On Spatial Processes and Asymptotic Inference under Near-Epoch Dependence

Abstract

The development of a general inferential theory for nonlinear models with cross-sectionally 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 first extends the notion of near-epoch dependent (NED) processes used in the time series literature to random fields. The class of processes that is NED on, say, an α-mixing process, is shown to be closed under infinite transformations, and thus accommodates models with spatial dynamics. This would generally not be the case for the smaller class of α-mixing processes. The paper then derives a central limit theorem and law of large numbers for NED random fields. These limit theorems allow for fairly general forms of heterogeneity including asymptotically unbounded moments, and accommodate arrays of random fields 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 spatial dependence.

1 Introduction

Models with spatially dependent data have recently attracted considerable attention in various fields of economics including labor and public economics, IO, political economy, international and urban economics. In these models, strategic interaction, neighborhood effects, shared resources and common shocks lead to interdependences in the dependent and/or explanatory variables, with the variables indexed by their location in some socioeconomic space.¹ Insofar as these locations are deterministic, observations can be modeled as a realization of a dependent heterogenous process indexed by a point in ℝ^d, d > 1, i.e., as a random field.

The aim of this paper is to define a class of random fields that is sufficiently 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.

To date, linear spatial autoregressive models, also known as Cliff Ord (1981) type models², have arguably been one of the most popular approaches to modeling spatial dependence in the econometrics literature. The asymptotic theory in these models is facilitated, loosely speaking, by imposing specific structural conditions on the data generating process, and by exploiting some underlying independence assumptions. Another popular approach to model dependence is through mixing conditions. Various mixing concepts developed for time series processes have been extended to random fields. However, the respective limit theorems for random fields have not been sufficiently 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 filling this gap, Jenish and Prucha (2009) have recently introduced a set of limit theorems (CLT, ULLN, LLN) for α-mixing random fields on unevenly spaced lattices that allow for nonstationary processes with trending moments.

However, some important classes of dependent processes are not necessarily mixing, including linear autoregressive (AR) and infinite moving average (MA(∞)) processes. Sufficient conditions for the α-mixing property of linear processes³ are fairly stringent, and involve three types of restrictions (i) smoothness of the density functions of the innovations, (ii) sufficiently fast rates of decay of the coefficients, and (iii) invertibility of the linear process. There are examples demonstrating that the mixing property can fail for any of these reasons. In particular, Andrews (1984) showed that a simple AR(1) process of independent Bernoulli innovations is not α-mixing. Similar examples have been constructed for random fields, see, e.g. Doukhan and Lang (2002). Thus, mixing may break down in the case of discrete innovations. Further, Gorodetskii (1977) showed that the strong mixing property may fail even in the case of continuously distributed (normal) innovations when the coefficients of the linear process do not decline sufficiently fast. As these examples suggest, the mixing property is generally not preserved under infinite transformations of mixing processes. Yet stochastic processes generated as functionals of some underlying process arise in a wide range of models, with autoregressive models being the leading example. Thus, it is important to develop an asymptotic theory for a generalized class of random fields that is “closed with respect to infinite transformations”.

To tackle this problem, the paper first 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 (1975), Bierens (1981), Wooldridge (1986), Gallant and White (1988), Andrews (1988), Pötscher and Prucha (1997), Davidson (1992, 1993, 1994) and de Jong (1997), among others. Doukhan and Louhichi (1999) introduced an alternative class of dependent processes called “θ-weakly dependent”.

In deriving our limit theorems we then only assume that the process is NED on a mixing input process, i.e., that the process can be approximated by a mixing input process in the NED sense, rather than to assume that the process itself is mixing. Of course, every mixing process is trivially also NED on itself, and thus the class of processes that are NED on a mixing input process includes the class of mixing processes. There are several advantages to working with the enlarged class of process that are NED on a mixing process. First, linear processes with discrete innovations, which results in the process to not satisfy the strong mixing property, will still be NED on the mixing input process of innovations, provided the latter are mixing. We note that, in particular, the NED property holds in both examples of Andrews (1984) and Gorodetskii (1977), by Proposition 1 of this paper. Second, as shown in this paper, nonlinear MA(∞) random fields are also NED under some mild conditions, while such conditions are not readily available for mixing. Third, the NED property is often easy to verify. For instance, the sufficient conditions for MA(∞) random fields involve only smoothness conditions on the functional form and absolute summability of the coefficients, which are not difficult to check, while verification of mixing is usually more difficult.

The paper derives a CLT and an LLN for spatial processes that are near epoch dependent on an α-mixing input process. These limit theorems allow for fairly general forms of heterogeneity including asymptotically unbounded moments, and accommodate arrays of random fields 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 confidence 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 α-mixing.⁴

The rest of the paper is organized as follows. Section 2 introduces the concept of NED spatial processes and gives of examples random fields satisfying this condition. Section 3 contains the LLN and CLT for NED spatial processes. Section 4 establishes the asymptotic properties of spatial GMM estimators. All proofs are relegated to the appendices.

2 NED Spatial Processes

Let D ⊂ ℝ^d, d ≥ 1, be a lattice of (possibly) unevenly placed locations in ℝ^d, and let Z = {Z_i,n, i ∈ D_n, n ≥ 1} and ε = {ε_i,n, i ∈ T_n, n ≥ 1} be triangular arrays of random fields defined on a probability space (Ω, , P) with D_n ⊆ T_n ⊆ D. The space ℝ^d is equipped with the metric ρ(i, j) = max_1≤l≤d |j_l − i_l|, where i_l is the l-th component of i. The distance between any subsets U, V ⊆ D is defined as ρ(U, V ) = inf {ρ(i, j): i ∈ U and j ∈ V }. Furthermore, let |U| denote the cardinality of a finite subset U ⊂ D.

The random variables Z_i,n and ε_i,n are possibly vector-valued taking their values in ℝ^p_z and ℝ^p_ε, respectively. We assume that ℝ^p_z and ℝ^p_ε are normed metric spaces equipped with the Euclidean norm, which we denote (in an obvious misuse of notation) as |.|. For any random vector Y, let ||Y||_p = [E |Y|_p]^1/p, p ≥ 1, denote its L_p-norm. Finally, let (s) = σ(ε_j,n; j ∈ T_n: ρ(i, j) ≤ s) be the σ-field generated by the random vectors ε_j,n located in the s-neighborhood of location i.

Throughout the paper, we maintain these notational conventions and the following assumption concerning D.

Assumption 1

The lattice D ⊂ ℝ^d, d ≥ 1, is infinitely countable. All elements in D are located at distances of at least ρ₀ > 0 from each other, i.e., for all i, j ∈ D: ρ(i, j) ≥ ρ₀; w.l.o.g. we assume that ρ₀ > 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 D_n and T_n 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 fields.

