:Denote P1n = q1n(q01nq1n)
�1q01n and P2n = q2n(q02nq2n)
�1q02n. Then, we have
b�� �0 = (bT 0nP2n bTn)�1 bT 0nP2n��n(T ) + ( bT 0nP2n bTn)�1 bT 0nP2nKn(K0nP1nKn)
�1K 0nP1n�"n(T )�0:
Details can be found from Appendix. For the consistency, it is su¢ cient to show that 1nq
02nbTn, 1
nq02nq2n,
1nq
02nKn, 1nq
01nKn, 1nq
01nq12n are Op(1),
1nq
02n��n(T ) = op(1); and
1nq
01n�"n(T ) = op(1):
We can express the reduced form of terms in Kn and bTn. For example,�(WY )nt = Gnt
t+t0Xh=0
B(h)nt (X1n;t�h�0 + cn0 + "n;t�h�0 + �n;t�h) +G
�1nt B
(t+t0)nt (con0 � cn0)
�Gn;t�1t�1+t0Xh=0
B(h)n;t�1(X1n;t�1�h�0 + cn0 + "n;t�1�h�0 + �n;t�1�h)�Gn;t�1B
(t�1+t0)n;t�1 (con0 � cn0);
2As c��ij0 decreases faster than ��c3d0ij , all the results hold for the case of 0 � wdij;n � c1c
��ij0 with some c1 � 0 and c0 > 1.
6
with
B(h)nt = (�0S
�1n;t�1+�20Gn;t�1)(�0S
�1n;t�2+�20Gn;t�2) � � � (�0S�1n;t�h+�20Gn;t�h) =
hYk=1
(�0S�1n;t�k+�20Gn;t�k)
and B(0)nt = In; where Snt = In � �1Wnt and Gnt = WntS�1nt . Therefore, in general, we want to show terms
like1
n
TXt=1
TXs=1
[U0nt;H1
Uns;H2� E(U0
nt;H1Uns;H2
)]p! 0;
where Unt;H = AntPH
h=0B(h)nt &n;t�hb; where &i;nt = fi(&
�i;nt; Xnt) with &
�nt = ("nt; �nt) being a vector-valued
function, Ant can be either Gnt or S�1nt , and b is any conformable vector of constants.
To show the consistency and asymptotic normality of the 2SIV estimator b�, in addition to the convergenceof each separated term, we need some rank conditions on relevant limiting matrices.
Assumption 5 5.1) limn!11nE(q
01nq1n) and limn!1
1nE(q
02nq2n) exists and are nonsingular.
5.2) limn!11nE(K
0nq1n) and limn!1
1nE(T
0nq2n) have full column rank.
Denote
0 =
0BBBBB@2�1 20 �1 2
0 0. . .
. . .0 0 0 �1 2
1CCCCCA(T�2)�(T�2)
:
Theorem 1 Under Assumptions 1-5, the 2SIV estimator b� is a consistent estimator of �0. Furthermore,pn (b�� �0) d! N(0;��0), where
��0 = plimn!1
1
n(T 0nP2nTn)
�1T 0nP2n�nP2nTn(T0nP2nTn)
�1; with
�n = �2�00 In +Kn(K0nP1nKn)
�1K 0nP1n(�
00�"0�00 In)P1nKn(K
0nP1nKn)
�1K 0n:
The estimated variance-covariance matrix isb�� =1
n( bT 0nP2n bTn)�1 bT 0nP2nb�nP2n bTn( bT 0nP2n bTn)�1; withb�n = Kn(K
0nP1nKn)
�1K 0nP1n(
b�0b�"b�0 In)P1nKn(K0nP1nKn)
�1K 0n + (b�2�0 In);
where b�" = 1
2n(T � 2)�b"n(T )0�b"n(T ); b�2� = 1
2n(T � 2)�b�n(T )0�b�n(T ):
Its consistency can be found in the following claim.
Claim 3.2.1 Suppose�b�2b� � is a consistent estimator of ��0�0�, then b�" = 1
2n(T�2)�b"n(T )0�b"n(T ) is a con-sistent estimator of �"0, where �b"n(T ) = �Zn(T ) � Kn
�b�2b� �: Similarly, if b� = (b�1; b�2;b�1; b�0;b�)0 is a con-sistent estimator of �0, then b�2� = 1
2n(T�2)�b�0n(T )�b�n(T ) is a consistent estimator of �2�0, where �b�n =
�Yn(T )� bTnb�. Furthurmore, if we replace �0 with a consistent estimator b�; �"n(T ) with �b"n(T ); �"0 withb�", and �2�0 with b�2�, to obtain the empirical estimate of b��, then b�� p! ��0.
7
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 endoge-nous spatial weight matrices. The data generating process (DGP) is
Ynt = (In � �1Wnt)�1(�2Wn;t�1Yn;t�1 + �1Yn;t�1 +X1nt� + cn + Vnt);
with the initial valueYn;�t0 = �10Wn;�t0Yn;�t0 +X1n;�t0�0 + c
on0 + 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.
2. Construct the spatial weight matrix as the Hadamard product Wnt = Wdn �W e
nt, i.e., wij;nt = wdij;n �
weij;nt, whereWdn is a predetermined matrix based on geographic distance (not time-varying): w
dij;n = 1
if the two locations are neighbors and otherwise 0; W ent is a time-varying matrix based on economic
similarity: weij;nt = 1=jzi;nt � zj;ntj if i 6= j and weii;nt = 0; where elements of zi;nt is generated byzi;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 , whereX2n � N(0; 1), di;n � U(0; 1), �2 = 0:7; and = 0:8.
3. Row-normalize Wn.
For the predeterminedW dn , we use four examples. First, the U.S. states spatial weight matrixWS(49�49);
based on the contiguity of the 48 contiguous states and D.C.; second, the Toledo spatial weight matrixWO(98�98); based on the 5 nearest neighbors of 98 census tracts in Toledo, Ohio; third, the Iowa �County"spatial weight matrix WC(361� 361), based on whether the school districts are in the same county in Iowain 2009; and lastly, the Iowa �Adjacency" spatial weight matrix WA(361� 361), based on the adjacency of361 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 (Zntequation is not estimated) since it treats Wnt as exogenous. For the simulation, we choose �(WX)nt toinstrument �(WY )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, �(WX)nt,�(WX)n;t�1, and �Xn;t�1 are used as instruments of �(WY )nt, �(WY )n;t�1, and �Yn;t�1 to estimatethe Ynt equation. Of particular interest, we want to see how large the bias is for this conventional estimationmethod when Wnt is indeed endogenous. To generate di¤erent degrees of endogeneity, we choose correlationcoe¢ cients � = 0:2, 0:5, and 0:8. We also let the spatial correlation to be � = 0:2 and 0:4 to investigate howthe 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
WS(49)� = 0:2 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:2IV 0:5069 0:0844 0:2048 0:9988
(0:0811) (0:1025) (0:0784) (0:0692)[0:0835] [0:1067] [0:0804] [0:0688]
2SIV 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]
� = 0:5 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:5IV 0:5251 0:0685 0:2052 0:9922
(0:0813) (0:1052) (0:0795) (0:0696)[0:0834] [0:1054] [0:0811] [0:0697]
2SIV 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]
� = 0:8 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:8IV 0:5432 0:0490 0:2064 0:9845
(0:0817) (0:1078) (0:0808) (0:0701)[0:0801] [0:1094] [0:0835] [0:0700]
2SIV 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]
Note: Observations n = 49: Estimated standard error based on an asymptotic variance-covariance matrix isin parentheses; and empirical standard deviation is in brackets.
