On Spatial Processes and Asymptotic Inference under Near$Epoch ...

On Spatial Processes and Asymptotic Inference
under Near-Epoch Dependence
Nazgul Jenish 1 and Ingmar R. Prucha 2
October 21, 2011
1
Department of Economics, New York University, 19 West 4th Street, New York,
NY 10012. Tel.: 212-998-3891, Email: nazgul.jenish@nyu.edu
2
Department of Economics, University of Maryland, College Park, MD 20742. Tel.:
301-405-3499, Email: prucha@econ.umd.edu

Abstract
The development of a general inferential theory for nonlinear models with crosssectionally
or spatially dependent data has been hampered by a lack of appropriate
limit theorems. To facilitate a general asymptotic inference theory relevant
to economic applications, this paper …rst extends the notion of near-epoch dependent
(NED) processes used in the time series literature to random …elds.
The class of processes that is NED on, say, an -mixing process, is shown to
be closed under in…nite transformations, and thus accommodates models with
spatial dynamics. This would generally not be the case for the smaller class of
-mixing processes. The paper then derives a central limit theorem and law
of large numbers for NED random …elds. These limit theorems allow for fairly
general forms of heterogeneity including asymptotically unbounded moments,
and accommodate arrays of random …elds on unevenly spaced lattices. The
limit theorems are employed to establish consistency and asymptotic normality
of GMM estimators. These results provide a basis for inference in a wide range
of models with cross-sectional or spatial dependence.
JEL Classi…cation: C10, C21, C31
Key words: Random …elds, near-epoch dependent processes, central limit
theorem, law of large numbers, GMM estimator

1 Introduction 1
Models with cross-sectionally or spatially dependent data have recently attracted
considerable attention in various …elds of economics including labor and
public economics, IO, political economy, macroeconomics, international economics,
regional science and urban economics. In these models cross sectional
dependence typically stems from agents’strategic interaction, neighborhood effects,
shared resources and common shocks. Furthermore, economic agents are
typically heterogeneous, e.g., vary in size. Mathematically, agents’observations
may thus be viewed as a realization of a dependent heterogenous process indexed
by a point in R d , d > 1, i.e., a random …eld. 2
The aim of this paper is to de…ne a class of random …elds that is su¢ -
ciently general to accommodate many applications of interest, and to establish
corresponding limit theorems that can be used for asymptotic inference. In
particular, we apply these limit theorems to prove consistency and asymptotic
normality of generalized method of moments (GMM) estimators for a general
class of nonlinear spatial models.
There have been di¤erent approaches to modeling cross-sectional dependence
in the econometrics literature. Two large strands of the literature have focused
on (i) linear spatial autoregressive models, also known as Cli¤-Ord (1981) type
models 3 , and (ii) common factor models 4 . In both cases, progress was facilitated,
loosely speaking, by imposing speci…c structural conditions on the data
generating process, and/or by exploiting some underlying (conditional) independence
assumptions. Another common approach to model dependence is through
mixing conditions. Various mixing concepts developed for time series processes
have been extended to random …elds. However, the respective limit theorems
for random …elds have not been su¢ ciently general to accommodate many of
the processes encountered in economics. This hampered the development of a
general asymptotic inference theory for nonlinear models with cross-sectional
dependence. Towards …lling this gap, Jenish and Prucha (2009) have recently
introduced a set of limit theorems (CLT, ULLN, LLN) for -mixing random
…elds on unevenly spaced lattices that allow for nonstationary processes with
trending moments.
Yet some dependent spatial processes do not satisfy mixing conditions. As
with time series, spatial processes may be generated as functions of in…nitely
many elements of an input process, with a spatial moving average process being
1 We thank the participants of the Cowles Foundation Conference, Yale, June 2009, as well
as the seminar participants at the Columbia University for helpful discussions. This research
bene…tted from a University of Maryland Ann G. Wylie Dissertation Fellowship for the …rst
author and from …nancial support from the National Institute of Health through SBIR grant
1 R43 AG027622 for the second author.
2 The space and metric are not restricted to physical space and distance.
3 For recent contributions see, e.g., Robinson (2009, 2007), Yu, de Jong and Lee (2008),
Kelejian and Prucha (2007a,b,2004), Lee (2007a,b, 2004), and Chen and Conley (2001).
4 For recent contributions, see, e.g., Andrews (2005), Bai and Ng (2004), Bai (2003), Phillips
and Sul (2003).
1

a leading example. As is well-known, the mixing property is not necessarily
preserved under transformations involving in…nite number of lags. For instance,
Andrews (1984) shows that a simple time series AR(1) process fails to be -
mixing though its input process is independent. Thus, it is important to develop
an asymptotic theory for a generalized class of random …elds that is “closed with
respect to in…nite transformations.”
To tackle this problem, the paper …rst extends the concept of near-epoch dependent
(NED) processes used in the time series literature to spatial processes.
The notion dates back to Ibragimov (1962), and Billingsley (1968). The NED
concept, or variants thereof, have been used extensively in the time series literature
by McLeish (1975a,b), Bierens (1981), Wooldridge (1986), Gallant (1987),
Gallant and White (1988), Andrews (1988), Pötscher and Prucha (1991,1997),
Davidson (1992, 1993, 1994) and de Jong (1997), among others. Doukhan and
Louhichi (1999), Ango Nze and Doukhan (2004) introduced an alternative class
of dependent processes called “ -weakly dependent.”We provide various examples
suggesting that the class of NED spatial processes, which subsumes mixing
processes, is su¢ ciently broad to cover many applications of interest. For
instance, we show that it covers Cli¤-Ord type processes, autoregressive and
moving average random …elds, and, more generally, nonlinear spatial Bernoulli
shifts.
The paper then derives a CLT and an LLN for spatial processes that are near
epoch dependent on an -mixing input process. These limit theorems allow
for fairly general forms of heterogeneity including asymptotically unbounded
moments, and accommodate arrays of random …elds on unevenly spaced lattices.
The LLN can be combined with the generic ULLN in Jenish and Prucha (2009)
to obtain an ULLN for NED spatial processes. In the time series literature,
CLTs for NED processes were derived by Wooldridge (1986), Davidson (1992,
1993), and de Jong (1997). Interestingly, our CLT contains as a special case the
CLT of Wooldridge (1986), Theorem 3.13 and Corollary 4.4.
In addition, we give conditions under which the NED property is preserved
under transformations. These results play a key role in verifying the NED property
in applications. Thus, the NED property is compatible with considerable
heterogeneity and dependence, invariant under transformations, and leads to
a CLT and LLN under fairly general conditions. All these features make it a
convenient tool for modeling spatial dependence.
As an application, we establish consistency and asymptotic normality of
spatial GMM estimators. These results provide a fundamental basis for constructing
con…dence intervals and testing hypothesis for GMM estimators in
nonlinear spatial models. Our results also expand on Conley (1999), who established
the asymptotic properties of GMM estimators assuming that the data
generating process and the moment functions are stationary and -mixing. 5
The rest of the paper is organized as follows. Section 2 introduces the concept
of NED spatial processes and gives some mild conditions that ensure preserva-
5 This important early contribution employs Bolthausen’s (1982) CLT for stationary -
mixing random …elds on the regular lattice Z 2 . However, the mixing and stationarity assumptions
may not hold in many applications. The present paper relaxes these critical assumptions.
2

tion of the NED property under transformations. Section 3 provides examples
of NED spatial processes. Section 4 contains the CLT and LLN for NED spatial
processes. Section 5 establishes the asymptotic properties of spatial GMM
estimators. All proofs are collected in the appendices.
2 De…nition of NED Spatial Processes
Let D R d , d 1, be a (possibly) unevenly spaced lattice of locations in R d ,
and let Z = fZi;n; i 2 Dn; n 1g and " = f"i;n; i 2 Tn; n 1g be triangular
arrays of random …elds de…ned on a probability space ( ; F; P ) with Dn Tn
D. The space R d is equipped with the metric (i; j) = max1 l d jjl ilj, where
il is the l-th component of i. The distance between any subsets U; V D is
de…ned as (U; V ) = inf f (i; j) : i 2 U and j 2 V g. Furthermore, let jUj denote
the cardinality of a …nite subset U D.
The random variables Zi;n and "i;n are possibly vector-valued taking their
values in R pz and R p" . We assume that R pz and R p" are normed metric spaces
equipped with the Euclidian norm, which we denote (in an obvious misuse of
notation) as j:j. For any random vector Y , let kY k p = [E jY j p ] 1=p , p 1;
denote its Lp-norm. Finally, let Fi;n(s) = ("j;n; j 2 Tn : (i; j) s) be the
-…eld generated by the random vectors "j;n located in the s-neighborhood of
location i.
Throughout the paper, we maintain the above notational conventions as well
as the following assumption concerning D.
Assumption 1 The lattice D R d , d 1, is in…nite countable. All elements
in D are located at distances of at least 0 > 0 from each other, i.e., for all i;
j 2 D : (i; j) 0; w.l.o.g. we assume that 0 > 1.
The assumption of a minimum distance has also been used by Conley (1999)
and Jenish and Prucha (2009). It ensures the growth of the sample size as the
sample regions Dn and Tn expand. The setup is thus geared towards what is
referred to in the spatial literature as increasing domain asymptotics.
We now introduce the notion of near-epoch dependent (NED) random …elds.
De…nition 1 Let Z = fZi;n; i 2 Dn; n 1g be a random …eld with kZi;nk p <
1, p 1, let " = f"i;n; i 2 Tn; n 1g be a random …eld, where jTnj ! 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 Z is said to be Lp(d)-near-epoch dependent on the random
…eld " if
kZi;n E(Zi;njFi;n(s))k p di;n (s) (1)
3