Definition 1

Let Z = {Z_i,n, i ∈ D_n, n ≥ 1} be a random field with ||Z_i,n||_p < ∞, p ≥ 1, let ε = {ε_i,n, i ∈ T_n, n ≥ 1} be a random field, where |T_n| → ∞ as n → ∞, and let d = {d_i,n, i ∈ D_n, n ≥ 1} be an array of finite positive constants. Then the random field Z is said to be L_p(d)-near-epoch dependent on the random field ε if

$${\left|\right|Z_{i,n}-E(Z_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\left|\right|}_p\le d_{i,n}\psi (s)$$ (1)

for some sequence ψ(s) ≥ 0 with lim_s→∞ ψ(s) = 0. The ψ(s), which are w.l.o.g. assumed to be non-increasing, are called the NED coefficients, and the d_i,n are called NED scaling factors. Z is said to be L_p-NED on ε of size −λ if ψ(s) = O(s^−μ) for some μ > λ > 0. Furthermore, if sup_n sup_i∈Dnd_i,n < ∞, then Z is said to be uniformly L_p-NED on ε.

Recall that D_n ⊆ T_n. Typically, T_n will be an infinite subset of D, and often T_n = D. However, as discussed in more detail in Jenish and Prucha (2011), to cover Cliff Ord type processes T_n is allowed to depend on n and to be finite provided that it increases in size with n.

The role of the scaling factors {d_i,n} is to allow for the the possibility of “unbounded moments”, i.e., sup_n sup_i∈Dnd_i,n = ∞. Unbounded moments may reflect 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 d_i,n. In this respect, observe that

$${\left|\right|Z_{i,n}-E(Z_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\left|\right|}_p\le {\left|\right|Z_{i,n}\left|\right|}_p+{\left|\right|E(Z_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\left|\right|}_p\le 2{\left|\right|Z_{i,n}\left|\right|}_p$$

by the Minkowski and the conditional Jensen inequalities. Given this, we may choose d_i,n ≤ 2 ||Z_i,n||_p, and consequently w.l.o.g. 0 ≤ ψ(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 Z_i,n is L_p-NED, then it is also L_q-NED with the same coefficients {d_i,n} and {ψ(s)} for any q ≤ p.

Our definition of NED for spatial processes is adapted from the definition of NED for time series processes. In the time series literature, the NED concept first appeared in the works of Ibragimov (1962) and Billingsley (1968), although they did not use the present term. The concept of time series NED processes was later formalized by McLeish (1975), Wooldridge (1986), Gallant and White (1988). These authors considered only L₂-NED processes. Andrews (1988) generalized it to L_p-NED processes for p ≥ 1. Davidson (1992, 1993, 1994) and de Jong (1997) further extended it to allow for trending time series processes.

We note that aside from the NED condition, a number of different notions of dependence have been used in the time series literature. For instance, Pötscher and Prucha (1997) considered a more general dependence condition (called L_p-approximability). They use more general approximating functions than the conditional mean in Definition 1 to describe the dependence structure of a process. Similar conditions are also used by Lu (2001), Lu and Linton (2007), among others. These conditions allow for more general choices of approximating functions than the conditional expectation. One of the main results in this paper is a central limit theorem, which requires the existence of second moments. Since for p = 2 the conditional mean is the best approximator in the sense of minimizing the mean squared error, our use the conditional mean as an approximating function is not restrictive. Still, in particular applications it may be convenient to work with some other (s)-measurable approximating function, say h_i,s,n. Of course, if one can show that ||Z_i,n − h_i,s,n||₂ ≤ d_i,nψ(s), then this also established (1) for p = 2.

In the spatial literature, NED processes were considered in the special context of density estimation by Hallin, Lu and Tran (2001), and Hallin, Lu and Tran (2004), albeit they did not use this term. The first paper proves asymptotic normality of the kernel density estimator for linear random fields, the second paper shows L₁-consistency of the kernel density estimator for nonlinear functionals of i.i.d. random fields. We note that neither of these papers establishes a central limit theorem for nonlinear NED random fields.

As discussed earlier, an important motivation for considering NED processes is that mixing is generally not preserved under transformations involving infinitely many arguments. However, as illustrated below, the output process is generated as a function of infinitely many input variables in a wide range of models. 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 approximation error declines “sufficiently fast” as the conditioning set of input variables expands.

We now give examples of NED spatial processes. First, spatial Cliff and Ord (1981) type autoregressive processes are NED under some weak conditions on the spatial weight coefficients. These models have been used widely in applications. For recent contributions on estimation strategies for these models see, e.g., Robinson (2010, 2009), Kelejian and Prucha (2010, 2007, 2004), and Lee (2007, 2004). The second example is linear infinite moving average (MA(∞)) random fields. In preparation of the example, we first give a more general result, which shows that the NED property is satisfied by random fields generated from nonlinear Lipschitz type functionals of some ℝ^p_ε-valued random field ε = {ε_in, i ∈ D}:

$$Z_{in}=H_{in}({({\epsilon }_{jn})}_{j\in D})$$ (2)

where H_in: → ℝ^p_z, ⊆ ℝ^p_ε, are measurable functions satisfying for all e, e′ ∈

$$\mid H_{in}(e)-H_{in}(e^{\prime })\mid \le \sum _{j\in D}{w}_{\mathit{ijn}}\mid e_j-e_j^{\prime }\mid \text{with}{w}_{\mathit{ijn}}\ge 0$$ (3)

with

$$\underset{s\to \infty }{lim}\underset{n,i\in D}{sup}\sum _{j\in D:\rho (i,j)>s}{w}_{\mathit{ijn}}=0,\text{and}{\left|\right|\epsilon \left|\right|}_2=\underset{n,i\in D}{sup}{\left|\right|{\epsilon }_{in}\left|\right|}_2<\infty .$$ (4)

Proposition 1

Under conditions (3)–(4),Z = {Z_in, i ∈ D_n, n ≥ 1} given by (2) is well-defined, and is L₂-NED on ε with ψ(s) = ||ε||₂ sup_n,i∈DΣ_{j∈D:ρ(i,j)>s} w_ijn.

We now use the above proposition to establish the NED property for linear MA(∞) random fields. Linear MA(∞) random fields may arise as solutions of autoregressive models. For any k ∈ ℕ and fixed vectors v_l ∈ ℤ^d, l = 1, …, k, consider the following autoregressive random field:

$$Z_i=\sum _{l=1}^k{a}_l{Z}_{i-v_l}+{\epsilon }_i$$ (5)

where $a={\sum }_{l=1}^k\mid a_l\mid <1$, {ε_i, i ∈ ℤ^d} are i.i.d. with ||ε_i||_q < ∞, q ≥ 1. Model (5) is also known as a k-nearest-neighbor or interaction model with the radius of interaction r = max_1≤l≤k |v_l|.