9
Table 2: Estimates from the Ohio State spatial weight matrix
WO(98)� = 0:2 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:2IV 0:5130 0:0812 0:2022 0:9950
(0:0615) (0:0764) (0:0549) (0:0483)[0:0609] [0:0747] [0:0528] [0:0476]
2SIV 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]
� = 0:5 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:5IV 0:5360 0:0602 0:2029 0:9874
(0:0623) (0:0792) (0:0559) (0:0488)[0:0621] [0:0764] [0:0544] [0:0483]
2SIV 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]
� = 0:8 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:8IV 0:5632 0:0373 0:2039 0:9767
(0:0627) (0:0820) (0:0571) (0:0492)[0:0631] [0:0813] [0:0563] [0:0491]
2SIV 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]
Note: Observations n = 98: Estimated standard error based on an asymptotic variance-covariance matrix isin parentheses; and empirical standard deviation is in brackets.
10
Table 3: Estimates from the Iowa State county spatial weight matrix
WC(361)� = 0:2 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:2IV 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 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:5IV 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 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:8IV 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 matrixis in parentheses; and empirical standard deviation is in brackets.
11
Table 4: Estimates from the Iowa State adjacency spatial weight matrix
WA(361)� = 0:2 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:2IV 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 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:5IV 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 �1 = 0:5 �2 = 0:1 �1 = 0:2 � = 1 �2 = 0:7 = 0:8 � = 0:8IV 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 matrixis in parentheses; and empirical standard deviation is in brackets.
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 empiricalstandard deviation of the estimator (in brackets) based on 1000 replications using WS; WO; WC; or WAas the predetermined spatial weight matrices. To see how the di¤erent estimation methods react to themagnitude of endogeneity, we have three sets of results. In each table, the upper panel shows the resultswith � = 0:2; the middle panel is for � = 0:5, and the lower panel is for � = 0:8, so the endogeneity isincreasing.There are several patterns in the estimation results:
1. For the accuracy of estimation, we consider the empirical standard deviation based on 1000 replicationsand the estimated standard error based on the asymptotic variance-covariance matrix. These two valuesare very close in all cases. Compared across di¤erent estimation methods, accuracy of IV is very closeto 2SIV. In the case of WS, when the endogeneity is high, 2SIV estimation has a slightly smallervariance. 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, themore bias IV method creates. Most bias comes from the estimator of the spatial correlation coe¢ cients�1 and �2, especially �2. In general, estimators for � and �1 are much less biased. b�2 from IV su¤erssevere downward bias when � achieves 0:5, in some cases exceeding 100%. Correspondingly, b�1 from IVhas the upword bias, but the bias is not as big as b�2. Although biases of � and �1 in the conventionalIV method can be neglectable, interpetation of these coe¢ cients signi�cantly depends on ��s. Also theexistense of spatial correlation indicates the multiplier e¤ects. In all cases, 2SIV has estimates veryclose 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 growsin a slower rate, we observe that the bias and standard error of estimators decrease. When the samplesize is 361, such as in the case of WA and WC, the estimates appear quite precise and close to thetrue parameters.
5 An empirical application
In this section, we provide an illustration of the use of our method. We employ the dynamic spatial panel datamodel to investigate the �scal interactions among the 91 non-oil countries used in Mankiw et al. (1992) andErtur and Koch (2007). Now, we will give some explanations why we use such a model to estimate the �scalreaction function. Suggested by Michela (2007), we believe that state�s �scal policies are dependent on theirneighbour�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 samepolicy 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 taxbases. 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
3We 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 inderdepedenceamong 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 theadjusted cost caused by the abrupt change in �scal system or objection from the interest group at the politicallevel. Similarly, we also want to know whether the serially correlated e¤ect of other countries will a¤ect thepolicy makers�decision. So, that is why we incorporate the last lagged period spatial term. In sum, it isreasonable 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 theweight matrix based on both �physical� and �economic� distance: inverse of the product of geographicdistance and distance between GDP per capita. In the �rst step, we estimate the growth equation by usingdynamic panel data model. Then, by noting equation (3), so we can study the �scal interdependence amongstates 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.
lnGit = �i + � lnGi;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., 1965-2010and the initial period is 1960. x1it is a list of country i�s characteristics at time period t including variablesmeasuring real income, investment and population, suggested by Ertur and Koch (2007). We list the speci�cvariables of x1it in Table (5). By using the estimated result �b"it from this step and noting equation (3), wecan estimate the regression model in the second step.Next, in the second step, we consider the following dynamic spatial panel data model:
Eit = ci + �Ei;t�1 + �1Xj 6=i
wijtEjt + �2Xj 6=i
wij;t�1Ej;t�1 + �>x2it + vit; t = 1; 2; :::; 10;
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 matrixWnt and Wn;t�1 respectively. x2it is a set of variables which are conventionally assumed to a¤ect thedetermination of the �scal choices. We use the speci�c variables in Michena (2007) and also list them inTable (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 canrewrite 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
wijt =
8<:1
jGit�GjtjdijPj
1jGit�Gjtjdij
; if j 6= i
0; Otherwise;
where dij denotes the geographic distance between county i and country j. Alternatively, we can constructthe weight matrix as follows
wijt =
8<:1
jGit�GjtjPj
1jGit�Gjtj
; if j 6= i and i, j country share a border
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 ifCountry 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 countriesin the supplemental materials. And all of the data we use are collected from World Bank-World DevelopmentIndicator.