for some sequence (s) 0 with lims!1 (s) = 0. The (s), which are w.l.o.g.
assumed to be non-increasing, are called the NED coe¢ cients, and the di;n are
called NED scaling factors. Z is said to be Lp-NED on " of size if (s) =
O(s ) for some > > 0. Furthermore, if sup n sup i2Dn di;n < 1, then Z is
said to be uniformly Lp-NED on ":
Recall that Dn Tn. Typically, Tn will be an in…nite subset of D, and
often Tn = D. However, to cover Cli¤-Ord type processes, see Section 3.3, Tn
is allowed to depend on n and to be …nite provided that it increases in size with
n.
The role of the scaling factors fdi;ng is to allow for the the possibility of
“unbounded moments”, i.e., sup n sup i2Dn di;n = 1. Unbounded moments may
re‡ect trends in the moments in certain directions, in which case we may also
use, as in the time series literature, the terminology of “trending moments”. The
NED property is thus compatible with a considerable amount of heterogeneity.
In establishing limit theorems for NED processes, we will have to impose restrictions
on the scaling factors di;n. In this respect, observe that
kZi;n E(Zi;njFi;n(s))k p kZi;nk p + kE(Zi;njFi;n(s))k p 2 kZi;nk p
by the Minkowski and the conditional Jensen inequalities. Given this, we may
choose di;n 2 kZi;nk p , and consequently w.l.o.g. 0 (s) 1; see, e.g.,
Davidson (1994), p. 262, for a corresponding discussion within the context of
time series processes. Note that by the Lyapunov inequality, if Zi;n is Lp-NED,
then it is also Lq-NED with the same coe¢ cients fdi;ng and f (s)g for any
q p.
Our de…nition of NED for spatial processes is adapted from the de…nition of
NED for time series processes. In the time series literature, the NED concept
…rst appeared in the works of Ibragimov (1962) and Billingsley (1968), although
they did use the present term. The concept of time series NED processes was
later formalized by McLeish (1975a,b), Wooldridge (1986), Gallant (1987), Gallant
and White (1988). These authors considered only L2-NED processes. Andrews
(1988) generalized it to Lp-NED processes for p 1: Davidson (1992,
1993, 1994) and de Jong (1997) further extended it to allow for trending time
series processes.
An important motivation for considering NED processes is that mixing is
not necessarily preserved under transformations involving in…nitely many arguments;
see, e.g., Andrews (1984) for simple examples in a time series context.
However, as illustrated below, for a wide range of models the output process
is obtained as a function of in…nitely many input variables. In those situations
mixing of the input process does not necessarily carry over to the output process,
and thus limit theorems for averages of the output process cannot simply be established
from limit theorems for mixing processes. Nevertheless, as with time
series processes, we show below that limit theorems can be extended to spatial
processes that are NED on a mixing input process, provided the approxima-
4

tion error declines “su¢ ciently fast” as the conditioning set of input variables
expands.
A further attractive feature of NED processes is that the NED property is
preserved under transformations. Econometric estimators are usually de…ned
either explicitly as functions of some underlying data generating processes or
implicitly as optimizers of a function of the data generating process. Thus, if
the data generating process is NED on some input process, the question arises
under what conditions functions of random …elds are also NED on the same
input process.
Various conditions that ensure preservation of the NED property under
transformations have been established in the time series literature by Gallant
and White (1988), and Davidson (1994). In fact, these results extend to random
…elds. In particular, the NED property is preserved under summation and multiplication,
and carries over from a random vector to its components and vice
versa. For future reference, we now state some results for generalized classes
of nonlinear functions. Their proofs are analogous to those in the time series
literature, and therefore omitted.
Consider transformations of Zi;n given by a family of functions
gi;n : R pz ! R:
The functions gi;n are assumed Borel-measurable for all n and i 2 D. They
are furthermore assumed to satisfy the following Lipschitz condition: For all
(z; z ) 2 R pz R pz and all i 2 Dn and n 1:
where
jgi;n(z) gi;n(z )j Bi;n(z; z ) jz z j (2)
Bi;n(z; z ) : R pz R pz ! R+;
is Borel-measurable. Of course, this condition would be devoid of meaning without
further restrictions on Bi;n(z; z ), which are given in the next propositions.
Proposition 1 Suppose gi;n( ) satis…es Lipschitz condition (2) with
jBi;n(z; z )j C < 1
for all (z; z ) 2 R pz R pz and all i and n. If for p 1 the fZi;ng are Lp-NED
of size on f"i;ng with scaling factors fdi;ng, then gi;n(Zi;n) is also Lp-NED
of size on f"i;ng with scaling factors f2Cdi;ng.
Proposition 2 Suppose gi;n( ) satis…es Lipschitz condition (2) with
sup
s
B (s)
i;n 2 < 1 and sup
s
5
B (s)
i;n Zi;n
eZ s i;n r < 1 (3)

for some r > 2; where B (s)
i;n = Bi;n(Zi;n; e Z s i;n ) and e Z s i;n = E [Zi;njFi;n(s)]. If
kgi;n(Zi;n)k 2 < 1 and Zi;n is L2-NED of size on f"i;ng with scaling factors
fdi;ng ; then gi;n(Zi;n) is L2-NED of size (r 2)=(2r 2) on f"i;ng with scaling
factors d0 2)=(2r
i;n = d(r i;n
2)
sups B (s)
i;n
(r
2
2)=(2r 2)
B (s)
i;n Zi;n
eZ s i;n
r=(2r
r
2)
Thus, the NED property is hereditary under reasonably weak conditions.
These conditions facilitate veri…cation of the NED property in practical application.
In particular, we will use them in the proof of asymptotic normality of
spatial GMM estimators in Section 5.
3 Examples of NED Spatial Processes
In this section, we give examples of some important classes of spatial processes,
which are widely used in applications, and show that they satisfy the NED
property. Of course, to establish limit theorems the NED property would be
accompanied by mixing assumptions on the input process.
3.1 Moving Average Random Fields
Consider the in…nite moving average (or linear) random …eld on Zd , say, Y =
fYi; i 2 Zdg de…ned as
Yi = X
gij"j; (4)
j2Z d
where " = f"i; i 2 Zdg is a zero mean vector-valued random …eld on Zd and the
gij are some real numbers. We assume furthermore that the coe¢ cients gij and
the innovations "i satisfy, respectively,
and
lim
s!1 sup
X
i2Zd j2Zd ; (i;j)>s
jgijj = 0; (5)
sup
i2Zd k"ikp < 1 where p 1: (6)
Linear stationary random …elds on Z 2 have been explored by Whittle (1954)
in a paper that signi…cantly in‡uenced the subsequent modeling of spatial dependencies.
More recently, linear random …elds on Z d , d 1, have been studied
by Doukhan and Guyon (1991), see also Doukhan (1994). Analogous to the
time series literature, Yi is de…ned as the limit in Lp as s ! 1 of the partial
sums Y s
i
= P
j2Z d ; (i;j) s gij"j. The following existence results is proven by
Doukhan (1994), pp. 75-81. 6
6 In fact, Doukhan (1994) proves a more general result that also covers the case p < 1.
6
:

Proposition 3 (Doukhan, 1994) Under conditions (5)-(6) the distribution of
linear random …eld Y in (4) is well de…ned. In particular, the …nite dimensional
distributions of Y are limits in Lp as s ! 1 of the those of the random …elds
Y s = fY s
i ; i 2 Zd g.
We next give a result that shows that under the same conditions, the linear
…eld Y is Lp-NED on the …eld ".
Proposition 4 Under conditions (5)-(6) the linear random …eld Y in (4) is
uniformly Lp-NED on ":
The nice feature of this result is that it does not impose any additional
conditions over and above those employed in Proposition 3 to establish the
existence of the linear …eld.
3.2 Autoregressive Random Fields
The linear random …elds of the previous example may arise as solutions of
autoregressive spatial models. For example, Whittle (1954) considered the following
simple autoregressive model on Z 2 :
Yi1;i2 = Yi1+1;i2 + Yi1;i2+1 + "i1;i2 (7)
where j j + j j < 1 and " = f"i1;i2 ; (i1; i2) 2 Z 2 g are i.i.d. random variables
with zero mean and …nite variance. Whittle (1954), p. 447, gives the following
the stationary solution of (7):
Yi1;i2 =
=
1X
1X
j1=0 j2=0
1X
1X
j1=i1 j2=i2
j1 + j2
Thus, (8) is a linear random …eld with
j1
j1 i1 + j2 i2
j1 i1
gij = j1 i1 + j2 i2
j1 i1
j1 j2 "i1+j1;i2+j2 (8)
j1 i1 j2 i2 "j1;j2
j1 i1 j2 i2 :
for j = (j1; j2) i = (i1; i2) and zero, otherwise. Furthermore, note that
the assumptions of Proposition 4 are satis…ed for p = 2, since P
P 1
k=0
P k
l=0
k
l j j l j j k l = P1 k=0 (j j + j j)k < 1, and hence
0 lim
s!1 sup
X
i2Z2 j i; (i;j)>s
observing that j j + j j < 1.
jgijj lim
7
s!1
k=s
1X
(j j + j j) k = 0;
j i jgijj =

3.3 Cli¤-Ord Type Spatial Processes
Consider the following Cli¤-Ord type model:
Yn = WnYn + Xn + un (9)
un = Mnun + vn
where Yn = (Y1;n; :::; Yn;n) 0 is an n 1 vector of endogenous variables, Xn =
(Xij;n) is n k matrix of regressors, Wn = (wij;n) and Mn = (mij;n) are n n
nonstochastic weight matrices, vn = (v1;n; : : : ; vn;n) 0 is an n 1 vector of disturbances,
and are scalar parameters typically referred to as spatial autoregressive
parameters, and is a k 1 parameter vector. To simplify presentation, we
assume in the following that k = 1 and use the notation Xn = (X1;n; :::; Xn;n) 0 .
The above model is frequently referred to as a spatial ARAR(1,1) model, and
WnYn and Mnun are referred to as spatial lags. For recent contributions on
estimation strategies for this model see, e.g., Robinson (2009, 2007), Kelejian
and Prucha (2007a,b, 2004), and Lee (2007a,b, 2004).
The spatial weights mij;n and wij;n are typically thought of as to depend
on some measure of distance, say dij, between units i and j, where the weights
decline as the distance increases. Thus although in Cli¤-Ord models observations
are indexed by natural numbers, i.e., i = 1; : : : ; n, a leading case would
be where i correspond to some location, say si 2 R d , and the distance between
i and j is given by dij = (si; sj). In the following, we assume that
Dn = fs1; : : : ; sng D, where the lattice D satis…es Assumption 1. However,
to simplify notation we will use (i; j) instead of (si; sj). We assume
furthermore that for all n
In Wn and In Mn are nonsingular. (10)
The reduced form of the model is then given by Yn = AnXn + Bnvn:with
An = (aij;n) = (In Wn) 1 and Bn = (bij;n) = (In Wn) 1 (In Mn) 1 .
In scalar notation we have for i = 1; :::; n:
Yi;n =
nX
j=1
aij;nXj;n +
nX
j=1
bij;nvj;n;
Although for …xed n; the output process Yi;n depends on only a …nite number of
elements of the input process "i;n = (Xi;n; vi;n) 0 , the mixing property of "i;n may
not carry over to Yi;n: The reason is that the number of elements composing the
spatial lags grows unboundedly with the sample size so that the mixing property
can break down in the limit. This is especially important when analyzing the
asymptotic properties of Cli¤-Ord type processes.
Towards establishing that Y = fYi;n; si 2 Dn; n 1g is NED on " =
f"i;n; si 2 Dn; n 1g, we maintain the following assumptions:
lim
s!1 sup sup
n
X
jaij;nj + jbij;nj = 0 (11)
1 i n
1 j n: (i;j)>s
8