As shown by Doukhan and Lang (2002), there exists a stationary solution of (5) given by:

$$Z_i=\sum _{m=0}^{\infty }\sum _{m_1+\dots +m_k=m}\frac{m!}{m_1!\dots m_k!}{a}_1^{m_1}\dots a_k^{m_k}{\epsilon }_{i-(m_1{v}_1+\dots +m_k{v}_k)}$$

with m_i ∈ ℕ. Thus, (5) can be represented as a linear random field Z_i = Σ_j∈ℤ^dw_jε_i−j, with

$$w_j=\sum _{m\ge \mid j\mid /r}^{\infty }\sum _{V(j,m)}\frac{m!}{m_1!\dots m_k!}{a}_1^{m_1}\dots a_m^{m_k},$$

where V (j, m) = {(m₁, …, m_k) ∈ ℕ^k: m₁ + … + m_k = m, m₁v₁ + … + m_kv_k = j}, observing that V (j, m) is empty if m < |j|/r. Observing further that

$$\sum _{m_1+\dots +m_k=m}\frac{m!}{m_1!\dots m_k!}\mid a_1^{m_1}\dots a_k^{m_k}\mid =a^m$$

the coefficients w_j can be bounded as

$$\mid w_j\mid \le \sum _{m\ge \mid j\mid /r}^{\infty }\sum _{m_1+\dots +m_k=m}\frac{m!}{m_1!\dots m_k!}\mid a_1^{m_1}\dots a_k^{m_k}\mid =\sum _{m\ge \mid j\mid /r}^{\infty }{a}^m={(1-a)}^{-1}{a}^{\mid j\mid /r}.$$

Rewriting the process Z_i as Z_i = Σ_j∈ℤ^dw_ijε_j with w_ij = w_i−j it follows from Proposition 1 that the random field (5) is L_p-NED on ε with the NED coefficients ψ(s) = ||ε||_q(1 − a)⁻¹(1 − a^1/r)⁻¹a^s/r.

The asymptotic theory of AR and MA(∞), satisfying the NED condition, can be useful in a variety of empirical applications where the data are cross sectionally correlated. For instance, Pinkse et al.’s (2002) study of spatial price competition among firms that produce differentiated products in one example of an empirical application with cross sectional dependence. They model the price charged by firm at location i in the geographic (or product characteristic) space as a linear spatial autoregressive process. Another example is Fogli and Veldkamp (2011) who investigate spatial correlation in the female labor force participation (LFP). In particular, they consider a spatial autoregression of county i’s LFP rate on LFP rates of its neighbors. Dell (2010) examines the impact of mita, the forced mining labor system in colonial Peru and Bolivia, on household consumption and child growth across different regions. Although her model is not spatially autoregressive, the regressors and errors exhibit persistent spatial correlation, which can be modeled as a spatial NED process.

As discussed, an attractive feature of NED processes is that the NED property is preserved under transformations. Econometric estimators are usually defined 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 fields 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 fields. 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 Z_i,n given by a family of functions g_i,n: ℝ^p_z → ℝ. The functions g_i,n are assumed Borel-measurable for all n and i ∈ D. They are furthermore assumed to satisfy the following Lipschitz condition: For all (z, z^•) ∈ ℝ^p_z × ℝ^p_z and all i ∈ D_n and n ≥ 1:

$$\mid g_{i,n}(z)-g_{i,n}(z^{•})\mid \le B_{i,n}(z,z^{•})\mid z-z^{•}\mid$$ (6)

where B_i,n(z, z^•): ℝ^p_z × ℝ^p_z → ℝ₊ is Borel-measurable. Of course, this condition would be devoid of meaning without further restrictions on B_i,n(z, z^•), which are given in the next propositions.

Proposition 2

Suppose g_i,n(·) satisfies Lipschitz condition (6) with |B_i,n(z, z^•)| ≤ C < ∞, for all (z, z^•) ∈ ℝ^p_z × ℝ^p_z and all i and n. If for p ≥ 1 the {Z_i,n} are L_p-NED of size −λ on {ε_i,n} with scaling factors {d_i,n}, then g_i,n(Z_i,n) is also L_p-NED of size −λ on {ε_i,n} with scaling factors {2Cd_i,n}.⁵

Proposition 3

Suppose g_i,n(·) satisfies Lipschitz condition (6) with

$$\underset{s}{sup}{\Vert B_{i,n}^{(s)}\Vert }_2<\infty \mathit{and}\underset{s}{sup}{\Vert B_{i,n}^{(s)}\left|Z_{i,n}-{\stackrel{\sim }{Z}}_{i,n}^s\right|\Vert }_r<\infty$$ (7)

for some r > 2, where $B_{i,n}^{(s)}=B_{i,n}(Z_{i,n},{\stackrel{\sim }{Z}}_{i,n}^s)$ and ${\stackrel{\sim }{Z}}_{i,n}^s=E[Z_{i,n}|{\mathfrak{F}}_{i,n}(s)]$. If ||g_i,n(Z_i,n)||₂ < ∞ and Z_i,n is L₂-NED of size −λ on {ε_i,n} with scaling factors {d_i,n}, then g_i,n(Z_i,n) is L₂-NED of size −λ(r − 2)/(2r − 2) on {ε_i,n} with scaling factors

$$d_{i,n}^{\prime }=d_{i,n}^{(r-2)/(2r-2)}\underset{s}{sup}{\Vert B_{i,n}^{(s)}\Vert }_2^{(r-2)/(2r-2)}{\Vert B_{i,n}^{(s)}|Z_{i,n}-{\stackrel{\sim }{Z}}_{i,n}^s\mid \Vert }_r^{r/(2r-2)}.$$

Thus, the NED property is hereditary under reasonably weak conditions. These conditions facilitate verification 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 4.

3 Limit Theorems

3.1 Law of Large Numbers

In this section, we present a LLN for real valued random fields Z = {Z_i,n, i ∈ D_n, n ≥ 1} that are L₁-NED on some vector-valued α-mixing random field ε = {ε_i,n, i ∈ T_n, n ≥ 1} with the NED coefficients {ψ(s)} and scaling factors {d_i,n}, where D_n ⊆ T_n ⊆ D and the lattice D satisfies Assumption 1. For ease of reference, we state below the definition of the α-mixing coefficients employed in the paper.