6 Conclusion
In this paper, we consider the speci�cation and estimation of a dynamic spatial panel data model withendogenous spatial weight matrices. First, we specify two sets of equations: one is for the dynamic outcomeinvolving spatial correlation, and the other is for entries of the spatial weight matrices. The source of endo-geneity is the correlation between the disturbances in the dynamic spatial outcome equation and the errorsin 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 employingthe theory of asymptotic inference under near-epoch dependence. We consider two types of spatial weightmatrices: one is sparse and another one has its entries decreasing su¢ ciently fast as the physical distanceincreases. To examine the behavior of our proposed estimator in �nite samples, we conduct a Monte Carlosimulation study. The simulation results indicate that the commonly used estimates under exogenous weightmatrix su¤er serious downward bias when the true weight matrix is endogenous. On the other hand, our2SIV estimates have good �nite sample properties. As the sample size increases and the number of neighborsgrows 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 endo-
geneity 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 islong. This could be a technical challenging issue as the near-epoch assumption needs to extend to the spaceand time dimension.
Appendix
The model
Based on the conditional moment conditions in Assumption 2, our model can be written as
Ynt = �1WntYnt + �2Wn;t�1Yn;t�1 + �1Yn;t�1 +X1nt� + cn + "nt� + �nt; (4)
where "nt = Znt�Zn;t�1�2�X2nt��dn: Taking di¤erence to eliminate the individual �xed e¤ects, we have
�Ynt = �1�(WY )nt + �2�(WY )n;t�1 + �1�Yn;t�1 +�X1nt� + (�"nt)� + �nt; (5)
where �"nt = �Znt ��Zn;t�1�2 ��X2nt�:In the 1st step, we estimate
�Zn(T ) = (�Zn(T � 1),�X2n(T ))��2�
�+�"n(T ) = Kn
��2�
�+�"n(T )
15
by �b�2b��= (K 0
nP1nKn)�1K 0
nP1n�Zn(T ) =
��0�0
�+ (K 0
nP1nKn)�1K 0
nP1n�"n(T );
where P1n = q1n(q01nq1n)�1q01n with q1n being a valid instrument, for example q1n = (�X2n(T�1);�X2n(T )).
In the 2nd step, we treat
�b"n(T ) = �Zn(T )��Zn(T � 1)b�2 ��X2n(T )b�= �"n(T )�Kn[K
0nq1n(q
01nq1n)
�1q01nKn]�1K 0
nq1n(q01nq1n)
�1q01n�"n(T )
as an approximate of �"n(T ) and estimate the equation
�Yn(T ) = [�(WY )n(T );�(WY )n(T � 1);�Yn(T � 1);�X1n(T );�"n(T )](�1; �2; �1; �0; �)0 +��n(T )= Tn�+��n(T ) = bTn�+Kn(K
0nP1nKn)
�1K 0nP1n�"n(T )�0 +��n(T );
whereTn = [�(WY )n(T );�(WY )n(T � 1);�Yn(T � 1);�X1n(T );�"n(T )]
and bTn = [�(WY )n(T );�(WY )n(T � 1);�Yn(T � 1);�X1n(T );�b"n(T )]:If q2n is choosen to be the instrument variable and denote P2n = q2n(q02nq2n)
�1q02n, then b� = ( bT 0nP2n bTn)�1 bT 0nP2n�Yn(T ).Some useful lemmas
Let &�i;n = fi("i;n; �i;n; Xn; �0). As Xn is deterministic, &�i;n is purely determined by the location i, indepen-
dent 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�d0�1 points in thespace Bi(�+ 1)nBi(�), where c4 and c5 are positive constants.
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 � 1g is L1-NED, the base f&i;ng�s arei.i.d., and fTi;ng�s are uniformly Lp bounded for some p > 1, then 1
n
Pni=1(Ti;n � ETi;n)
L1! 0:
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 &. IfAssumption 1 and the following conditions are met:(1) fTi;n; i 2 Dn; n � 1g is uniformly L2+�-bounded for some � > 0;(2) infn 1
n�2n > 0 where �
2n = V ar(
Pni=1 Ti;n);
(3) NED coe¢ cients satisfyP1
r=1 rd0�1'(r) <1;
(4) NED scaling factors satisfy supn;i2D di;n <1,then ��1n
Pni=1(Ti;n � ETi;n)
d! N(0; 1).
Claim 6.0.1 If for any t; r = 1; :::T , jjt1i;nt�E(t1i;ntjFi;n(s))jj4 � C1'1(s) and jjt2i;nr�E(t2i;nrjFi;n(s))jj4 �C2'2(s); with maxi;n;t(jjt1i;ntjj4; jjt2i;ntjj4) � C, then jjt1i;ntt2i;nr � E(t1i;ntt2i;nrjFi;n(s))jj2 � 2C(C1 +C2)'(s); where '(s) = max('1(s); '2(s)):
16
Consider the product of t1i;ntt2i;nr,
jjt1i;ntt2i;nr � E(t1i;ntt2i;nrjFi;n(s))jj2 � jjt1i;ntt2i;nr � E(t1i;ntjFi;n(s))E(t2i;nrjFi;n(s))jj2� jjt2i;nr[t1i;nt � E(t1i;ntjFi;n(s))]jj2 + jjE(t1i;ntjFi;n(s))[t2i;nr � E(t2i;nrjFi;n(s))]jj2� jjt2i;nrjj4 � jjt1i;nt � E(t1i;ntjFi;n(s))jj4 + jjt1i;ntjj4 � jjt2i;nr � E(t2i;nrjFi;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, jjqQl=1
Wn;tl jj1 � (q �
1)cu(L+ T )Kcq�1w + cuc
q�1w � qcu(L+ T )Kcq�1w , where cu = supn;t jjWntjj1 and cw = supn;t jjWntjj1.
Proof of Claim 6.0.2. Denote an index set Vn with at least one period t satis�es cw �Pn
j=1 wji;nt < cuif i 2 Vn and
Pnj=1 wji;nt < cw for all t if i =2 Vn. Then Assumption 6.1) constrains that jVnj � (L + T )K
for any n, where (L + T ) is the total number of periods. Consider the kth column sum ofqQl=1
Wn;tl , i.e.,
e0nqQl=1
Wn;tlek;n, where en = (1; :::; 1)0 and ek;n is the unit column vector with one in its kth entry and zeros
in its other entries. As In =Pn
i=1 ei;ne0i;n,
e0n
qYl=1
Wn;tlek;n =
nXi=1
e0nWn;t1ei;ne0i;n
qYl=2
Wn;tlek;n
=Xi2Vn
e0nWn;t1ei;ne0i;n
qYl=2
Wn;tlek;n +Xi=2Vn
e0nWn;t1ei;ne0i;n
qYl=2
Wn;tlek;n
� (L+ T )K
�maxi2Vn
e0nWn;t1ei;n
� maxi2Vn
e0i;n
qYl=2
Wn;tlek;n
!+
�maxi=2Vn
e0nWn;t1ei;n
� Xi=2Vn
e0i;n
qYl=2
Wn;tlek;n
� (L+ T )KcujjqYl=2
Wn;tl jj1 + cwjjqYl=2
Wn;tl jj1 � (L+ T )Kcucq�1w + cwjjqYl=2
Wn;tl jj1:
Since this inequality holds for any k = 1; :::; n, we have jjqQl=1
Wn;tl jj1 � cu(L + T )Kcq�1w + cwjjqQl=2
Wn;tl jj1:
By deduction, we have jjqQl=1
Wn;tl jj1 � (q � 1)cu(L+ T )Kcq�1w + cucq�1w � qcu(L+ T )Kcq�1w .