and
sup
n
sup
1 i n
We have the following result.
k"i;nk p < 1 for some p 1: (12)
Proposition 5 Under conditions (10)-(12), the process Y = fYi;n; si 2 Dn; n
1g, where Yi;n is de…ned by (9), is uniformly Lp-NED on the process " =
f"i;n; si 2 Dn; n 1g, where "i;n = (Xi;n; vi;n) 0 :
since
It is readily seen that a su¢ cient condition for (11) is that for some > 0,
sup
n
sup
n
sup
n
sup
sup
1 i n
j=1
nX
(jaij;nj + jbij;nj) (i; j) < 1 (13)
X
jaij;nj + jbij;nj
1 i n
1 j n; (i;j)>s
sup
X
(jaij;nj + jbij;nj)
1 i n
1 j n; (i;j)>s
1
s sup sup
n
1 i n
j=1
(i; j)
s
nX
(jaij;nj + jbij;nj) (i; j) ! 0 as s ! 1:
A condition analogous to (13) has been used recently by Kelejian and Prucha
(2007a), and should be satis…ed in a wide range of applications. It is slightly
stronger than the typical assumption in the Cli¤-Ord literature which maintains
assumptions that imply that the row and column sums of the absolute elements
of the matrices An and Bn are uniformly bounded; compare, e.g., Kelejian and
Prucha (2007b) and Lee (2007a,b, 2004).
3.4 Spatial Bernoulli Shifts
Consider a real-valued random …eld Y = fYi; i 2 Z d g de…ned as:
Yi = fi("i+j; j 2 Z d ) (14)
where " = f"i; i 2 Z d g is a real-valued random …eld and the fi : R Zd
! R
are measurable functions. The process (14) is a generalization of one-parameter
Bernoulli shifts considered by Doukhan and Louhichi (1999). A simple example
of a spatial Bernoulli shift is the in…nite moving average random …eld of the
…rst example. More generally, the fi may be nonlinear functions. We assume
that the fi satisfy the following Lipschitz-type regularity condition:
fi(xj; j 2 Z d ) fi(yj; j 2 Z d ) X
wij jxj yjj (15)
9
j2Z d

for some nonnegative constants wij such that
lim
s!1 sup
X
wij = 0: (16)
i2Zd j2Zd : (i;j)>s
Finally, we assume that the input process " = f"i; i 2 Z d g has uniformly
bounded second moments, i.e.,
Then, one can establish the following result.
sup
i2Zd k"ik2 < 1 (17)
Proposition 6 Under conditions (15)-(17), the random …eld Y = fYi; i 2 Z d g
given by (14) is well-de…ned and uniformly L2-NED on the random …eld " =
f"i; i 2 Z d g.
Conditions (15)-(17) are analogous to those used by Doukhan and Louhichi
(1999) for time-series Bernoulli shifts. Condition (15) is ful…lled if the functions
fi have bounded partial derivatives, i.e., j@fi=@xjj wij.
Thus, the class of NED spatial processes covers fairly large classes of linear
and nonlinear transformations of random …elds.
4 Limit Theorems
4.1 Central Limit Theorem
In the following, we introduce our CLT for real valued random …elds Y =
fYi;n; i 2 Dn; n 1g, where Z is assumed to be L2-NED on some vector-valued
-mixing random …eld " = f"i;n; i 2 Tn; n 1g with the NED coe¢ cients
f (s)g, where Dn Tn D and the lattice D satis…es Assumption 1. In the
sequel, we will use the following notation:
Sn = X
i2Dn
Yi;n;
2
n = var(Sn):
For ease of reference, we state below the de…nition of the -mixing coe¢ cients
employed in the paper.
De…nition 2 Let A and B be two -algebras of F, and let
(A; B) = sup(jP (AB) P (A)P (B)j; A 2 A; B 2 B);
For U Dn and V Dn, let n(U) = ("i;n; i 2 U) and n(U; V ) =
( n(U); n(V )). Then, the -mixing coe¢ cients for the random …eld " are
de…ned as:
(u; v; r) = sup
n
sup(
n(U; V ); jUj u; jV j v; (U; V ) r):
U;V
10

Dobrushin (1968) showed that weak dependence conditions based on the
above mixing coe¢ cients are satis…ed by broad classes of random …elds including
Markov …elds. In contrast to standard mixing numbers for time-series processes,
the mixing coe¢ cients for random …elds depend not only on the distance between
two datasets but also their sizes. To explicitly account for such dependence, it
is furthermore assumed that
(u; v; r) '(u; v)b(r) (18)
where the function '(u; v) is nondecreasing in each argument, and b(r) ! 0
as r ! 1. The idea is to account separately for the two di¤erent aspects of
dependence: (i) decay of dependence with the distance, and (ii) accumulation of
dependence as the sample region expands. The two common choices of '(u; v)
in the random …elds literature are
and
'(u; v) = (u + v) ; 0; (19)
'(u; v) = min fu; vg : (20)
The above mixing conditions have been used extensively in the random …elds
literature including Neaderhouser (1978), Takahata (1983), Nahapetian (1987),
Bulinskii (1989), Bulinskii and Doukhan (1990), Tran (1990) and Bradley (1993).
They are satis…ed by fairly large classes of random …elds. Bradley (1993) provides
examples of random …elds satisfying conditions (18)-(19) with u = v and
= 1. Furthermore, Bulinskii (1989) constructs in…nite moving average random
…elds satisfying the same conditions with = 1 for any given decay rate of coe¢
cients b(r): Clearly, standard mixing coe¢ cients in the time series literature
are covered by conditions (18)-(19) when = 0.
Following the literature, we employ the above mixing conditions for the
input random …eld, and impose the following restriction on the decay rates of
the mixing coe¢ cients.
Assumption 2 The -mixing coe¢ cients of " satisfy (18) and (19) for some
0 and b(r), such that for some > 0
where = =(2 + ).
1X
r=1
r d( +1) 1 b 2(2+ ) (r) < 1; (21)
Note that if = 1 this assumption also covers the case where '(u; v) is given
by (20). Furthermore, the CLT relies on the following conditions regarding
moments, the NED coe¢ cients and the NED scaling factors.
Assumption 3 (a) (Uniform L2+ integrability)
lim
k!1 sup sup E[jYi;nj 2+ 1(jYi;nj > k)] = 0;
n
i2Dn
where 1( ) is the indicator function and > 0 is as in Assumption 2.
11

(b) infn jDnj 1 2 n > 0, where fDng is a sequence of …nite subsets of D such
that jDnj ! 1 as n ! 1:
(c) NED coe¢ cients satisfy P 1
r=1 rd 1 (r) < 1:
We can now state the following CLT for L2-NED random …elds.
Theorem 1 Suppose fDng is a sequence of …nite subsets such that jDnj ! 1
as n ! 1 and fTng is a sequence of subsets such that Dn Tn D of the
lattice D satisfying Assumption 1: Let Y = fYi;n; i 2 Dn; n 1g be a real valued
zero-mean random …eld that is L2-NED on a vector-valued -mixing random
…eld " = f"i;n; i 2 Tn; n 1g. Suppose Assumptions 2 and 3 hold, then
1
n Sn =) N(0; 1):
The above CLT contains the CLT for -mixing random …elds given in Jenish
and Prucha (2009) as a special case. As shown earlier, the class of NED random
…elds is an important extension of the class of -mixing random …elds, in that the
class includes important and widely considered …elds, e.g., spatial autoregressive
processes, that do not generally satisfy mixing conditions. It is for that reason
that an estimation theory limited to the class of -mixing random …elds is not
fully satisfactory; see, e.g., Gallant and White (1988) and Pötscher and Prucha
(1997) for an analogous discussion in the time series literature. We note that
Theorem 1 also contains as a special case the CLT for time series NED processes
of Wooldridge (1986), see Theorem 3.13 and Corollary 4.4.
Assumption 2 restricts the dependence structure of the input process ":
Assumptions 3(a),(b) are standard in the limit theory of mixing processes,
e.g., Wooldridge (1986), Davidson (1992), and de Jong (1997), and Jenish and
Prucha (2009).
Assumption 3(a) is satis…ed if the Yi;n are uniformly Lp-bounded for some
p > 2 + , i.e., sup n sup i2Dn kYi;nk p < 1. Assumption 3(b) is an asymptotic
negligibility condition that ensures that no single summand in‡uences disproportionately
the entire sum. Assumption 3(c) controls the size of the NED
coe¢ cients which measure the error in the approximation of Yi;n by ". Intuitively,
the approximation errors have to decline su¢ ciently fast with each
successive approximation. Assumption 3(c) is satis…ed if (r) = O(r d ) for
some > 0; i.e., (r) is of size d.
Theorem 1 can be easily extended to vector-valued …elds using the standard
Cramér-Wold device.
Corollary 1 Suppose fDng is a sequence of …nite subsets such that jDnj ! 1
as n ! 1 and fTng is a sequence of subsets such that Dn Tn D of the
lattice D satisfying Assumption 1: Let Y = fYi;n; i 2 Dn; n 1g with Yi;n 2
12