Definition 2

Let and be two σ-algebras of , and let

$$\alpha (\mathfrak{A},\mathfrak{B})=sup(\mid P(AB)-P(A)P(B)\mid ,A\in \mathfrak{A},B\in \mathfrak{B}),$$

For U ⊆ D_n and V ⊆ D_n, let σ_n(U) = σ(ε_i,n; i ∈ U) and α_n(U, V) = α(σ_n(U), σ_n(V)). Then, the α-mixing coefficients for the random field ε are defined as:

$$\overline{\alpha }(u,v,r)=\underset{n}{sup}\underset{U,V}{sup}({\alpha }_n(U,V),\mid U\mid \le u,\mid V\mid \le v,\rho (U,V)\ge r).$$

Dobrushin (1968) showed that weak dependence conditions based on the above mixing coefficients are satisfied by broad classes of random fields including Markov fields. In contrast to standard mixing numbers for time-series processes, the mixing coefficients for random fields depend not only on the distance between two datasets but also their sizes. To explicitly account for such dependence, it is furthermore assumed that

$$\overline{\alpha }(u,v,r)\le \varphi (u,v)\widehat{\alpha }(r)$$ (8)

where the function ϕ(u, v) is nondecreasing in each argument, and α̂(r) → 0 as r → ∞. The idea is to account separately for the two different 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 ϕ(u, v) in the random fields literature are

$$\varphi (u,v)={(u+v)}^{\tau },\tau \ge 0,$$ (9)

$$\varphi (u,v)=min\{u,v\}.$$ (10)

The above mixing conditions have been used extensively in the random fields literature including Takahata (1983), Nahapetian (1987), Bulinskii (1989), Bulinskii and Doukhan (1990), and Bradley (1993). They are satisfied by fairly large classes of random fields. Bradley (1993) provides examples of random fields satisfying conditions (8)–(9) with u = v and τ = 1. Furthermore, Bulinskii (1989) constructs moving average random fields satisfying the same conditions with τ = 1 for any given decay rate of coefficients α̂(r). Clearly, standard mixing coefficients in the time series literature are covered by conditions (8)–(9) when τ = 0.

Following the literature, we employ the above mixing conditions for the input random field, and impose further restrictions on the decay rates of the mixing coefficients.

Assumption 2

1. There exist nonrandom positive constants {c_i,n, i ∈ D_n, n ≥ 1} such that Z_i,n/c_i,n is uniformly L_p-bounded for some p > 1, i.e., sup_nsup_i∊DnE|Z_i,n/c_i,n|^p < ∞

2. The α-mixing coefficients of the input field ε satisfy (8) for some function ϕ(u, v) which is nondecreasing in each argument, and some α̂(r) such that ${\sum }_{r=1}^{\infty }{r}^{d-1}\widehat{\alpha }(r)<\infty$.

Theorem 1

Let {D_n} be a sequence of arbitrary finite subsets of D such that |D_n| → ∞ as n → ∞, where D ⊂ ℝ^d, d ≥ 1 is as in Assumption 1, and let T_n be a sequence of subsets of D such that D_n ⊆ T_n. Suppose further that Z = {Z_i,n, i ∈ D_n, n ≥ 1} is L₁-NED on ε = {ε_i,n, i ∈ T_n, n ≥ 1} with the scaling factors d_i,n. If Z and ε satisfy Assumption 2, then

$$\frac{1}{M_n\mid D_n\mid }\sum _{i\in D_n}(Z_{i,n}-{EZ}_{i,n})\stackrel{L_1}{\to }0,$$

where M_n = max_i∈Dnmax(c_i,n, d_i,n).

This LLN can 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.

Assumption 2(a) is a standard moment condition employed in weak LLNs for dependent processes. It requires existence of moments of order slightly greater than 1. As in Theorem 2 below, c_i,n and d_i,n are the scaling factors that reflect the magnitudes of potentially trending moments. The case of variables with uniformly bounded moments is covered by setting c_i,n = d_i,n = 1. The LLN does not require any restrictions on the NED coefficients. 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.

3.2 Central Limit Theorem

In this section, we present a CLT for real valued random fields Z = {Z_i,n, i ∈ D_n, n ≥ 1} that are L₂-NED on some vector-valued α-mixing random field ε = {ε_i,n, i ∈ T_n, n ≥ 1} with the NED coefficients {ψ(s)} and scaling factors {d_i,n}, where D_n ⊆ T_n ⊆ D and the lattice D satisfies Assumption 1. In the following, we will use the following notation:

$$S_n=\sum _{i\in D_n}{Z}_{i,n};{\sigma }_n^2=\mathit{var}(S_n).$$

The CLT relies on the following assumptions.

Assumption 3

The α-mixing coefficients of ε satisfy (8) and (9) for some τ ≥ 0 and α̂(r), such that for some δ > 0

$$\sum _{r=1}^{\infty }{r}^{d({\tau }_{\ast }+1)-1}{\widehat{\alpha }}^{\frac{\delta }{2(2+\delta )}}(r)<\infty ,$$ (11)

where τ_* = δτ/(2 + δ).

Assumption 3 restricts the dependence structure of the input process ε. Note that if τ = 1 this assumption also covers the case where ϕ(u, v) is given by (10).

Assumption 4

1. (Uniform L_2+δ integrability) There exists an array of positive constants {c_i,n} such that

   $$\underset{k\to \infty }{lim}\underset{n}{sup}\underset{i\in D_n}{sup}E[{\mid Z_{i,n}/c_{i,n}\mid }^{2+\delta }\mathbf{1}(\mid Z_{i,n}/c_{i,n}\mid >k)]=0,$$

   where 1 (·) is the indicator function and δ > 0 is as in Assumption 3.

2. ${inf}_n{\mid D_n\mid }^{-1}{M}_n^{-2}{\sigma }_n^2>0$, where M_n = max_i∈Dnc_i,n.

3. NED coefficients satisfy ${\sum }_{r=1}^{\infty }{r}^{d-1}\psi (r)<\infty$.

4. NED scaling factors satisfy ${sup}_n{sup}_{i\in D_n}{c}_{i,n}^{-1}{d}_{i,n}\le C<\infty$.

Assumptions 4(a),(b) are standard in the limit theory of mixing processes, e.g., Wooldridge (1986), Davidson (1992), de Jong (1997), and Jenish and Prucha (2009). Assumption 4(a) is satisfied if Z_i,n/c_i,n are uniformly L_p-bounded for p > 2+δ, i.e., sup_n,i∈Dn||Z_i,n/c_i,n||_p < ∞.

Assumption 4(b) is an asymptotic negligibility condition that ensures that no single summand influences disproportionately the entire sum. In the case of uniformly L_2+δ-bounded fields, 4(b) reduces to $lim{inf}_{n\to \infty }{\mid D_n\mid }^{-1}{\sigma }_n^2>0$, as is, e.g., maintained in Bolthausen (1982). Assumption 4(c) controls the size of the NED coefficients which measure the error in the approximation of Z_i,n by ε. Intuitively, the approximation errors have to decline sufficiently fast with each successive approximation. Assumption 4(c) is satisfied if ψ(r) = O(r^−d−γ) for some γ > 0, i.e., ψ(r) is of size −d. Finally, Assumption 4(d) is a technical condition, which ensures that the order of magnitude of the NED scaling factors does not exceed that of the 2 + δ moments. For instance, suppose the constant c_i,n can be chosen as c_i,n = ||Z_i,n||_2+δ, and the NED scaling numbers as d_i,n ≤ 2 ||Z_i,n||₂. Then Assumption 4(d) is satisfied, since by Lyapunov’s inequality, ||Z_i,n||₂ ≤ ||Z_i,n||_2+δ. This condition has also been used by de Jong (1997) and Davidson (1992).