Claim 6.0.3 Suppose Wn;l is an n � n square matrix which can be decomposed into the sum of two n � nmatrices such that Wn;tl = An;tl+Bn;tl . Denote jAjmax = maxfjaij;nlj : i; j = 1; :::; ng: Then for any positiveinteger k and any i; j = 1; :::; n,
(kYl=1
Wn;tl �kYl=1
Bn;tl)ij � jAjmaxk�1Xm=0
jjmYl1=1
Bn;tl1 jj1 � jjkY
l2=m+2
Wn;tl2jj1:
17
Proof of Claim 6.0.3. By expansion,
kYl=1
Wn;tl �kYl=1
Bn;l = (An;t1 +Bn;t1)kYl=2
Wn;tl = An;t1
kYl=2
Wn;tl +Bn;t1An;t2
kYl=3
Wn;tl + :::+k�1Yl=1
Bn;tlAn;tk
=k�1Xm=0
mYl1=1
Bn;tl1
!An;tm+1
kY
l2=m+2
Wn;tl2
!:
Denote ei;n = (0; :::0; 1; 0; :::; 0)0, which is the ith unit vector of order n; then (kQl=1
Wn;tl �kQl=1
Bn;l)ij =
Pk�1m=0 e
0i;n
mQl1=1
Bn;tl1
!An;tm+1
kQ
l2=m+2
Wn;tl2
!ej;n. For any matrix M and vector e of dimension n, it is
easy to see that jjMejj1 � jM jmaxjjejj1: Thus, for any integer m = 0; :::; k � 1,
e0i;n
mYl1=1
Bn;tl1
!An;tm+1
kY
l2=m+2
Wn;tl2
!ej;n � jje0i;n
mYl1=1
Bn;tl1
!jj1 � jjAn;tm+1
kY
l2=m+2
Wn;tl2
!ej;njj1
� jjmYl1=1
Bn;tl1 jj1 � jAn;tm+1
kYl2=m+2
Wn;tl2jmax � jAjmax � jj
mYl1=1
Bn;tl1 jj1 � jjkY
l2=m+2
Wn;tl2jj1
Together, we have the result.
Claim 6.0.4 Let ti;nt(m) be the ith element of the vectorYm
l=1Wn;tl&
�n;tma, where &
�i;nt = fi(&i;nt; Xnt) with
&nt = ("nt; �nt) is a vector-valued function and a is any conformable vector of constants. Under Assumptions1, 3.1), and 4.1), suppose supi;n;t jj&�i;ntjjp <1, then supi;n;t jjti;nt(m)jjp � mc3d0+2cmwCap andsupi;n;t jjti;nt(m)� E(ti;nt(m)jFi;n;T (s))jjp � Capcmwm3+c3d0s(2�c3)d0 with Cap being a �nite constant.
Proof. Note that for any integer 1 � k � m and ik =2 Bi(s), we can show that
(kYl=1
Wn;tl)iik � C0mc3d0+2ckw��c3d0iik
andX
ik =2Bi(s)
(kYl=1
Wn;tl)iik � C1ckwmc3d0+2s(1�c3)d0 (6)
with C0 and C1 being positive constants, not depend on k. To show this, we construct two matrices An;tl andBn;tl as follows: aij;ntl = wij;ntlI(wij;ntl � c1(�iik=m)
�c3d0) and bij;ntl = wij;ntlI(wij;ntl > c1(�iik=m)�c3d0),
then Wn;tl = An;tl +Bn;tl and aij;ntlbij;ntl = 0: As
e0i;n
kYl=1
Wn;tleik;n =nX
i1=1
nXi2=1
� � �nX
ik=1
wii1;nt1wi1i2;nt2 � � �wik�1ik;ntk ;
if ik =2 Bi(s), at least one of the items wii1;nt1 , wi1i2;nt2 ; � � � ; wik�1ik;ntk , say wiq�1iq;ntq , satis�es thatwiq�1iq;ntq � c1(�iik=k)
�c3d0 � c1(�iik=m)�c3d0 , because there exist at least two neighboring nodes in
the chain i ! i1 ! � � � ! ik such that their distance is at least �iik=k. Hence, (Yk
l=1Bn;tl)iik =
18
Pi1� � �P
ik�1wii1;nt1wi1i2;nt2 � � �wik�1ik;ntkI(all w�s> c1(�iik=m)
�c3d0) = 0, and we have
(kYl=1
Wn;tl)iik = (kYl=1
Wn;tl �kYl=1
Bn;tl)iik � supn;tl
jAn;tl jmaxk�1Xq=0
jjqY
l1=1
Bn;l1 jj1 � jjkY
l2=q+2
Wn;l2 jj1
� c1(�iikm)�c3d0
k�1Xq=0
cqw(k � 1� q)K(L+ T )cuck�q�2w
� K(L+ T )cuc1(�iik=m)�c3d0k2ck�2w � C0mc3d0+2ckw�
�c3d0iik
;
where the �rst inequality is from Claim 6.0.3; the second one is from Claim 6.0.2 and all elements in An;tlare � c1(�iik=m)
�c3d0 . And hence, for any i;
Xik =2Bi(s)
(kYl=1
Wn;tl)iik � C0mc3d0+2ckwX
ik =2Bi(s)
��c3d0iik� C0mc3d0+2ckwc4
1X�=[s]
�(1�c3)d0�1 � C1ckwmc3d0+2s(1�c3)d0 :
Any element of e0i;nYm
l=1Wn;tl&
�n;tma has a chain of w�s starting from i and ending in im with m steps.