R k be a zero-mean random …eld that is L2-NED on a vector-valued -mixing
random …eld " = f"i;n; i 2 Tn; n 1g. Suppose Assumptions 2 and 3 hold with
jYi;nj denoting the Euclidean norm of Yi;n and 2 n replaced by min( n), where
n = V ar(Sn) and min(:) is the smallest eigenvalue, then
1=2
n Sn =) N(0; Ik):
Furthermore, sup n jDnj 1 max ( n) < 1, where max(:) denotes the largest
eigenvalue.
4.2 Law of Large Numbers
To prove consistency of spatial estimators, we also derive a weak LLN for random
…elds which are for L1-NED on some mixing …eld. This LLN can then be
used to establish uniform convergence of random functions by combining it
with the generic ULLN given in Jenish and Prucha (2009), which transforms
pointwise LLNs (at a given parameter value) into ULLNs. The LLN maintains
the following moment and mixing assumptions.
Assumption 4 (a) sup n sup i2Dn E jYi;nj p < 1:; p > 1:
(b) The -mixing coe¢ cients of the input …eld " satisfy (18) for some function
'(u; v) which is nondecreasing in each argument, and some b(r) such that
P 1
r=1 rd 1 b (r) < 1.
We now have the following weak LLN.
Theorem 2 Let fDng be a sequence of arbitrary …nite subsets of D such that
jDnj ! 1 as n ! 1; where D Rd , d 1 is as in Assumption 1, and let
Tn be a sequence of subsets of D such that Dn Tn. Suppose further that
Y = fYi;n; i 2 Dn; n 1g is L1-NED on " = f"i;n; i 2 Tn; n 1g. If Y and "
satisfy Assumption 4, then
1
jDnj
X
i2Dn
(Yi;n EYi;n) L1
! 0;
We note that Y and " can be vector-valued random …elds. In this case, jYi;nj
should be understood as the Euclidean norm in the respective vector-space.
Assumption 4(a) is a standard moment condition employed in weak LLNs for
dependent processes. It requires existence of moments of order slightly greater
than 1. Assumption 4(b) is weaker than the mixing conditions used for the CLT.
Furthermore, the LLN does not require any restrictions on the NED coe¢ cients.
In the time series literature, weak LLNs for NED processes have been obtained
by Andrews (1988) and Davidson (1993), among others. Andrews (1988)
derives an L1-law for triangular arrays of L1-mixingales. He then shows that
NED processes are L1-mixingales, and hence, satisfy his LLN. Davidson (1993)
extends the latter result to processes with trending moments.
13

5 Large Sample Properties of Spatial GMM Estimators
In this section, we apply the above developed limit theorems to establish the
large sample properties of spatial GMM estimators under a reasonably general
set of assumptions that should cover a wide range of empirical problems. More
speci…cally, our consistency and asymptotic normality results (i) maintain only
that the spatial data process is NED on an -mixing basis process to accommodate
spatial lags in the data process as discussed above, (ii) allow for the data
process to be located on an unevenly spaced grid, and (iii) allow for the data
process to be non-stationary, which will frequently be the case in empirical applications.
We also give our results under a set of primitive su¢ cient conditions
for easier interpretation by the applied researcher. 7
We continue with the basic set-up of Section 2. Consider the moment function
qi;n : R pz ! R pq , where denotes the parameter space, and let 0 n 2
denote the parameter vector of interest (which we allow to depend on n for reasons
of generality). Suppose the following moment conditions hold
Eqi;n(Zi;n; 0 n) = 0; (22)
then the corresponding spatial GMM estimator is de…ned as
where Qn: ! R,
b n = arg min
2 Qn(!; ); (23)
Qn(!; ) = Rn( ) 0 PnRn( );
Rn( ) = jDnj
X
1
qi;n(Zi;n; );
i2Dn
where the Pn are a positive de…nite weighting matrices. To prove that b n is
0
a consistent estimator for n consider the following non-stochastic analogue of
Qn, say
Qn( ) = [ERn( )] 0 P [ERn( )] ; (24)
where P denotes the probability limit of Pn. Then given the moment condition
(22) clearly Rn( o o
n) = 0, and the functions Qn are minimized at n with
Qn( o n) = 0. In proving consistency, we follow the classical approach; see, e.g.,
7 In an important contribution, Conley (1999) gives a …rst set of results regarding the
asymptotic properties of GMM estimators under the assumption that the data process is stationary
and -mixing. Conley also maintains some high level assumption such as …rst moment
continuity of the moment function, which in turn immediately implies uniform convergence
- see, e.g., Pötscher and Prucha (1989) for a discussion. Our results extend Conley (1999)
in several important directions, as indicated above. We establish uniform convergence from
primitive su¢ cient conditions via the generic uniform law of large numbers given in Jenish
and Prucha (2009) and the law of large numbers given as Theorem 2 above.
14

Gallant and White (1988) or Pötscher and Prucha (1997) for more recent expo-
o
sitions. In particular, given identi…able uniqueness of n we establish, loosely
speaking, convergence of the minimizers b o
n to the minimizers n by establishing
convergence of the objective function Qn(!; ) to its non-stochastic analogue
Qn( ) uniformly over the parameter space.
Throughout the sequel, we maintain the following assumptions regarding the
parameter space, the GMM objective function and the unknown parameters 0 n.
Assumption 5 (a) The parameter space is a compact metric space with
metric .
(b) The functions qi;n : R pz ! R pq are B pz =B pq -measurable for each
2 , and continuous on for each z 2 R pz :
(c) The elements of the pq pq real matrices Pn are B-measurable, and Pn is
positive de…nite a.s. Furthermore P = p lim Pn exists and P is positive
de…nite.
(d) The minimizers o n are identi…ably unique in the sense that for every " > 0,
lim infn!1 inf 2 : ( ; o n ) " [ERn( )] 0 [ERn( )] > 0:
Compactness of the parameter space as maintained in Assumption 5(a) is
typical for the GMM literature. Assumptions 5(b),(c) imply that Qn( ; ) is
measurable for all 2 , and Qn(!; ) is continuous on . Given those assumptions
the existence of measurable functions b n that solves (23) follows, e.g., from
Lemma A3 of Pötscher and Prucha (1997).
Since P is positive de…nite, it is readily seen that Assumption 5(d) implies
that for every " > 0 :
lim inf
n!1
inf
2 : ( ; o n ) " Qn( ) Qn( o n) > 0;
observing that Qn( o n) = 0. Thus under Assumption 5(d) the minimizers
are identi…ably unique; compare, e.g., Gallant and White (1988), p.19. For
interpretation consider the important special case where o n = o , Rn( ) = R( )
o
and R( ) is continuous. In this case identi…able uniqueness of is equivalent
to the assumption that
o
is the unique solution of the moment conditions, i.e.,
Rn( ) 6= 0 for all 6= o ; compare, e.g., Pötscher and Prucha (1997), p. 16.
5.1 Consistency
o
Given the minimizers n are identi…ably unique, b n is a consistent estimator
o
p
for n if Qn converges uniformly to Qn, i.e., if sup 2 Qn(!; ) Qn( ) ! 0
as n ! 1; this follows immediately from, e.g, Pötscher and Prucha (1997),
Lemma 3.1.
We now proceed by giving a set of primitive domination and Lipschitz type
conditions for the moment functions that ensure uniform convergence of Qn to
15
o
n

Q n. The conditions are in line with those maintained in the general literature
on M-estimation, e.g., Andrews (1987), Gallant and White (1988), and Pötscher
and Prucha (1989,1994).
De…nition 3 Let fi;n : R pz ! R pq be B pz =B pq -measurable functions for
each 2 , then:
(a) The random functions fi;n(Zi;n; ) are said to be p-dominated on for
some p > 1 if sup n sup i2Dn E sup 2 jfi;n(Zi;n; )j p < 1.
(b) The random functions fi;n(Zi;n; ) are said to be Lipschitz in the parameter
on if
jfi;n(Zi;n; ) fi;n(Zi;n; )j Li;n(Zi;n)h( ( ; )) a.s., (25)
for all ; 2 and i 2 Dn; n 1, where h is a nonrandom function
with h(x) # 0 as x # 0, and Li;n are random variables with
< 1 for some > 0:
lim sup n!1 jDnj 1 P
i2Dn EL i;n
Towards establishing consistency of b n we furthermore maintain the following
moment and mixing assumptions.
Assumption 6 The moment functions qi;n(Zi;n; ) have the following properties:
(a) They are p-dominated on for p = 2.
(b) They are uniformly L1-NED on " = f"i;n; i 2 Tn; n 1g, where Dn Tn
D, and " is -mixing with -mixing coe¢ cients the conditions stated in
Assumption 4(b).
(c) They are Lipschitz in the parameter on .
Assumption 6(a) implies that sup n;i2Dn E jqi;n(Zi;n; )j p < 1 for each 2
. Assumptions 6(b) then allow us to apply the LLN given as Theorem 2 above
the sample moments Rn( ) = jDnj 1 P
i2Dn qi;n(Zi;n; ).
To verify Assumption 6(b) one can use either Proposition 1 or Proposition
2 to imply this condition from the lower level assumption that the data Zi;n are
L1-NED. For example, the qi;n are L1-NED, if the Zi;n are L1-NED and satisfy
the Lipschitz condition of Proposition 1. Note that no restrictions on the sizes
of the NED coe¢ cients are required.
Assumption 6(c) ensures stochastic equicontinuity of qi;n w.r.t. . Stochastic
equicontinuity jointly with Assumption 6(a) and the pointwise LLN enable us
to invoke the ULLN of Jenish and Prucha (2009) to prove uniform convergence
of the sample moments, which in turn is used to establish that Qn converges
16

uniformly to Q n. A su¢ cient condition for Assumption 6(c) is existence of
integrable partial derivatives of qi;n w.r.t. if 2 R k .
Our consistency results for the spatial GMM estimator given by (23) is summarized
by the next theorem.
Theorem 3 (Consistency) Suppose fDng is a sequence of …nite sets of D such
that jDnj ! 1 as n ! 1; where D R d ; d 1 is as in Assumption 1. Suppose
further that Assumptions 5 and 6 hold. Then
( b n; o n) p ! 0 as n ! 1;
and Q n( ) is uniformly equicontinuous on .
5.2 Asymptotic Normality
We next establish that the spatial GMM estimators de…ned by (23) is asymptotically
normally distributed. For that purpose, we need a stronger set of assumptions
than for consistency, including di¤erentiability of the moment functions in
. It proofs helpful to adopt the notation r in place of @=@ . 8
o
Assumption 7 (a) The minimizers n lie uniformly in the interior of with
Rk . Furthermore E [Rn( o n)] = 0.
(b) The functions qi;n : R pz ! R pq are continuously di¤erentiable w.r.t.
in the interior of for each z 2 R pz :
(c) The functions qi;n(Zi;n; o n) are uniformly L2-NED on " of size d, and
supn;i2Dn E jqi;n(Zi;n; o 0
2+
n)j < 1 for some
r qi;n(Zi;n; ) are uniformly L1-NED on ".
0
> 0. The functions
(d) The input process " = f"i;n; i 2 Tn; n 1g, where Dn Tn D, is -
mixing and the mixing coe¢ cients satisfy Assumption 2 for some < 0 ,
where 0 is the same as in Assumption 7(c).
(e) The functions r qi;n are p-dominated on for some p > 1:
(f) The functions r qi;n are Lipschitz in on .
(g) infn min(jDnj 1 n) > 0 where n = V ar P
(h) infn min [Er Rn( o n) 0 r Rn( o n)] > 0.
i2Dn qi;n(Zi;n; o n) .
8 To ensure that the derivatives are de…ned on the border of , we assume in the following
that the moment functions are de…ned on an open set containing , and that the qi;n and
r qi;n are restrictions to .
17