Theorem 2

Suppose {D_n} is a sequence of finite subsets such that |D_n| → ∞ as n → ∞ and {T_n} is a sequence of subsets such that D_n ⊆ T_n ⊆ D of the lattice D satisfying Assumption 1. Let Z = {Z_i,n, i ∈ D_n, n ≥ 1} be a real valued zero-mean random field that is L₂-NED on a vector-valued α-mixing random field ε = {ε_i,n, i ∈ T_n, n ≥ 1}. Suppose Assumptions 3 and 4 hold, then

$${\sigma }_n^{-1}{S}_n\Rightarrow N(0,1).$$

Theorem 2 contains the CLT for α-mixing random fields given in Jenish and Prucha (2009) as a special case. It also contains as a special case the CLT for time series NED processes of Wooldridge (1986), see Theorem 3.13 and Corollary 4.4.

Theorem 2 can be easily extended to vector-valued fields using the standard Cramér-Wold device.

Corollary 1

Suppose {D_n} is a sequence of finite subsets such that |D_n| → ∞ as n → ∞ and {T_n} is a sequence of subsets such that D_n ⊆ T_n ⊆ D of the lattice D satisfying Assumption 1. Let Z = {Z_i,n, i ∈ D_n, n ≥ 1} with Z_i,n ∈ ℝ^k be a zero-mean random field that is L₂-NED on a vector-valued α-mixing random field ε = {ε_i,n, i ∈ T_n, n ≥ 1}. Suppose Assumptions 3 and 4 hold with |Z_i,n| denoting the Euclidean norm of Z_i,n and ${\sigma }_n^2$ replaced by λ_min(Σ_n), where Σ_n = Var(S_n) and λ_min(.) is the smallest eigenvalue, then

$${\sum }_n^{-1/2}{S}_n\Rightarrow N(0,I_k).$$

Furthermore, sup_n |D_n|⁻¹ λ_max (Σ_n) < ∞, where λ_max(.) denotes the largest eigenvalue.

4 Large Sample Properties of Spatial GMM Estimators

We now apply the limit theorems of the previous section 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 specifically, our consistency and asymptotic normality results (i) maintain only that the spatial data process is NED on an α-mixing basis process to accommodate spatial lags in the data process as discussed above, (ii) allow for unevenly placed locations, 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 sufficient conditions for easier interpretation by the applied researcher.⁶

We continue with the basic set-up of Section 2. Consider the moment function q_i,n: ℝ^p_z × Θ → ℝ^p_q, where Θ denotes the parameter space, and let θ_0n ∈ Θ denote the parameter vector of interest (which we allow to depend on n for reasons of generality). Suppose the following moment conditions hold

$${Eq}_{i,n}(Z_{i,n},{\theta }_{0n})=0.$$ (12)

Then, the corresponding spatial GMM estimator is defined as

$${\widehat{\theta }}_n=arg\underset{\theta \in \mathrm{\Theta }}{min}{Q}_n(\omega ,\theta ),$$ (13)

where Q_n: Ω × Θ → ℝ,

$$Q_n(\omega ,\theta )=R_n{(\theta )}^{\prime }{P}_n{R}_n(\theta ),$$

with R_n(θ) = |D_n|⁻¹Σ_i∈Dnq_i,n(Z_i,n, θ), and where the P_n are some positive semidefinite weighting matrices. To show consistency, consider the following non-stochastic analogue of Q_n, say

$${\overline{Q}}_n(\theta )={[{ER}_n(\theta )]}^{\prime }P[{ER}_n(\theta )],$$ (14)

where P denotes the probability limit of P_n. Given the moment condition (12), E [R_n(θ_0n)] = 0, the functions Q̄_n are minimized at θ_0n. In proving consistency, we follow the classical approach; see, e.g., Gallant and White (1988) or Pötscher and Prucha (1997) for more recent expositions. In particular, given identifiable uniqueness of θ_0n we establish, loosely speaking, convergence of the minimizers θ̂_n to the minimizers θ_0n by establishing convergence of the objective function Q_n(ω, θ) to its non-stochastic analogue Q̄_n(θ) 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 θ_0n.

Assumption 5

1. The parameter space Θ is a compact metric space with metric ν.

2. The functions q_i,n: ℝ^p_z × Θ → ℝ^p_q are / -measurable for each θ ∈ Θ, and continuous on Θ for each z ∈ ℝ^p_z.

3. The elements of the p_q × p_q real matrices P_n are -measurable, and P_n is positive semidefinite. Furthermore P = p lim P_n exists and P is positive definite.

4. The minimizers θ_0n are identifiably unique in the sense that every ε > 0, lim inf_n→∞ [inf_{θ∈Θ:ν(θ, θ0n)≥ε} [ER_n(θ)]′[ER_n(θ)]] > 0.

Compactness of the parameter space as maintained in Assumption 5(a) is typical for the GMM literature. Assumptions 5(b),(c) imply that Q_n(·, θ) is measurable for all θ ∈ Θ, and Q_n(ω, ·) is continuous on Θ. Given those assumptions the existence of measurable functions θ̂_n that solves (13) follows, e.g., from Lemma A3 of Pötscher and Prucha (1997).

Since P is positive definite, it is readily seen that Assumption 5(d) implies that for every ε > 0:

$$lim\underset{n\to \infty }{inf}\left[\underset{\theta \in \mathrm{\Theta }:\nu (\theta ,{\theta }_{0n})\ge \epsilon }{inf}\mid {\overline{Q}}_n(\theta )-{\overline{Q}}_n({\theta }_{0n})\mid \right]>0,$$

observing that Q̄_n(θ_0n) = 0. Thus, under Assumption 5(d) the minimizers θ_0n are identifiably unique; compare, e.g., Gallant and White (1988), p.19. For interpretation, consider the important special case where θ_0n = θ₀, ER_n(θ) does not depend on n, and is continuous in θ. In this case, identifiable uniqueness of θ₀ is equivalent to the assumption that θ₀ is the unique solution of the moment conditions, i.e., E [R_n(θ)] ≠ 0 for all θ ≠ θ₀; compare, e.g., Pötscher and Prucha (1997), p. 16.

4.1 Consistency

Given the minimizers θ_0n are identifiably unique, θ̂_n is a consistent estimator for θ_0n if Q_n converges uniformly to Q̄_n, i.e., if ${sup}_{\theta \in \mathrm{\Theta }}\mid Q_n(\omega ,\theta )-{\overline{Q}}_n(\theta )\mid \stackrel{p}{\to }0$ as n → ∞; 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 Q_n to Q̄_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).