We can divide these chains into two sets: one set has all its paths staying within Bi(s), and the other sethas 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:
(i)P
im =2Bi(s)(Ym
l=1Wn;tl)iim&
�im;ntm
a;
(ii)P
im�1 =2Bi(s)
Pim2Bi(s)
(Ym�1
l=1Wn;tl)iim�1wim�1im;ntm&
�im;ntm
a; etc.Such subset can be written as
Xim�k =2Bi(s)
Xim�k+12Bi(s)
� � �X
im2Bi(s)
(
m�kYl=1
Wn;tl)iim�kwim�kim�k+1;ntm�k+1� � �wim�1im;ntm&
�im;ntma
for k = 0; :::m� 1.Consider (i): As
jX
im =2Bi(s)
(mYl=1
Wn;tl)iim&�im;ntmaj �
Xim =2Bi(s)
(mYl=1
Wn;tl)iim j&�im;ntmaj �X
im =2Bi(s)
j&�im;najC0mc3d0+2cmw �
�c3d0iim
;
we have
jjX
im =2Bi(s)
(Wmn )iim&
�im;ntmajjp � capC0m
c3d0+2cmw
1X�=[s]
c4�(1�c3)d0�1 � C2capmc3d0+2cmw s
(1�c3)d0 ; (7)
where cap = supi;n;t(Ej&i;ntajp)1=p. For (ii),
jX
im�1 =2Bi(s)
Xim2Bi(s)
(m�1Yl=1
Wn;tl)iim�1wim�1im;ntm j � jjWn;tm jj1X
im�1 =2Bi(s)
(m�1Yl=1
Wn;tl)iim�1 � cmwmc3d0+2s(1�c3)d0C1;
19
and hence,
jjX
im�1 =2Bi(s)
Xim2Bi(s)
(m�1Yl=1
Wn;tl)iim�1wim�1im;ntm&�im;ntmajjp � C1c
mwm
c3d0+2s(1�c3)d0X
im2Bi(s)
cap
� C1cmwmc3d0+2s(1�c3)d0c5sd0cap � C3cmwmc3d0+2s(2�c3)d0cap by (6).
And similarly, for subset (k) with im�k =2 Bi(s) where 1 � k � m� 1; we have
Xim�k =2Bi(s)
Xim�k+12Bi(s)
� � �X
im2Bi(s)
(m�kYl=1
Wn;tl)iim�kwim�kim�k+1;ntm�k+1� � �wim�1im;ntm
� jjmY
l=m�k+1Wn;tl jj1
Xim�k =2Bi(s)
(m�kYl=1
Wn;tl)iim�k � cmwmc3d0+2s(1�c3)d0C1 as jjmY
l=m�k+1Wn;tl jj1 � ckw:
Hence,
jjX
im�k =2Bi(s)
Xim�k+12Bi(s)
� � �X
im2Bi(s)
(m�kYl=1
Wn;tl)iim�kwim�kim�k+1;ntm�k+1� � �wim�1im;ntm&
�im;ntmajjp
� C3cmwm
c3d0+2s(2�c3)d0cap:
These together imply
jjti;nt(m)� E(ti;nt(m)jFi;n(s))jjp � 2jjpaths in ti;nt(m) with at least one node im�k =2 Bi(s)jjp
� 2 C2m
c3d0+2cmw s(1�c3)d0cap + C3
m�1Xk=1
cmwmc3d0+2s(2�c3)d0cap
!� cmwmc3d0+3s(2�c3)d0Cap:
To conclude, for any integerm, jjti;nt(m)�E(ti;nt(m)jFi;n(s))jjp � Capcmwm3+c3d0'(s) with '(s) = s(2�c3)d0 .Now we show jjti;nt(m)jjp � mc3d0+2cmwCap. 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
jjX
im2Bi(�+1)nBi(�)
(mYl=1
Wn;tl)iim&�im;ntmajjp � C2capm
c3d0+2cmw �(1�c3)d0 :
For Bi(1); there are two cases: im = i and im 6= i: For the case im = i, we have jj(Ym
l=1Wn;tl)ii&
�i;najjp �
capcmw : For the case im 6= i, it must be �iim = 1 from Assumption 1. Hence, jj
Pim;�iim=1
(Ym
l=1Wn;tl)iim&
�im;n
ajjp �capc5c
mw : Since
ti;nt(m) = e0i;n
mYl=1
Wn;tl&�ntma =
1X�=1
Xim2Bi(�+1)nBi(�)
(
mYl=1
Wn;tl)iim&�im;ntma+
Xim2Bi(1)
(mYl=1
Wn;tl)iim&�im;ntma;
20
we have
jjti;nt(m)jjp �1X�=1
jjX
im2Bi(�+1)nBi(�)
(mYl=1
Wn;tl)iim&�im;ntmajjp + jj
Xim2Bi(1)
(mYl=1
Wn;tl)iim&�im;ntmajjp
� C2capmc3d0+2cmw
1X�=1
�(1�c3)d0 + capc5cmw + c
mw cap � mc3d0+2cmwCap:
Claim 6.0.5 Let gi;nt(m) = e0i;nYm
l=1Gn;tl(�1)&
�n;tma, where &
�nt and a are the same as Claim 6.0.4.
Under Assumptions 1, 3.1), and 4.1), suppose supi;n;t jj&�i;ntjjp < 1, then supi;n;t jjgi;nt(m)jjp < 1 andsupi;n;t jjgi;nt(m)� E(gi;nt(m)jFi;n(s))jjp � Capms(2�c3)d0 with Capm being a �nite constant.
As
mYl=1
Gn;tl(�1) =1Xl1=0
1Xl2=0
� � �1X
lm=0
�l1+l2+���+lm1 W l1+1n;t1 W
l2+1n;t2 � � �W
lm+1n;tm =
1Xq=0
�q +m� 1
m
��q1
q+mYl=1
Wn;tl ;
where q = l1 + l2 + � � �+ 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 �1Xq=0
qmj�q1j � jjti;nt(q +m)jjp � cmwCap1Xq=0
(q +m)m+c3d0+2j�1cwjq <1
and
jjgi;nt(m)� E(gi;nt(m)jFi;n(s))jjp �1Xq=0
qmj�1jqjjti;nt(q +m)� E(ti;nt(q +m)jFi;n(s))jjp
� Cap
1Xq=0
j�1jqcq+mw (q +m)m+3+c3d0s(2�c3)d0 � Capms(2�c3)d0 :
Claim 6.0.6 Denote Unt;H = GntPH
h=0B(h)nt &1n;t�h. Under Assumptions 1, 2.1), 3.1), and 4.1), we have
the NED property of ui;nt, i.e., jje0i;nGntPH
h=0B(h)nt &1n;t�hjjp < Cu1 and
jje0i;nGntHXh=0
B(h)nt &1n;t�h � E(e0i;nGnt
HXh=0
B(h)nt &1n;t�hjFi;n(s))jjp � d'(s)
such that lims!1 '(s) = 0.