The …rst part of Assumption 7(a) is needed to ensure that the estimator
b n lies in the interior of with probability tending to one, and facilitates the
application of the mean value theorem to Rn( b n) around o n. The second part
states in essence that the moment conditions are correctly speci…ed. Its violation
will generally invalidate the limiting distribution result.
Assumptions 7(c),(d),(g) enable us to apply the CLT for vector-valued NED
processes given above as Corollary 1 to Rn( o n). Some low level su¢ cient conditions
for Assumption 7(c) are given below. To establish asymptotic normality,
we also need uniform convergence of r Rn on , which is implied via Assumptions
7(c),(d),(e),(f).
Given the above assumptions, we have the following asymptotic normality
result for the spatial GMM estimator de…ned by(23).
Theorem 4 Suppose fDng is a sequence of …nite sets of D such that jDnj ! 1
as n ! 1; where D R d ; d 1 is as in Assumption 1. Suppose further that
Assumptions 5- 7 hold. Then
where
A 1
n BnB 0 nA 10
n
1=2 jDnj 1=2 b n
o
n =) N(0; Ik);
An = [Er Rn( o n)] 0 P [Er Rn( o n)] and Bn = [Er Rn( o n)] 0 P
h
jDnj 1 i1=2 n :
Moreover, jAnj = O(1); A 1
n = O(1); jBnj = O(1); (BnB 0 n) 1 = O(1) and
hence, b n is jDnj 1=2 -consistent for
o
n.
As remarked above, relative to the existing literature Theorem 4 allows for
nonstationary processes and only assumes that qi;n and r qi;n are NED on
an -mixing input process, rather than postulating that qi;n and r qi;n are -
mixing. The latter assumption is restrictive in that the -mixing property is not
preserved under transformation. Since the NED property is preserved under a
wide range of transformations it accommodates a much larger class of dependent
spatial processes, including in…nite moving average and spatial autoregressive
models. As such, Theorem 4 should provide a basis for constructing con…dence
intervals and hypothesis testing in a wider range of spatial models.
Using Proposition 2, we now give some su¢ cient conditions for Assumption
7(c).
Assumption 8 The process fZi;n; i 2 Dn Tn; n 1g is uniformly L2-NED
on f"i;n; i 2 Tn; n 1g of size 2d(r 1)=(r 2) for some r > 2:
Assumption 9 For every sequence f ng on , the functions qi;n(Zi;n; n) and
r qi;n(Zi;n; n) satisfy Lipschitz condition (2) in z, that is, for gi;n = qi;n or
r qi;n:
jgi;n(z; n) gi;n(z ; n)j Bi;n(z; z ) jz z j :
18

Furthermore, for the r > 2 as speci…ed in Assumption 8,
sup sup
n;i2Dn s
B (s)
i;n < 1 and sup sup
2
n;i2Dn s
B (s)
i;n Zi;n
where B (s)
i;n = Bi;n(Zi;n; e Zs i;n ) with e Zs i;n = E [Zi;njFi;n(s)].
6 Conclusion
eZ s i;n r
< 1
The paper develops an asymptotic inference theory for a class of dependent
nonstationary random …elds that could be used in a wide range of econometric
models with cross-sectional or spatial dependence, e.g., social interaction models.
More speci…cally, the paper extends the notion of near-epoch dependent
(NED) processes used in the time series literature to spatial processes. This
allows to accommodate larger classes of dependent processes than mixing random
…elds. The class of NED random …elds is “closed with respect to in…nite
transformations ” and thus should be su¢ ciently broad for many applications
of interest. In particular, it covers autoregressive and in…nite moving average
random …elds as well as nonlinear spatial Bernoulli shifts. The NED property
is also compatible with considerable heterogeneity and preserved under transformations
under fairly mild conditions. Furthermore, a CLT and an LLN are
derived for spatial processes that are NED on an -mixing process. Apart from
covering a larger class of dependent processes, these limit theorems also allow
for arrays of nonstationary random …elds on unevenly spaced lattices. Building
on these limit results, the paper develops an asymptotic theory of spatial GMM
estimators, which provides a basis for inference in a broad range of models with
cross-sectional or spatial dependence.
Much of the random …elds literature assumes that the process resides on an
equally spaced grid. In contrast, and as in Jenish and Prucha (2009), we allow
for locations to be unequally spaced. The implicit assumption of …xed locations
seems reasonable for a large class of applications, especially in the short run.
Still, an important direction for future work would be to extend the asymptotic
theory to spatial processes with endogenous locations, while maintaining a set
of assumptions that are reasonably easy to interpret. 9 One possible approach
may be to augment the contributions of the present paper with theory from
point processes.
A Appendix: Proofs for Section 3
Proof of Proposition 4: Let Fi(s) = ("j; j 2 Z d : (i; j) s). By the
Minkowski inequality for in…nite sums, the conditional Jensen inequality, and
9 Pinkse et al. (2007) made an interesting contribution in this direction. Their catalogue
of assumption is at the level of Bernstein blocks. Without further su¢ cient conditions, veri-
…cation of those assumptions would typically be challenging in practical situations.
19

(6), we have
kYi E[YijFi(s)]k p = X
X
j2Z d : (i;j)>s
j2Z d ; (i;j)>s
jgijj k"j E ["jjFi(s)]k p K (s):
gij f"j E ["jjFi(s)]g
P
with K = 2 supj2Zd k"jkp < 1 and (s) = supi2Zd j2Zd : (i;j)>s jgijj. Since
(s) ! 0 as s ! 1 by condition (5) we have kYi E[YijFi(s)]kp ! 0 as s ! 1,
which proves the claim. Q.E.D.
Proof of Proposition 5: Let Fi;n(s) = ("j;n; 1 j n: (i; j) s). By the
Minkowski inequality, the conditional Jensen inequality, and (12), we have
X
kYi;n E[Yi;njFi;n(s)]k p
+
X
1 j n; (i;j)>s
1 j n; (i;j)>s
jbij;nj kvj;n E[vj;njFi;n(s)]k p K (s)
jaij;nj kXj;n E[Xj;njFi;n(s)]k p
P
with K = maxf2 kXj;nkp ; 2 kvj;nkpg < 1 and (s) = supn;1 i n 1 j n; (i;j)>s (jaij;nj + jbij;nj).
Since (s) ! 0 as s ! 1 by condition (11) we have kYi E[YijFi(s)]kp ! 0 as
s ! 1, which proves the claim. Q.E.D.
Proof of Proposition 6: We …rst show that Yi is well de…ned as the limit in
L2 as s ! 1 of the random variables
where
Y s
i = fi(" (s)
i+j ; j 2 Zd );
" (s)
i+j = "i+j for (i; j) s
0 for (i; j) > s :
In light of condition (15) we have for any s, m 2 N:
Y s+m
i
Y s
i 2
= fi(" (s+m)
i+j ) fi(" (s)
i+j ) 2
sup
i2Zd k"ik2
X
j2Z d :s< (i;j) s+m
X
j2Z d :s< (i;j) s+m
wij K sup
p
wij k"i+jk 2
X
i2Zd j2Zd wij
: (i;j)>s
with K = supi2Zd k"ik2. In light of (16)-(17) the r.h.s. converges monotonically
to zero as s ! 1. Thus clearly Y s
i is a Cauchy series in L2, and hence Yi is
well de…ned since the L2 space is a Banach space and as such complete.
Now, let Fi(s) = ("i+j; j 2 Zd : (i; j) s). By the minimum mean-squared
error property of the conditional expectation, we have
20

with
kYi E[YijFi(s)]k 2 fi("i+j; Z d ) fi(" (s)
i+j ; j 2 Zd ) 2
(s) = sup
K
X
j2Z d : (i;j)>s
X
i2Zd j2Zd : (i;j)>s
wij K (s);
wij ! 0
as s ! 1 by condition (16). This completes the proof of the lemma. Q.E.D.
B Appendix: Proofs for Section 4
Throughout this appendix let Fi;n(s) = ("j;n; j 2 Tn : (i; j) s) be the
-…eld generated by the random vectors "j;n located in the s-neighborhood of
location i. Furthermore, C denotes a generic constant that does not depend on
n and may be di¤erent from line to line.
The proof of the CLT builds on Ibragimov and Linnik (1971), pp. 352-355,
and makes use of the following lemmata:
Lemma B.1 (Brockwell and Davis (1991), Proposition 6.3.9). Let Yn, n =
1; 2; ::: and Vns; s = 1; 2; :::; n = 1; 2; :::, be random vectors such that
(i) Vns =) Vs as n ! 1 for each s = 1; 2; :::
(ii) Vs =) V as s ! 1, and
(iii) lims!1 lim sup n!1 P (jYn Vnsj > ) = 0 for every > 0:
Then Yn =) V as n ! 1:
Lemma B.2 (Ibragimov and Linnik (1971)) Let Lp(F1) and Lp(F2) denote,
respectively, the class of F1-measurable and F2-measurable random variables
satisfying k k p < 1. Let X 2 Lp(F1) and Y 2 Lq(F2): Then, for any 1
p; q; r < 1 such that p 1 + q 1 + r 1 = 1,
jCov(X; Y )j < 4 1=r (F1; F2) kXk p kY k q
where (F1; F2) = sup A2F1;B2F2 (jP (AB) P (A)P (B)j).
To prove the CLT for NED random …elds, we …rst establish some moment
inequalities and a slightly modi…ed version of the CLT for mixing …elds
developed in Jenish and Prucha (2009). It is helpful to introduce the following
notation. Let X = fXi;n; i 2 Dn; n 1g be a random …eld, then
kXk q := sup n;i2Dn kXi;nk q for q 1.
21