Definition 3

Let f_i,n: ℝ^p_z × Θ → ℝ^p_q be / -measurable functions for each θ ∈ Θ, then:

1. The random functions f_i,n(Z_i,n; θ) are said to be p-dominated on Θ for some p > 1 if sup_n sup_i∈DnE sup_θ∈Θ|f_i,n(Z_i,n; θ)|^p < ∞.

2. The random functions f_i,n(Z_i,n; θ) are said to be Lipschitz in the parameter θ on Θ if

   $$\mid f_{i,n}(Z_{i,n},\theta )-f_{i,n}(Z_{i,n},{\theta }^{•})\mid \le L_{i,n}(Z_{i,n})h(\nu (\theta ,{\theta }^{•}))a.s.,$$ (15)

   for all θ, θ^• ∈ Θ and i ∈ D_n, n ≥ 1, where h is a nonrandom function with h(x) ↓ 0 as x ↓ 0, and L_i,n are random variables with $lim{sup}_{n\to \infty }{\mid D_n\mid }^{-1}{\sum }_{i\in D_n}{EL}_{i,n}^{\eta }<\infty$ for some η > 0

Towards establishing consistency of θ̂_n we furthermore maintain the following moment and mixing assumptions.

Assumption 6

The moment functions q_i,n(Z_i,n; θ) have the following properties:

1. They are p-dominated on Θ for p = 2.

2. They are uniformly L₁-NED on ε = {ε_i,n, i ∈ T_n, n ≥ 1}, where D_n ⊆ T_n ⊆ D, and ε is α-mixing with α-mixing coefficients the conditions stated in Assumption 2(b).

3. They are Lipschitz in the parameter θ on Θ.

Assumption 6(a) implies that sup_n,i∈DnE |q_i,n(Z_i,n; θ)|^p < ∞ for each θ ∈ Θ. Assumptions 6(b) then allow us to apply the LLN given as Theorem 1 above the sample moments R_n(θ) = |D_n|⁻¹Σ_i∈Dnq_i,n(Z_i,n, θ).

To verify Assumption 6(b) one can use either Proposition 2 or Proposition 3 to imply this condition from the lower level assumption that the data Z_i,n are L₁-NED. For example, the q_i,n are L₁-NED, if the Z_i,n are L₁-NED and satisfy the Lipschitz condition of Proposition 2. Note that no restrictions on the sizes of the NED coefficients are required.

Assumption 6(c) ensures stochastic equicontinuity of q_i,n w.r.t. θ. 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 Q_n converges uniformly to Q̄_n. A sufficient condition for Assumption 6(c) is existence of integrable partial derivatives of q_i,n w.r.t. θ if θ ∈ ℝ^k.

Our consistency results for the spatial GMM estimator given by (13) is summarized by the next theorem.

Theorem 3 (Consistency)

Suppose {D_n} is a sequence of finite sets of D such that |D_n| → ∞ as n → ∞, where D ⊂ ℝ^d, d ≥ 1 is as in Assumption 1. Suppose further that Assumptions 5 and 6 hold. Then

$$\nu ({\widehat{\theta }}_n,{\theta }_{0n})\stackrel{p}{\to }0asn\to \infty ,$$

and Q̄_n(θ) is uniformly equicontinuous on Θ.

4.2 Asymptotic Normality

We next establish that the spatial GMM estimators defined by (13) is asymptotically normally distributed. For that purpose, we need a stronger set of assumptions than for consistency, including differentiability of the moment functions in θ. It proofs helpful to adopt the notation ∇_θ in place of ∂/∂θ.⁷

Assumption 7

1. The minimizers θ_0n lie uniformly in the interior of Θ with Θ ⊆ ℝ^k. Furthermore E [R_n(θ_0n)] = 0.

2. The functions q_i,n: ℝ^p_z × Θ → ℝ^p_q are continuously differentiable w.r.t. θ for each z ∈ ℝ^p_z.

3. The functions q_i,n(Z_i,n; θ_0n) are uniformly L₂-NED on ε of size −d, and for some δ′ > 0 sup_n,i∈DnE |q_i,n(Z_i,n; θ_0n)|^2+δ′< ∞. The functions ∇_θq_i,n(Z_i,n; θ) are uniformly L₁-NED on ε.

4. The input process ε = {ε_i,n, i ∈ T_n, n ≥ 1}, where D_n ⊆ T_n ⊆ D, is α-mixing and the mixing coefficients satisfy Assumption 3 for some δ < δ′, where δ′ is the same as in Assumption 7(c).

5. The functions ∇_θq_i,n are p-dominated on Θ for some p > 1.

6. The functions ∇_θq_i,n are Lipschitz in θ on Θ.

7. inf_n λ_min(|D_n|⁻¹ Σ_n) > 0 where Σ_n = Var [Σ_i∈Dnq_i,n(Z_i,n, θ_0n)].

8. inf_n λ_min [E∇_θR_n(θ_0n)′ ∇_θR_n(θ_0n)] > 0.

The first part of Assumption 7(a) is needed to ensure that the estimator θ̂_n lies in the interior of Θ with probability tending to one, and facilitates the application of the mean value theorem to R_n(θ̂_n) around θ_0n. The second part states in essence that the moment conditions are correctly specified. 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 R_n(θ_0n). Some low level sufficient conditions for Assumption 7(c) are given below. To establish asymptotic normality, we also need uniform convergence of ∇_θR_n on Θ, which is implied via Assumptions 7(c),(d),(e),(f). Finally, Assumption 7(h) ensures positive-definitness of the variance-covariance matrix of the GMM estimator.

Given the above assumptions, we have the following asymptotic normality result for the spatial GMM estimator defined by (13).

Theorem 4

Suppose {D_n} is a sequence of finite sets of D such that |D_n| → ∞ as n → ∞, where D ⊂ ℝ^d, d ≥ 1 is as in Assumption 1. Suppose further that Assumptions 5–7 hold. Then

$${(A_n^{-1}{B}_n{B}_n^{\prime }{A}_n^{-1^{\prime }})}^{-1/2}{\mid D_n\mid }^{1/2}\left({\widehat{\theta }}_n-{\theta }_{0n}\right)\Rightarrow N(0,I_k),$$

where

$$A_n={[E{\nabla }_{\theta }{R}_n({\theta }_{0n})]}^{\prime }P[E{\nabla }_{\theta }{R}_n({\theta }_{0n})]\mathit{and}{B}_n={[E{\nabla }_{\theta }{R}_n({\theta }_{0n})]}^{\prime }P{\left[{\mid D_n\mid }^{-1}{\mathrm{\sum }}_n\right]}^{1/2}.$$

Moreover, $\mid A_n\mid =O(1);\mid A_n^{-1}\mid =O(1);\mid B_n\mid =O(1);\left|{(B_n{B}_n^{\prime })}^{-1}\right|=O(1)$ and hence, θ̂_n is |D_n|^1/2-consistent for θ_0n.