Proof. First we show jje0i;nGntB(h)nt &1n;t�h � E(e0i;nGntB
(h)nt &1n;t�hjFi;n(s))jjp � d1'(s) holds for any h =
0; :::H. DenoteYhnl
k=1An;k =
Yl�1
k=1An;k
Yh
k=l+1An;k,
21
Yhnl1;l2
k=1An;k =
Yl1�1
k=1An;k
Yl2�1
k=l1+1An;k
Yl2+1
k=l1+1An;k
Yh
k=l3+1An;k, and so on. As
B(h)nt =
hYk=1
(�0S�1n;t�k + �20Gn;t�k) =
hYk=1
[�0In + (�0�10 + �20)Gn;t�k]
= (�0�10 + �20)h
hYk=1
Gn;t�k + (�0�10 + �20)h�1�0
hXl=1
hnlYk=1
Gn;t�k
+(�0�10 + �20)h�2�20
hXl1=1
Xl2 6=l1
hnl1;l2Yk=1
Gn;t�k + � � �+ (�0�10 + �20)�h�10
hXk=1
Gn;t�k + �h0In;
we have
jje0i;nGntB(h)nt &1n;t�h � E(e0i;nGntB
(h)nt &1n;t�hjFi;n(s))jjp
� j�0�10 + �20jhCap(h+1)s(2�c3)d0 + hj�0�20 + �20jh�1j�0jCaphs(2�c3)d0
+
�h
2
�j�0�20 + �20jh�2j�0j2Cap(h�1)s(2�c3)d0 + � � �
+
�h
h� 1
�j�0�20 + �20j � j�0jh�1Cap(2)s(2�c3)d0 + j�0jhCap(1)s(2�c3)d0
�hXq=0
�h
q
�j�0jqj�0�20 + �20jh�qCapHs(2�c3)d0
= (j�0j+ j�0�10 + �20j)hCapHs(2�c3)d0 ; (8)
where CapH = max0�h�H Caph < 1 is a �nite constant not depending on n: The �rst inequality is from
Claim 6.0.5 that jje0i;nYm
l=1Gn;tl(�1)&
�n;tma�E(e
0i;n
Ym
l=1Gn;tl(�1)&
�n;tmajFi;n;T (s))jjp � Capms
(2�c3)d0 : As8 holds for any h = 0; :::H, we have
jje0i;nGntHXh=0
B(h)nt &1n;t�h � E(e0i;nGnt
HXh=0
B(h)nt &1n;t�hjFi;n(s))jjp
�HXh=0
jje0i;nGntB(h)nt &1n;t�h � E(e0i;nGntB
(h)nt &1n;t�hjFi;n(s))jjp
� CapHs(2�c3)d0
HXh=0
(j�0j+ j�0�20 + �20j)h � Cp;Hs(2�c3)d0 :
Similarly,
jje0i;nGntHXh=0
B(h)nt &1n;t�hjjp �
HXh=0
jje0i;nGntB(h)nt &1n;t�hjjp � CapH
HXh=0
(j�0j+ j�0�20 + �20j)h � Cp;H :
NED properties in the case under Assumption 4.2)
22
Claim 6.0.7 If the i; jth element ofYm
l=1Wn;tl is not zero, then �ij � m�c.
Proof of Claim 6.0.7. The i; jth element ofYm
l=1Wn;tl is
Pi1
Pi2� � �P
im�1wii1;nt1wi1i2;nt2 � � �wim�1j;ntm :
If it is not zero, then there exists at least one path i! i1 ! � � � im�1 ! j such that all wii1;nt1 , wi1i2;nt2 ; � � � ;wim�1j;ntm are positive. As wij;nt = 0 if j =2 Bi(�c) holds for any period t, it must be i1 2 Bi(�c),i2 2 Bi1(�c),..., j 2 Bim�1(�c). Therefore, �ij � �ii1 + �i1i2 + :::+ �im�1j � m�c.
Claim 6.0.8 Under Assumptions 1, 3.1), and 4.2), for any positive integer q, supn;t jjqQl=1
Wn;tl jj1 � cqwc5(q�c)d0 .
Proof of Claim 6.0.8. Consider the kth column sum ofqQl=1
Wn;tl , as all elements in Wnt are non-negative,
e0n
qYl=1
Wn;tlek;n =nX
i2Bk(q�c)
e0i;n
qYl=1
Wn;tlek;n =nX
i2Bk(q�c)
nXj=1
e0i;n
q�1Yl=1
Wn;tlej;ne0j;nWn;tqek;n
� supi;n
������nXj=1
e0i;n
q�1Yl=1
Wn;tlej;n
������ � supj;k;n;tq
nXi2Bk(q�c)
��e0j;nWn;tqek;n�� � cqwc5(q�c)d0 :
The �rst �=� holds because of Claim 6.0.7. The last ��� holds because supn;t jjmQl=1
Wn;tl jj1 � cmw and
supj;k;n;tq��e0j;nWn;tqek;n
�� < cw.Claim 6.0.9 For any positive integer p and 0 < q < 1, if s � p=(� ln q) + 1, then there exists a �niteconstant c such that
Pl=[s] l
pql < cspqs, where [s] denotes the largest integer less than or equal to s.
This is from Qu and Lee (2015).
Claim 6.0.10 Let ti;nt(m) = e0i;nYm
l=1Wn;tl&
�n;tma; where &
�nt and a are the same as Claim 6.0.4. Under
Assumptions 1, 3.1) and 4.2), suppose supi;n;t jj&�i;ntjjp < 1, then supi;n;t jjti;nt(m)jjp � Capmd0cmw 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 ifs � m�c and '(s) = 0 if s > m�c.
Proof of Claim 6.0.10. As
jti;nt(m)j = jXk
e0i;n
mYl=1
Wn;tlek;ne0k;n&
�n;tmaj = j
Xk2Bi(m�c)
e0i;n
mYl=1
Wn;tlek;ne0k;n&
�n;tmaj
� maxk;n;t
je0i;nmYl=1
Wn;tlek;njX
k2Bi(m�c)
j&�k;ntmaj;
we havejjti;nt(m)jjp � cmw
Xk2Bi(m�c)
jj&�k;ntmajjp � cmw c5(m�c)
d0cap = Capcmwm
d0 ;
where cap = supi;n;t jj&�i;ntajjp and Cap = capc5�d0c .
23
Next, we show the NED property. For the spatial weight matrix without row-normalization, wij;nt is afunction of &i;nt and &j;nt, and wij;nt = 0 if j =2 Bi(�c). For the row-normalized case, wij;nt may be relatedto all points in Bi(�c) and in general is a function of &�s at those locations. In both cases, all the locations
of nodes in the chains of e0i;nYm
l=1Wn;tl&
�n;tma related to ti;nt(m) are within the ball Bi(m�c). Hence, when
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 � 2jjti;nt(m)jjp � 2Capcmwmd0 :
Therefore, the NED property follows if we choose '(s) = 1 for s � m�c and '(s) = 0 for s > m�c.
Claim 6.0.11 Denote gi;nt(m) = e0i;nYm
l=1Gn;tl(�1)&
�n;tma, where &
�nt and a are the same as Claim 6.0.4.