Lemma B.3 Let fXi;ng be uniformly L2-NED on a random …eld f"i;ng with
-mixing coe¢ cients (u; v; r) (u + v) b(r), 0. Let Sn = P
i2Dn Xi;n
and suppose that the NED coe¢ cients of fXi;ng satisfy P1 r=1 rd 1 (r) < 1
and kXk2+ < 1 for some > 0. Then,
(a)
(b)
jCov (Xi;nXj;n)j kXk2+
n
C1kXk2+ [h=3] d
b =(2+ ) o
([h=3]) + C2 ([h=3])
where h = (i; j) and = =(2+ ). If in addition P 1
r=1 rd( +1) 1 b =(2+ ) (r) <
1, then for some constant C < 1, not depending on n
jCov (Xi;nXj;n)j kXk2
V ar (Sn) C jDnj :
n
C3kXk2+ [h=3] d
b =(4+2 ) o
([h=3]) + C4 ([h=3])
where h = (i; j) and = =(4+2 ). If, in addition, P 1
r=1 rd( +1) 1 b =(4+2 ) (r) <
1 where = =(4 + 2 ), then for some constant C < 1, not depending
on n
V ar (Sn) CkXk2 jDnj :
Proof of Lemma B.3: (a) For any i 2 Dn and m > 0, let
m
i;n = E(Xi;njFi;n(m));
m
i;n = Xi;n
By the Jensen and Lyapunov inequalities, we have for all i 2 Dn; n; m 2 N
and any 1 q 2 + :
E m i;n
and thus
m
i;n q
m
i;n
q = EfjE(Xi;njFi;n(m))j q g EfE(jXi;nj q jFi;n(m))g = E jXi;nj q
kXi;nk q kXk 2+ ;
m
i;n q
2 kXi;nk q 2 kXk 2+ :
Thus, both m i;n and m i;n are uniformly L2+ bounded. Also, note that
sup
n;i2Dn
m
i;n 2
(m);
given that the fXi;ng is uniformly L2-NED on f"i;ng and thus the NED-scaling
factors can be chosen w.l.g. to be one. Furthermore, let ( m i;n) denote the
-…eld generated by m i;n: Since ( m i;n) Fi;n(m), the mixing coe¢ cients of m i;n
satisfy
(1; 1; r)
1; r 2m
(Mm d ; Mm d ; r 2m); r > 2m
22

where (u; v; r) are the mixing coe¢ cients of the input process ", since the mneighborhood
of any point on D contains at most Mm d points of D for some M
that does not depend on m, see the proof of Lemma A.1 of Jenish and Prucha
(2009).
Now, decompose Xi;n and and Xj;n as
Xi;n = [h=3]
i;n
where h = (i; j). Then,
jCov (Xi;nXj;n)j = Cov
[h=3]
+ i;n ; and Xj;n = [h=3] [h=3]
j;n + j;n ;
Cov
+ Cov
[h=3]
i;n
[h=3]
i;n ;
[h=3]
j;n ;
+ [h=3]
i;n ;
[h=3]
j;n
[h=3]
i;n
[h=3]
j;n
+ Cov
+ Cov
+ [h=3]
j;n
[h=3]
i;n ;
[h=3]
i;n ;
[h=3]
j;n
[h=3]
j;n
:
(B.1)
We will now bound separately each term on the r.h.s. of the last inequality.
First, using Lemma B.2 with p = q = 2 + ; and r = (2 + )= yields the
following bound on the …rst term:
Cov
[h=3]
i;n ;
[h=3]
j;n
4
[h=3]
i;n
4kXk 2 2+
2+
C1kXk 2 2+ [h=3] d
[h=3]
j;n
2+
=(2+ ) (1; 1; [h=3]) (B.2)
=(2+ ) M [h=3] d ; M [h=3] d ; h 2 [h=3]
b =(2+ ) ([h=3])
where = =(2 + ).
Second, the Cauchy-Schwartz inequality gives the following bound on the
second and third terms:
Cov
[h=3]
i;n ;
[h=3]
j;n
4
[h=3]
i;n
2
[h=3]
j;n
Similarly, the fourth term can be bounded as:
Cov
[h=3]
i;n ;
[h=3]
j;n
4
[h=3]
i;n
2
[h=3]
j;n
Collecting (B.2)-(B.4), we have
n
jCov (Xi;nXj;n)j kXk2+ C1kXk2+ [h=3] d
2
2
4kXk2 ([h=3]) (B.3)
8kXk2 ([h=3]) (B.4)
b =(2+ ) o
([h=3]) + C2 ([h=3])
which proves the …rst inequality.
Using this inequality as well as the bounds on the sizes of the sets given in
23

Lemma A.1 of Jenish and Prucha (2009), we have
V ar (Sn)
X
V ar (Xi;n) +
X
i2Dn
i;j2Dn;i6=j
2jDnj kXk 2
2+ + C1kXk 2 2+
+C2kXk2+
X
1X
jCov (Xi;nXj;n)j
X
1X
i2Dn r=1 j2Dn: (i;j)2[r;r+1)
X
i2Dn r=1 j2Dn: (i;j)2[r;r+1)
X
2jDnj kXk 2
2+ + C jDnj kXk2+
for some constant C < 1, not depending on n.
r=1
([ (i; j)=3])
[ (i; j)=3] d
b =(2+ ) ([ (i; j)=3])
"
1X
r d( +1) 1 b =(2+ ) 1X
(r) + r d 1 #
(r)
(b) To prove the second part of the lemma, apply Lemma B.2 with p = 2+ ;
q = 2, and r = 2(2 + )= to obtain the following bound on the …rst term:
Cov
[h=3]
i;n ;
[h=3]
j;n
C3kXk2+ kXk2 [h=3] d
r=1
b =(4+2 ) ([h=3]) ; (B.5)
where = =(4 + 2 ). The other terms on the r.h.s. of (B.1) are bounded as
in part (a). Collecting (B.3)-(B.5) gives
jCov (Xi;nXj;n)j
n
kXk2 C3kXk2+ [h=3] d
b =(4+2 ) o
([h=3]) + C4 ([h=3])
as required. Finally, using similar arguments as in the proof of part (a), we can
bound V ar (Sn) as
V ar (Sn) CkXk2jDnj
for some constant C < 1, not depending on n. Q.E.D.
Theorem B.1 Suppose fDng is a sequence of …nite subsets of D, satisfying
Assumption 1, with jDnj ! 1 as n ! 1: Suppose further that f"i;n; i 2 Dn; n 2
Ng is an array of zero-mean random variables with -coe¢ cients (u; v; r)
C(u + v) b(r) for some constants C < 1 and 0. Suppose for some > 0
and > 0
and
lim
k!1 sup E[j"i;nj
n;i2Dn
2+ 1(j"i;nj > k)] = 0
b(r) = O(r
d(2 +1)
with = max f ; 1= g, and suppose lim infn!1 jDnj 1 2 n > 0, then
where 2 n = V ar P
1
n
i2Dn "i;n .
X
"i;n =) N(0; 1):
i2Dn
24
)
CjDnj

The above CLT is in essence a variant of CLT for -mixing random …elds
given as Corollary 1 of Theorem 1 in Jenish and Prucha (2009), applied to
mixing coe¢ cients of the type (u; v; r) C(u + v) b(r); 0.
Proof of Theorem B.1: The proof of the theorem is largely the same as the
proof of Theorem 1 in Jenish and Prucha (2009). We will …rst show that all
assumptions of that theorem except Assumption 3(c) are satis…ed. We will then
show that the entire proof goes through if Assumption 3(c) is replaced by the
condition b(r) = O(r d(2 +1) ) with = max f ; 1= g.
Clearly, Assumptions 1, 2 and 5 of that theorem with ci;n = 1 are satis…ed.
To verify Assumptions 3(a) note that
1X
(1; 1; r)r [d(2+ )= ] 1 1X
1X
2d( 1= ) 1
C r C r 1 < 1;
r=1
r=1
since = max f ; 1= g. As shown in the proof of Corollary 1 in Jenish and
Prucha (2009), the latter condition implies Assumption 3(a) of Theorem 1 in
Jenish and Prucha (2009). Furthermore, it is easy to see that Assumption 3 (b)
is also satis…ed. Indeed, for any u + v 4, we have
1X
d
(u; v; r)r
1
1X
4 C r d 1 b(r)
1X
C r d 1 r d(2 +1) < 1:
r=1
r=1
Thus, all assumptions, except Assumption 3(c), of Theorem 1 in Jenish and
Prucha (2009) hold, and hence, all steps of its proof which do not rely on that
assumptions remain valid in our case. Assumption 3(c) is only used in step 5
of that proof. Speci…cally, all arguments in that step continue to hold given we
show that there exists sequence mn such that
and
r=1
r=1
m d njDnj 1=2 ! 0 as n ! 1 (B.6)
(1; jDnj ; mn)jDnj 1=2 ! 0 as n ! 1 (B.7)
(Though 1;1 is used instead of 1;jDnj in the proof of Theorem 1 of Jenish and
Prucha (2009), in fact the proof only relies on the coe¢ cient 1;jDnj; see Step 9
(jEA3;nj ! 0) of the proof in the working version of the paper).
The desired sequence mn can be chosen as
It is immediate that (B.6) holds,
To verify (B.7), observe
(1; jDnj ; mn)jDnj 1=2
mn = jDnj 1=2 = log jDnj 1=d
:
m d njDnj 1=2 = (log jDnj) 1 ! 0:
CjDnj +1=2 m
CjDnj +1=2 jDnj
CjDnj
d(2 +1)
n
1=2 jDnj
=(2d) [log jDnj] (2 +1)+ =d ! 0:
25
=(2d) (2 +1)+ =d
[log jDnj]