As remarked above, relative to the existing literature Theorem 4 allows for nonstationary processes and only assumes that q_i,n and ∇_θq_i,n are NED on an α-mixing input process, rather than postulating that q_i,n and ∇_θq_i,n are α-mixing. As such, Theorem 4 should provide a basis for constructing confidence intervals and hypothesis testing in a wider range of spatial models.

Using Proposition 3, we now give some sufficient conditions for Assumption 7(c).

Assumption 8

The process {Z_i,n, i ∈ D_n ⊂ T_n, n ≥ 1} is uniformly L₂-NED on {ε_i,n, i ∈ T_n, n ≥ 1} of size − 2d(r − 1)/(r − 2) for some r > 2.

Assumption 9

For every sequence {θ_0n} on Θ, the functions q_i,n(Z_i,n; θ_0n) and ∇_θq_i,n(Z_i,n; θ_0n) satisfy Lipschitz condition (6) in z, that is, for g_i,n = q_i,n or ∇_θq_i,n:

$$\mid g_{i,n}(z;{\theta }_n)-g_{i,n}(z^{•};{\theta }_n)\mid \le B_{i,n}(z,z^{•})\mid z-z^{•}\mid .$$

Furthermore, for the r > 2 as specified in Assumption 8,

$$\underset{n,i\in D_n}{sup}\underset{s}{sup}{\Vert B_{i,n}^{(s)}\Vert }_2<\infty \mathit{and}\underset{n,i\in D_n}{sup}\underset{s}{sup}{\Vert B_{i,n}^{(s)}\left|Z_{i,n}-{\stackrel{\sim }{Z}}_{i,n}^s\right|\Vert }_r<\infty$$

where $B_{i,n}^{(s)}=B_{i,n}(Z_{i,n},{\stackrel{\sim }{Z}}_{i,n}^s)$ with ${\stackrel{\sim }{Z}}_{i,n}^s=E[Z_{i,n}\mid {\mathfrak{F}}_{i,n}(s)]$.

5 Conclusion

The paper develops an asymptotic inference theory for a class of dependent nonstationary random fields that could be used in a wide range of econometric models with spatial dependence. More specifically, 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 fields. The class of NED random fields is “closed with respect to infinite transformations” and thus should be sufficiently broad for many applications of interest. In particular, it covers autoregressive and infinite moving average random fields as well as nonlinear functionals of mixing processes. 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 α-mixing process. Apart from covering a larger class of dependent processes, these limit theorems also allow for arrays of nonstationary random fields 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 fields 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 fixed 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.⁸ One possible approach may be to augment the contributions of the present paper with theory from point processes.

A Appendix: Proofs for Sections 2 and 3

Throughout, let (s) = σ(ε_j,n; j ∈ T_n: ρ(i, j) ≤ s) be the σ-field generated by the random vectors ε_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 different from line to line.

Proof of Proposition 1

The proof is available online on the authors’ webpages.

Proof of Theorem 1

Define Y_i,n = Z_i,n/M_n, then to prove the theorem, it suffices to show that ${\mid D_n\mid }^{-1}{\sum }_{i\in D_n}(Y_{i,n}-{EY}_{i,n})\stackrel{L_1}{\to }0$. We first establish moment and mixing conditions for Y_i,n from those for Z_i,n. Observe that in light of the definition of M_n and Assumption 2(a)

$$\underset{n,i\in D_n}{sup}E{\mid Y_{i,n}\mid }^p\le \underset{n,i\in D_n}{sup}E{\mid Z_{i,n}/c_{i,n}\mid }^p<\infty .$$ (A.1)

Thus, Y_i,n is uniformly L_p-bounded for p > 1. Let (s) = σ(ε_j,n; j ∈ T_n: ρ(i, j) ≤ s). Since Z_i,n is L₁-NED on ε = {ε_i,n, i ∈ T_n, n ≥ 1}:

$$\underset{n,i\in D_n}{sup}{\left|\right|Y_{i,n}-E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\left|\right|}_1\le \underset{n,i\in D_n}{sup}{M}_n^{-1}{d}_{i,n}\psi (s)\le \psi (s),$$ (A.2)

observing that M_n = max_i∈Dnmax(c_i,n, d_i,n). Thus Y_i,n is also L₁-NED on ε.

Next we show that for each given s > 0, the conditional mean $V_{i,n}^s=E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s))$ satisfies 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 ∈ D_n, n ≥ 1:

$$E{\mid V_{i,n}^s\mid }^p\le E\{E({\mid Y_{i,n}\mid }^p\mid {\mathfrak{F}}_{i,n}(s))\}\le \underset{n,i\in D_n}{sup}E{\mid Y_{i,n}\mid }^p<\infty .$$

So, $V_{i,n}^s$ is uniformly L_p-bounded for p > 1 and hence uniformly integrable. For each fixed s, $V_{i,n}^s$ is a measurable function of {ε_j,n; j ∈ T_n: ρ(i, j) ≤ s}. Observe that under Assumption 1 there exists a finite constant C such that the cardinality of the set {j ∈ T_n: ρ(i, j) ≤ s} is bounded by Cs^d; compare Lemma A.1 in Jenish and Prucha (2009). Hence,

$${\overline{\alpha }}_{V^s}(1,1,r)\le \{\begin{array}{c}1,r\le 2s\\ \overline{\alpha }({Cs}^d,{Cs}^d,r-2s),r>2s\end{array}$$

and thus in light of Assumption 2(b)

$$\sum _{r=1}^{\infty }{r}^{d-1}{\overline{\alpha }}_{V^s}(1,1,r)\le \sum _{r=1}^{2s}{r}^{d-1}+\varphi ({Cs}^d,{Cs}^d)\sum _{r=1}^{\infty }{(r+2s)}^{d-1}\widehat{\alpha }(r)<\infty .$$

The above shows that indeed, for each fixed s, $V_{i,n}^s$ satisfies the assumptions of the L₁-norm LLN of Jenish and Prucha (2009, Theorem 3). Therefore, for each s, we have

$${\Vert {\mid D_n\mid }^{-1}\sum _{i\in D_n}[E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s))-{EY}_{i,n}]\Vert }_1\to 0\text{as}n\to \infty .$$ (A.3)

Furthermore observe that from (A.2) and the Minkowski inequality

$${\Vert {\mid D_n\mid }^{-1}\sum _{i\in D_n}(Y_{i,n}-E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s)))\Vert }_1\le \psi (s).$$ (A.4)

Given (A.3) and (A.4), and observing that lim_s→∞ ψ(s) = 0 it now follows that

$$\begin{array}{l}\underset{n\to \infty }{lim}{\Vert {\mid D_n\mid }^{-1}\sum _{i\in D_n}(Y_{i,n}-{EY}_{i,n})\Vert }_1=\underset{s\to \infty }{lim}\underset{n\to \infty }{lim}{\Vert {\mid D_n\mid }^{-1}\sum _{i\in D_n}(Y_{i,n}-{EY}_{i,n})\Vert }_1\\ \le \underset{s\to \infty }{lim}lim\underset{n\to \infty }{sup}{\Vert {\mid D_n\mid }^{-1}\sum _{i\in D_n}(Y_{i,n}-E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s)))\Vert }_1+\underset{s\to \infty }{lim}\underset{s\to \infty }{lim}{\Vert {\mid D_n\mid }^{-1}\sum _{i\in D_n}(E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s))-{EY}_{i,n})\Vert }_1=0.\end{array}$$

This completes the proof of the LLN.