Under Assumptions 1, 3.1), and 4.2), suppose supi;n;t jj&�i;ntjjp < 1, then supi;n;t jjgi;nt(m)jjp < 1 andsupi;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�cand '(s) = sd0+mj�cwjs=�c if s > m�c.
Proof of Claim 6.0.11. From the proof of Claim 6.0.5,
mYl=1
Gn;tl(�1) =1Xq=0
�q +m� 1
m
��q1
q+mYl=1
Wn;tl ;
where q = l1 + l2 + � � �+ lm. If �1 = 0, then gi;nt(m) = ti;nt(m) and the Claim follows directly from Claim6.0.10. For �1 6= 0, by Claim 6.0.10, for any i; n, and t; jjti;nt(m+ q)jjp � Capcm+qw (m+ q)d0 and
jjgi;nt(m)jjp � cmwCap1Xq=0
j�1cwjq(q +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 thecase 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 forany nonnegative integer q such that s > (m + q)�c. Such a set of q will be determined by q < ( s�c
�m).Therefore, when s > m�c,
jjgi;nt(m)� E(gi;nt(m)jFi;n(s))jjp
� 21X
q=[ s�c�m]
(q +m)mj�1jqjjti;nt(q +m)jjp � 2Capcmw1X
q=[ s�c�m]
j�1cwjq(q +m)m+d0 ;
where the last inequality follows from Claim 6.0.10. By the inequality in Claim 6.0.9, as s=�c > m, we have
1Xq=[ s�c
�m]
j�1cwjq+m(q +m)m+d0=j�1jm =1X
q=[ s�c]
j�1cwjqqm+d0=j�1jm = O(sm+d0 j�1cwjs=�c):
The Claim would follow if we set '(s) = 1 if s � m�c and '(s) = sd0+mj�1cwjs=�c if s > m�c.
24
Claim 6.0.12 Denote Unt;H = GntPH
h=0B(h)nt &1n;t�h. Under Assumptions 1, 2.1), 3.1), and 4.2), we have
the NED property of ui;nt, i.e., jje0i;nGntPH
h=0B(h)nt &1n;t�hjjp < Cu1 and
jje0i;nGntHXh=0
B(h)nt &1n;t�h � E(e0i;nGnt
HXh=0
B(h)nt &1n;t�hjFi;n(s))jjp � d'(s)
such that '(s) = 1 if s � (H + 1)�c and '(s) = sd0+H j�1cwjs=�c if s > (H + 1)�c.
Proof. Similarly to the proof of Claim 6.0.6, From Claim 6.0.11,
jje0i;nGntHXh=0
B(h)nt &1n;t�hjjp �
HXh=0
jje0i;nGntB(h)nt &1n;t�hjjp � C(H+1)
HXh=0
(j�0j+ j�0�20 + �20j)h � Cp;H+1:
If s � (h+ 1)�c � (H + 1)�c;
jje0i;nGntB(h)nt &1n;t�h � E(e0i;nGntB
(h)nt &1n;t�hjFi;n;T (s))jjp
� Cj�0�10 + �20jh + hCj�0�10 + �20jh�1j�0j+�h
2
�j�0�20
+�20jh�2j�0j2C + � � �+�
h
h� 1
�j�0�10 + �20j � j�0jh�1C + j�0jhC
�hXq=0
�h
q
�j�0jqj�0�10 + �20jh�qC = (j�0j+ j�0�10 + �20j)hC (9)
and
jje0i;nGntHXh=0
B(h)nt &1n;t�h � E(e0i;nGnt
HXh=0
B(h)nt &1n;t�hjFi;n(s))jjp �
HXh=0
(j�0j+ j�0�10 + �20j)hC = CH :
If s > (H + 1)�c;
jje0i;nGntB(h)nt &1n;t�h � E(e0i;nGntB
(h)nt &1n;t�hjFi;n(s))jjp
� j�0�10 + �20jhCsd0+hj�1cwjs=�c + hj�0�10 + �20jh�1j�0jCsd0+h�1j�1cwjs=�c
+
�h
2
�j�0�10 + �20jh�2j�0j2Csd0+h�2j�1cwjs=�c + � � �
+
�h
h� 1
�j�0�10 + �20j � j�0jh�1Csd0+1j�1cwjs=�c + j�0jhCsd0 j�1cwjs=�c
� (j�0j+ j�0�10 + �20j)hCsd0+hj�1cwjs=�c (10)
and
jje0i;nGntHXh=0
B(h)nt &1n;t�h � E(e0i;nGnt
HXh=0
B(h)nt &1n;t�hjFi;n(s))jjp
�HXh=0
(j�0j+ j�0�10 + �20j)hCsd0+hj�1cwjs=�c � CHsd0+H j�1cwjs=�c :
25
Lemma 1 Denote U1nt;H = A1ntPH
h=0B(h)nt &1n;t�h and U2nt;H = A2nt
PHh=0B
(h)nt &1n;t�h. Under Assump-
tions 1, 2.1), 3.1), and 4), 1n [�U01nt;H1
�U2nr;H2 �E(�U 01nt;H1�U2nr;H2)] = op(1), where H1 and H2 can be
any integer from 0 to T + L:
Proof. Denote 1n [�U
0nt;H1
�Unr;H2 � E(�U 0nt;H1�Unr;H2)] =
1n
Pni=1 qi;n with
qi;n = u1i;nt;H1u2i;nr;H2
� u1i;n;t�1;H1u2i;nr;H2
� u1i;nt;H1u2i;n;r�1;H2
+ u1i;n;t�1;H1u2i;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+Ls(2�c3)d0 and
supi;n;t;H jju2i;nt;H�E(u2i;nt;H jFi;n(s))jj4 � C4;T+Ls(2�c3)d0 . Together withmaxi;n;t;H(jju1i;nt;H jj4; jju2i;nt;H jj4) �C1 from Claim 6.0.6, we have jjqi;n � E(qi;njFi;n;T (s))jj2 � 8C1C4;T+Ls(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+Lj�1cwjs=�c if s > (T+L)�c.And maxi;n;t;H(jju1i;nt;H jj4; jju2i;nt;H jj4) � Cp;T+L from Claim 6.0.12. Therefore, 1n
Pni=1[qi;n�E(qi;n)]
L1! 0as 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 = (bT 0nP2n bTn)�1 bT 0nP2n�Yn(T )� �0= (bT 0nP2n bTn)�1 bT 0nP2n��n(T ) + ( bT 0nP2n bTn)�1 bT 0nP2nKn(K
0nP1nKn)