The rest of the proof is the same, word-by-word, as the proof of Theorem 1 of
Jenish and Prucha (2009). Q.E.D.
Proof of Theorem 1: Since the proof is lengthy it is broken into steps.
1. Decomposition of Yi;n
For any …xed s > 0, decompose Xi;n as
where
Let
Yi;n = s i;n + s i;n
s
i;n = E(Yi;njFi;n(s)),
Sn;s = X
i2Dn
s
i;n = Yi;n
s
i;n; e Sn;s = X
i2Dn
s
i;n
s
i;n
2
n;s = V ar [Sn;s] ; e 2
h i
n;s = V ar eSn;s
Repeated use of the Minkowski inequality yields:
Observe that
and hence
j n n;sj en;s, j n en;sj n;s: (B.8)
E [E(Yi;njFi;n(s))jFi;n(m))] =
E(Yi;njFi;n(s)); m s;
E(Yi;njFi;n(m)); m < s:
s
i;n E( s i;njFi;n(m))
2
= kYi;n E[Yi;njFi;n(s)] E[Yi;njFi;n(m)] + E[(Yi;njFi;n(s))jFi;n(m)]k2 =
kYi;n E(Yi;njFi;n(m))k2 C (m); if m s;
kYi;n E(Yi;njFi;n(s))k2 C (s) C (m); if m < s:
since by de…nition the sequence (m) is non-increasing. Thus, for any …xed
s > 0, s i;n is uniformly L2-NED on " with the same NED coe¢ cients (m)
as the random …eld fYi;ng. Furthermore, as shown in the proof of Lemma B.3,
s
i;n is also uniformly L2+ bounded.
2. Bounds for the Variances of P Yi;n and P s i;n
First note that exists 0 < B < 1 such that for all n
BjDnj
2
n: (B.9)
Furthermore, in light of Assumption 2, and observing that = =(4 + 2 )
= =(2 + ) and b =(2+ ) (r) b 2(2+ ) (r) we have
1X
r d( +1) 1 b
r=1
=(2+ )
1X
r d( +1) 1 b =(4+2 ) (r)
r=1
26
1X
r=1
1X
r=1
r d( +1) 1 b 2(2+ ) (r) < 1;
r d( +1) 1 b 2(2+ ) (r) < 1:

Using part (a) of Lemma B.3 with Xi;n = Yi;n, we have
BjDnj
2
n = V ar (Sn) C jDnj .
for some B > 0. Using part (b) of Lemma B.3 with Xi;n = s i;n we have
Hence,
e 2
n;s = V ar e Sn;s C jDnj k s i;nk2 = CjDnj (s) (B.10)
lim lim sup
s!1 n!1
Furthermore, by (B.8) we have
lim lim sup
s!1 n!1
and hence for all s 1 and n 1
1
e 2
n;s
2
n
n;s
n
n;s
n
C lim
s!1
(s) = 0: (B.11)
en;s
lim lim sup = 0 (B.12)
s!1 n!1 n
C < 1: (B.13)
3. CLT for P s
i2Dn i;n
We now show that for any …xed s > 0, s i;n satis…es the CLT given in Theorem
B.1.
h
si;n 2+
First, since supn;i2Dn E
i
< 1, the process is uniformly
L2+
0-integrable for
Second, note that since
and r > 2s
0
= =2, i.e.,
lim
k!1 sup E[
n;i2Dn
s 2+ =2
i;n 1(
s
i;n > k)] = 0:
s
i;n
s
i;n is a measurable function of "i;n for any u; v 2 N
(u; v; r) (uMs d ; vMs d ; r 2s) C (u + v) b (r 2s)
We next to show that b(r) = O(r d(2 +1) ) for = max f ; 2= g and some
> 0. By assumption,
1X
r=1
where = =(2 + ), which implies
r d( +1) 1 b 2(2+ ) (r) < 1;
b (r) = o(r 2d(2+ )( +1)= ) = o(r d[2( +2= )+1] d ) = o(r d[2 +1] d )
since +2= for = max f ; 2= g. Thus, b(r) = O(r d(2 +1) ) for = d.
27

We next show that for su¢ ciently large s;
By (B.9),
0 < lim inf
n!1 jDnj 1 2 n;s:
B 1=2
inf jDnj 1=2 n
Since lims!1 (s) = 0; there exists s such that in light of (B.10) for all s s ,
Hence by (B.8) for all s s ;
jDnj 1=2 en;s C 1=2 (s) B 1=2 =2: (B.14)
jDnj 1=2 ( n en;s) jDnj 1=2 n;s
and thus infn jDnj 1=2 n;s infn jDnj 1=2 n sup n jDnj 1=2 en;s. Using (??)
and (B.14), we have
Thus, for all s s ,
lim inf
n!1 jDnj 1=2 1=2 B1=2 B1=2
n;s B = > 0
2 2
1
n;s
X
i2Dn
s
i;n =) N(0; 1) as n ! 1: (B.15)
Since the …rst s terms do not a¤ect the analysis below we take in the following
s = 1:
P 1
n i2Dn Yi;n
4. CLT for
Finally, using Lemma B.1 we now show that, given the maintained NED
assumption, the just established CLT in (B.15) for the approximators
be carried over to the the Yi;n. De…ne
s
i;n can
X
X
X
Wn =
1
n
i2Dn
Yi;n; Vns = 1
n
i2Dn
s
i;n, Wn Vns = 1
n
so that we can exploit Lemma B.1 to prove that
X
Wn = Yi;n =) V N(0; 1):
1
n
i2Dn
i2Dn
We …rst verify condition (iii) of Lemma B.1. By Markov’s inequality and (B.11),
for every > 0 we have
lim lim sup P (jWn Vnsj > ) = lim lim sup P (
s!1 n!1
s!1 n!1
lim lim sup
s!1 n!1
28
e 2
n;s
= 0:
2 2
n
1
n
X
i2Dn
s
i;n
s
i;n > )

Next observe that
Vns = n;s
n
"
1
n;s
X
i2Dn
We proceed to show Wn =) V by contradiction. For that purpose let M be the
set of all probability measures on (R;B), and observe that we can metrize M by,
e.g., the Prokhorov distance d(:; :). Let n and be the probability measures
corresponding to Wn and V , respectively, then Wn =) V , or n =) , i¤
d( n; ) ! 0 as n ! 1. Now suppose n does not converge to . Then for
some > 0 there exists a subsequence fn(m)g such that d( n(m); ) > for
all n(m). By (B.13), we have 0 n;s= n C < 1 for all s; n 1. Hence,
0 n(m);s= n(m) C < 1 for all n(m). Consequently, for s = 1 there
exists a subsubsequence fn(m(l1))g such that n(m(l1));1= n(m(l1)) ! p(1) as
l1 ! 1. For s = 2, there exists a subsubsubsequence fn(m(l1(l2)))g such that
n(m(l1(l2)));2= n(m(l1(l2))) ! p(2) as l2 ! 1. The argument can be repeated
for s = 3; 4:::. Now construct a subsequence fnlg such that n1 corresponds
to the …rst element of fn(m(l1))g, n2 corresponds to the second element of
fn(m(l1(l2)))g, and so on, then
lim
l!1
nl;s
nl
s
i;n
for s = 1; 2; : : : Given (B.15), it follows that as l ! 1
Then, it follows from (B.12) that
lim jp(s) 1j lim
s!1 s!1 lim
l!1 p(s)
#
Vnls =) Vs N(0; p 2 (s)):
:
= p(s) (B.16)
nl;s
nl
+ lim
s!1 sup
n 1
n;s
n
1 = 0:
Thus Vs =) V and thus by Lemma B.1 Wnl =) V N(0; 1) as l ! 1.
Since fnlg fn(m)g this contradicts the assumption that d( n(m); ) > for
all n(m). This completes the proof of the CLT. Q.E.D.
Proof of Corollary 1: To prove the theorem, we apply the Cramer-Wold
device, and verify that for every 2 Rk P 1
with j j = 1, n Vi;n ) N(0; 1),
where Vi;n = 0 Yi;n. Observe that using the properties of norms, we have
and
jVi;nj = 0 Yi;n j j jYi;nj = jYi;nj
1(jVi;nj > k) = 1(
0 Yi;n > k) 1(jYi;nj > k);
and thus limk!1 sup n;i2Dn E[jVi;nj 2+ 1(jVi;nj > k)] = 0. Furthermore, observe
that
kVi;n E(Vi;n j Fi;n(s))k 2 j j kYi;n EYi;n j Fi;n(s))k 2 di;n (s)
29

and that for 2 n = V ar( P
i2Dn Vi;n) = 0 n we have
inf
n jDnj 1 2 n = inf
n jDnj 1 0 n inf
n jDnj 1 min( n) > 0:
From this we see that under the maintained assumptions Vi;n satis…es all assumptions
P
of the CLT for scalar-valued random …elds (Theorem 1) and, there-
1 fore, n Vi;n ) N(0; 1) as claimed.
Next de…ne Xi;n = Vi;n, then by analogous arguments as above
jXi;nj jYi;nj :
From the maintained uniform L2+ integrability of jYi;nj it then follows that
kXk 2+ kY k 2+ , which shows that the 2+ moments of Xi;n can be bounded
by a constant that does not depend on . Consequently if follows from the last
inequality in the proof of part (a) of Lemma B.3 that
V ar( X
i2Dn
Xi;n) = 0 n CjDnj
where C < 1 does not depend on n and . Hence
sup
n
jDnj 1 max( n) = sup jDnj
n
1 sup
j j=1
0 n C < 1.
This proves the second claim of the lemma. Q.E.D.
Proof of Theorem 2: To prove the theorem, it su¢ ces to show that jDnj 1 P
0. First we show that for each given s > 0, the conditional mean V s
i;n =
E(Yi;njFi;n(s)) satis…es the assumptions of the L1-norm LLN of Jenish and
Prucha (2009, Theorem 3). Using the Jensen and Lyapunov inequalities gives
for all s > 0, i 2 Dn; n 1:
E V s p
i;n EfE(jYi;nj p jFi;n(s))g sup E jYi;nj
n;i2Dn
p < 1:
So, V s
i;n is uniformly Lp-bounded for p > 1 and hence uniformly integrable.
For each …xed s; V s
i;n is a measurable function of f"j;n; j 2 Tn : (i; j) sg.
Observe that under Assumption 1 there exists a …nite constant C such that the
cardinality of the set fj 2 Tn : (i; j) sg is bounded by Csd ; compare Lemma
A.1 in Jenish and Prucha (2009). Hence,
V s(1; 1; r)
and thus in light of Assumption 6(b)
1X
r=1
r d 1 V s(1; 1; r)
X2s
r=1
1; r 2s
Cs d ; Cs d ; r 2s ; r > 2s
r d 1 + '(Cs d ; Cs d )
30
1X
(r + 2s) d 1 b(r) < 1:
r=1
!
i2Dn (Yi;n EYi;n) L1