�1K 0nP1n�"n(T )�0:
For the consistency, we need to show that 1nq02nbTn, 1nq02nq2n, 1nq02nKn, 1nq
01nKn, 1nq
01nq12n areOp(1),
1nq
02n��n(T ) =
op(1); and 1nq
01n�"n(T ) = op(1):
Since bTn = [�(WY )n(T );�(WY )n(T � 1);�Yn(T � 1);�X1n(T );�b"n(T )] with�b"n(T ) = �"n(T )�Kn[K
0nq1n(q
01nq1n)
�1q01nKn]�1K 0
nq1n(q01nq1n)
�1q01n�"n(T );
if we choose q2n = [�(WX)n(T );�(WX)n(T � 1);�X1n(T � 1);�X1n(T );�b"n(T )], then the most compli-cated term we need to show is
1
n�(WX)n(T )
0�(WY )n(T ) =1
n
TXt=2
(WntXnt �Wn;t�1Xn;t�1)0(WntYnt �Wn;t�1Yn;t�1) = Op(1):
In our setting, the reduced form is
Znt =
t+t0�1Xh=0
�h20(X2n;t�h�0 + dn0 + "n;t�h) + �t+t020 Zn;�t0
=
t+t0Xh=0
�h20(X2n;t�h�0 + dn0 + "n;t�h) + �t+t020 (don0 � dn0);
Ynt = S�1nt
t+t0�1Xh=0
B(h)nt (X1n;t�h�0 + cn0 + "n;t�h�0 + �n;t�h) + S
�1nt B
(t+t0�1)nt (�10In + �20Wn;�t0)Yn;�t0
= S�1nt
t+t0Xh=0
B(h)nt (X1n;t�h�0 + cn0 + "n;t�h�0 + �n;t�h) + S
�1nt B
(t+t0)nt (con0 � cn0):
26
where B(h)nt = (�0S�1n;t�1 + �20Gn;t�1)(�0S
�1n;t�2 + �20Gn;t�2) � � � (�0S�1n;t�h + �20Gn;t�h) =
Yh
k=1(�0S
�1n;t�k +
�20Gn;t�k) withB(0)nt = In and Yn;�t0 can be treated either exogenous or Yn;�t0 = (In��1Wn;�t0)
�1(X1n;�t0�+con0 + "n;�t0�0 + �n;�t0).Therefore,
�Znt =
t+t0Xh=0
�h20(X2n;t�h�0 + dn0 + "n;t�h) + �t+t020 (don0 � dn0)
�t+t0�1Xh=0
�h20(X2n;t�1�h�0 + dn0 + "n;t�1�h)� �t+t0�120 (don0 � dn0);
�Ynt = S�1nt
t+t0Xh=0
B(h)nt (X1n;t�h�0 + cn0 + "n;t�h�0 + �n;t�h) + S
�1nt B
(t+t0)nt (con0 � cn0)
�S�1n;t�1t�1+t0Xh=0
B(h)n;t�1(X1n;t�1�h�0 + cn0 + "n;t�1�h�0 + �n;t�1�h)� S�1n;t�1B
(t�1+t0)n;t�1 (con0 � cn0);
�(WY )nt = Gnt
t+t0Xh=0
B(h)nt (X1n;t�h�0 + cn0 + "n;t�h�0 + �n;t�h) +GntB
(t+t0)nt (con0 � cn0)
�Gn;t�1t�1+t0Xh=0
B(h)n;t�1(X1n;t�1�h�0 + cn0 + "n;t�1�h�0 + �n;t�1�h)�Gn;t�1B
(t�1+t0)n;t�1 (con0 � cn0);
to show 1nq
02n��n(T ) = op(1) and
1nq
01n�"n(T ) = op(1);
1nq
02nbTn and 1
nq0nKn are Op(1), in general, we need
to show terms like
1
n
"& 02n;tW
0ntGnt
t+t0Xh=0
B(h)nt &1n;t�h � E
& 02n;tW
0ntGnt
t+t0Xh=0
B(h)nt &1n;t�h
!#= op(1);
where & can be either X or " or � or a constant. From Claim 6.0.1, it is su¢ cient to show that
supi;n;t
jje0inGntt+t0Xh=0
B(h)nt &1n;t�h � E(e0inGnt
t+t0Xh=0
B(h)nt &1n;t�hjFi;n(s))jj4 � C1'1(s);
and supi;;n;t
jje0inGntt+t0Xh=0
B(h)nt &1n;t�hjj4 < 1:
For the asymptotic distribution of b�, we havepn(b�� �0) d! N(0;��0):
��0 = plimn!1
1
n(T 0nP2nTn)
�1T 0nP2n�nP2nTn(T0nP2nTn)
�1; with
�n = �2�00 In +Kn(K0nP1nKn)
�1K 0nP1n(�
00�"0�00 In)P1nKn(K
0nP1nKn)
�1K 0n:
27
We can write
pn(b�� �0) = (
1
nbT 0nP2n bTn)�1 1pn bT 0nP2n��n(T ) + ( 1n bT 0nP2n bTn)�1 1n bT 0nP2nKn(
1
nK 0nP1nKn)
�1 1pnK 0nP1n�"n(T )�0
=1pn
nXi=1
ri;n + op(1); with ri;n =TXt=3
(q1i;nt��i;nt + q2i;nt�"i;nt):
With the NED property of ri;n, we can show the asymptotic normality ofpn(b�� �0).
Proof of Claim 3.2.1. Estimated variance-covariance matrix
b�� =1
n( bT 0nP2n bTn)�1 bT 0nP2nb�nP2n bTn( bT 0nP2n bTn)�1; withb�n = Kn(K
0nP1nKn)
�1K 0nP1n(
b�0b�"b�0 In)P1nKn(K0nP1nKn)
�1K 0n + (b�2�0 In);
where b�" = 1
2n(T � 2)�b"n(T )0�b"n(T ); b�2� = 1
2n(T � 2)�b�n(T )0�b�n(T ):
First we show that b�" p! �"0. This holds because
�b"n(T ) = �Zn(T )�Kn
�b�2b��= �"n(T ) +Kn
���20�0
���b�2b�
��;
1
n�b"n(T )0�b"n(T ) p! 1
n�"n(T )
0�"n(T ) = 2(T � 2)�"0:
Then we show that b�2� p! �2�0. As
�b�nt = �Ynt � bTntb� = ��nt(T ) + �(WY )nt(�10 � b�1) + �(WY )n;t�1(�20 � b�2) + �Yn;t�1(�10 � b�1)+�X1nt(�0 � b�) + �"nt�0 ��b"nt(T )b�
= �Ynt � bTntb� = ��nt(T ) + �(WY )nt(�10 � b�1) + �(WY )n;t�1(�20 � b�2) + �Yn;t�1(�10 � b�1)+�X1nt(�0 � b�) + �"nt ��0 � b���Kn
���20�0
���b�2b�
��;
we have1
n�b�n(T )0�b�n(T ) p! lim
n!1
1
nE(��nt(T )
0��nt(T )) = �2�0tr(0 In) = 2(T � 2)�2�0
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