The above shows that indeed, for each …xed s, V s
i;n satis…es the assumptions
of the L1-norm LLN of Jenish and Prucha (2009, Theorem 3). Therefore, for
each s; we have
jDnj
X
1
[E(Yi;njFi;n(s)) EYi;n]
i2Dn
1
! 0 as n ! 1: (B.17)
Furthermore observe that from (??) and the Minkowski inequality
jDnj
1 X
i2Dn
(Yi;n E(Yi;njFi;n(s)))
1
(s): (B.18)
Given (B.17) and (B.18), and observing that lims!1 (s) = 0 it now follows
that
lim
n!1
jDnj 1 X
lim lim sup
s!1 n!1
+ lim
s!1 lim
n!1
i2Dn
jDnj
(Yi;n EYi;n)
1 X
i2Dn
jDnj 1 X
i2Dn
1
= lim
s!1 lim
n!1
(Yi;n E(Yi;njFi;n(s)))
(E(Yi;njFi;n(s)) EYi;n)
jDnj 1 X
1
1
= 0:
i2Dn
(Yi;n EYi;n)
This completes the proof of the LLN. Q.E.D.
C Appendix: Proofs for Section 5
Proof of Theorem 3: We show that
sup
2
Qn( ) Q n( )
p
! 0 (C.1)
as n ! 1. As discussed in the text, given that the o n are identi…ably unique it
then follows immediately from, e.g., Pötscher and Prucha (1997), Lemma 3.1,
that ( b n; o n) p ! 0 as n ! 1 as claimed.
We start by proving that
jDnj
1 X
i2Dn
[qi;n(Zi;n; ) Eqi;n(Zi;n; )] p ! 0 (C.2)
for each 2 , by applying the LLN given as Theorem 2 in the text to
qi;n(Zi;n; ). First note that by Assumption 6(a), we have sup n;i2Dn E jqi;n(Zi;n; )j p <
1 for each 2 and p = 2, which veri…es Assumption 4(a) for qi;n(Zi;n; ) with
ci;n = 1. By Assumption 6(b), the qi;n(Zi;n; ) are uniformly L1-NED on ", and
31
1

hence w.l.g. we can take di;n = 1. Furthermore, by Assumption 6(b) the input
process " is -mixing, and the -mixing coe¢ cients satisfy Assumption 4(b).
Consequently (C.2) follows directly from Theorem 2 applied to qi;n(Zi;n; ).
Next observe that by Proposition 1 of Jenish and Prucha (2009), Assumption
6(c) implies that qi;n is L0 stochastically equicontinuous on , i.e., for every
" > 0
1
lim sup
n!1 jDnj
X
P
i2Dn
sup
( ; )
jqi;n(Zi;n; ) qi;n(Zi;n; )j > "
!
! 0 as ! 0:
Furthermore, in light of Assumption 6(a) the qi;n(Zi;n; ) clearly satisfy the
domination condition postulated by the ULLN in Jenish and Prucha (2009),
stated as Theorem 2 in that paper. Given that we have already veri…ed the
pointwise LLN in (C.2) it now follows directly from that theorem that
1 P
sup jRn( ) ERn( )j
2
p ! 0 (C.3)
with Rn( ) = jDnj i2Dn qi;n(Zi;n; ), and that the ERn( ) are uniformly
equicontinuous on in the sense that
lim sup sup
n!1 2
sup
( ; )
To prove (C.1) observe that
sup
2
sup
2
sup
2
jERn( ) ERn( )j ! 0 as ! 0:
Qn( ) Q n( ) (C.4)
jRn( ) 0 P Rn( ) ERn( )P ERn( )j + sup jRn( )
2
0 (Pn P )Rn( )j
jRn( ) 0 P Rn( ) ERn( )P ERn( )j + 2 sup jRn( )j
2
2 jPn P j :
Furthermore observe that Assumption 6(a) we have E [sup 2 jqi;n(Zi;n; )j]
K and E [sup 2 jqi;n(Zi;n; )j] 2 K for some …nite constant K. Thus
and
sup
2
E jRn( )j E sup jRn( )j jDnj
2
E sup jRn( )j
2
2
jDnj
jDnj
2 X
i;j2Dn
2 X
i;j2Dn
E sup
2
"
1 X
i2Dn
E sup jqi;n(Zi;n; )j K (C.5)
2
jqi;n(Zi;n; )j sup jqj;n(Zj;n; )j (C.6)
2
E sup jqi;n(Zi;n; )j
2
2 # 1=2 "
E sup jqj;n(Zj;n; )j
2
Now consider the …rst terms on the r.h.s. of the last inequality of (C.4). From
(C.5) we see that E jRn( )j takes on its values in a compact set. Given (C.3) it
32
2 # 1=2
K:

now follows immediately from part (a.I) of Lemma 3.3 of Pötscher and Prucha
(1997) that
sup jRn( )
2
0 P Rn( ) ERn( )P ERn( )j p ! 0: (C.7)
Next we show that also the second term on the r.h.s. of the last inequality
of (C.4) converges in probability to zero. To see that this is indeed the case
observe that sup 2 jRn( )j 2 = Op(1) in light of (C.6) and jPn P j p ! 0 by
assumption. This completes the proof of (C.1).
Having established that ERn( ) are uniformly equicontinuous on , the
uniform equicontinuity of Q n( ) on follows immediately from Lemma 3.3(b)
of Pötscher and Prucha (1997). Q.E.D.
Proof of Theorem 4: Clearly by Theorem 3 we have b o
n n = op(1).
Step 1. The estimators b n corresponding to the objective function (23)
satisfy the following …rst order conditions:
h
jDnj 1=2 Rn( b i
n) = op(1): (C.8)
r Rn( b n) 0 Pn
The op(1) term on the r.h.s. re‡ects that the …rst order conditions may not
hold if b n falls onto the boundary of , and that the probability of that event
goes to zero as n ! 1, since the o n are uniformly in the in the interior of by
Assumption 7(a). If b n is in the interior of , then the l.h.s. of (C.8) is zero.
Taking the mean value expansion of Rn( b n) about
Rn( b n) = Rn( o n) + r Rn( e n)( b n
o
n yields
o
n) (C.9)
where e n 2 is between b n and o n (component-by-component). Let
bAn = r Rn( b n) 0 Pnr Rn( e n) and b Bn = r Rn( b n) 0 Pn
then combining (C.8) and (C.9) gives:
=
=
jDnj 1=2 b n
h
I
h
I
i
A b+ b
n An
i
A b+ b
n An
o
n
jDnj 1=2 b n
jDnj 1=2 b n
where b A + n denotes the generalized inverse of b An.
o
n
o
n
h
jDnj 1 i1=2 n ;
(C.10)
bA + n r Rn( b n) 0 h
Pn jDnj 1=2 Rn( o i
n) + b A + n op(1)
bA + n b h
1=2
Bn n jDnj Rn( o i
n) + b A + n op(1);
Step 2. By Assumptions 7(c) the qi;n(Zi;n; o n) are uniformly L2-NED and
uniformly L2+ -integrable with ci;n = 1. Given Assumptions 7(d),(g) it is now
readily seen that the process fqi;n(Zi;n; o n); i 2 Dng satis…es all assumptions
of the CLT for vector-valued NED processes, given as Corollary 1 in the text,
33

with cin = 1. (Note that Assumption 3(d) is satis…ed automatically since the
qi;n(Zi;n; o n) are uniformly L2-NED.) Hence,
1=2
n jDnj Rn( o X
1=2
n) = n qi;n(Zi;n; o n) =) N(0; Ipq ); (C.11)
with n = V ar P
i2Dn
i2Dn qi;n(Zi;n; o n) and sup n max
h
jDnj 1 i
n < 1.
Step 3. By Assumptions 7(c),(d),(e) the functions r qi;n(Zi;n; ) satisfy for
each 2 the LLN given as Theorem 2 in the text with ci;n = 1, observing
that Assumption 4(b) is implied by 2. By argumentation analogous as used in
the proof of consistency we have
jDnj
1 X
i2Dn
(r qi;n(Zi;n; ) Er qi;n(Zi;n; )) p ! 0:
By Proposition 1 of Jenish and Prucha (2009), Assumption 7(f) implies that the
r qi;n(Zi;n; ) are uniformly L0-equicontinuous on . Given L0-equicontinuity
and Assumption 7(e), we have by the ULLN of Jenish and Prucha (2009), Theorem
2,
sup jr Rn( )
2
Er Rn( )j p ! 0: (C.12)
and furthermore, the Er Rn( ) are uniformly equicontinuous on
that
in the sense
sup jEr Rn( ) Er Rn( )j ! 0 (C.13)
lim sup sup
n!1 02 j
as ! 0. In light of (C.12) and (C.13), and given that b n
hence e o
n n = op(1), if follows further that
0 j<
o
n = op(1) and
r Rn( b n) Er Rn( o n) p ! 0; and r Rn( e n) Er Rn( o n) p ! 0:
Hence,
bAn An
where b An and b Bn are as de…ned above, and
p
p
! 0 and Bn
b Bn ! 0; (C.14)
An = [Er Rn( o n)] 0 P [Er Rn( o n)] and Bn = [Er Rn( o n)] 0 P
h
jDnj 1 i1=2 n :
Step 4. Given Assumptions 7(e),(f), and since P is positive de…nite, we
have jAnj = O(1) and A 1
n = O(1), respectively. Hence by, e.g., Lemma
F1 in Pötscher and Prucha (1997) we have b An = Op(1), b A + n = Op(1), b An is
nonsingular with probability tending to one, and b A + n A 1 p
n ! 0. In light of the
above it follows from (C.10) that
jDnj 1=2 b
n
o
n =
h
A b+ b 1=2
n Bn n jDnj Rn( o =
i
n) + op(1)
A 1
h
1=2
n Bn n jDnj Rn( o i
n) + op(1)
34

Recalling that supn h
max jDnj 1 i
n < 1, Assumptions 7(e) implies that
jBnj = Op(1). In light of Assumption 7(g),(h) BnB0 n is invertible and fur-
thermore (BnB0 n) 1 = O(1). Thus A 1
n BnB0 nA 10
n
O(1) and therefore
A 1
n BnB 0 nA 10
n
= A 1
n BnB 0 nA 10
n
The claim that A 1
n BnB0 nA 10
n
1=2 jDnj 1=2 b n
h
1=2 1
An Bn
1=2 jDnj 1=2 b n
o
n
1
1=2
n jDnj Rn( o n)
o
n
jAnj 2 (BnB 0 n) 1 =
i
+ op(1):
=) N(0; Ik) now fol-
lows, e.g., from Corollary F4(b) in Pötscher and Prucha (1997): Q.E.D.
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