The proof of the CLT builds on Ibragimov and Linnik (1971), pp. 352–355, and makes use of the following lemmata:

Lemma A.1 (Brockwell and Davis (1991), Proposition 6.3.9)

Let Y_n, n = 1, 2, … and V_ns, s = 1, 2, …; n = 1, 2, …, be random vectors such that

1. V_ns ⇒ V_s as n → ∞ for each s = 1; 2; …

2. V_s rr; V as s → ∞, and

3. lim_s→∞ lim sup_n→∞ P(|Y_n − V_ns| > ε) = 0 for every ε > 0.

   Then Y_n ⇒ V as n → ∞.

Lemma A.2 (Ibragimov and Linnik (1971))

Let L_p( ) and L_p( ) denote, respectively, the class of -measurable and -measurable random variables ξ satisfying ||ξ||_p < ∞. Let X ∈ L_p( ) and Y ∈ L_q( ). Then, for any 1 ≤ p, q, r < ∞ such that p⁻¹ + q⁻¹ + r⁻¹ = 1,

$$\mid \mathit{Cov}(X,Y)\mid <4{\alpha }^{1/r}({\mathfrak{F}}_1,{\mathfrak{F}}_2){\left|\right|X\left|\right|}_p{\left|\right|Y\left|\right|}_q$$

where α( ; ) = sup_{A∈ ,B∈} (|P(AB) − P(A)P(B)|).

To prove the CLT for NED random fields, we first establish some moment inequalities and a slightly modified version of the CLT for mixing fields developed in Jenish and Prucha (2009). It is helpful to introduce the following notation. Let X = {X_i,n, i ∈ D_n, n ≥ 1} be a random field, then ||X||_q: = sup_n,i∈Dn||X_i,n||_q for q ≥ 1.

Lemma A.3

Let {X_i,n} be uniformly L₂-NED on a random field {ε_i,n} with α-mixing coefficients ᾱ(u, v, r) ≤ (u + v)^τ α̂(r), τ ≥ 0. Let S_n = Σ_i∈DnX_i,n and suppose that the NED coefficients of {X_i,n} satisfy ${\sum }_{r=1}^{\infty }{r}^{d-1}\psi (r)<\infty$ and ||X||_2+δ < ∞ for some δ > 0. Then,

1. |Cov (X_i,nX_j,n)| ≤ ||X||_2+δ {C₁||X||_2+δ [h/3]^dτ_*α̂^δ/(2+δ) ([h/3]) + C₂ψ([h/3])}, where h = ρ(i, j) and τ_* = δτ/(2 + δ). If, ${\sum }_{r=1}^{\infty }{r}^{d({\tau }_{\ast }+1)-1}{\widehat{\alpha }}^{\delta /(2+\delta )}(r)<\infty$, then for some C < ∞, not depending on n

   $$\mathit{Var}(S_n)\le C\mid D_n\mid .$$
2. |Cov (X_i,nX_j,n)| ≤ ||X||₂ {C₃||X||_2+δ [h/3]^dτ* α̂^δ/(4+2δ) ([h/3]) + C₄ψ([h/3])}, where h = ρ(i, j) and τ^* = δτ/(4 + 2δ). If, ${\sum }_{r=1}^{\infty }{r}^{d({\tau }^{\ast }+1)-1}{\widehat{\alpha }}^{\delta /(4+2\delta )}(r)<\infty$ where τ^* = δτ/(4 + 2δ), then for some C < ∞, not depending on n

   $$\mathit{Var}(S_n)\le C{\left|\right|X\left|\right|}_2\mid D_n\mid .$$

Proof of Lemma A.3

The proof is available online on the authors’ webpages.

Theorem A.1

Suppose {D_n} is a sequence of finite subsets of D, satisfying Assumption 1, with |D_n| → ∞ as n → ∞. Suppose further that {ε_i,n; i ∈ D_n, n ∈ ℕ} is an array of zero-mean random variables with α-coefficients ᾱ(u, v, r) ≤ C(u + v)^τ α̂(r) for some constants C < ∞ and τ ≥ 0. Suppose for some δ > 0 and γ > 0

$$\underset{k\to \infty }{lim}\underset{n,i\in D_n}{sup}E[{\mid {\epsilon }_{i,n}\mid }^{2+\delta }\mathbf{1}(\mid {\epsilon }_{i,n}\mid >k)]=0$$

and

$$\widehat{\alpha }(r)=O(r^{-d(2\mu +1)-\gamma })$$

with μ = max {τ, 1/δ}, and suppose $lim{inf}_{n\to \infty }{\mid D_n\mid }^{-1}{\sigma }_n^2>0$, then

$${\sigma }_n^{-1}\sum _{i\in D_n}{\epsilon }_{i,n}\Rightarrow N(0,1).$$

where ${\sigma }_n^2=\mathit{Var}({\sum }_{i\in D_n}{\epsilon }_{i,n})$.

Proof of Theorem A.1

The proof is available online on the authors’ webpages.

The above CLT is in essence a variant of CLT for α-mixing random fields given as Corollary 1 of Theorem 1 in Jenish and Prucha (2009), applied to mixing coefficients of the type ᾱ(u, v, r) ≤ C(u + v)^τα̂(r), τ ≥ 0.

Proof of Theorem 2

Since the proof is lengthy it is broken into steps.

1. Transition from Z_i,n to Y_i,n = Z_i,n/M_n

Let M_n = max_i∈Dnc_i,n and Y_i,n = Z_i,n/M_n. Also, let ${\sigma }_{Z,n}^2=\mathit{Var}[\sum Z_{i,n}]$ and ${\sigma }_{Y,n}^2=\mathit{Var}[\sum Y_{i,n}]=M_n^{-2}{\sigma }_{Z,n}^2$. Since

$${\sigma }_{Y,n}^{-1}\sum _{i\in D_n}{Y}_{i,n}={\sigma }_{Z,n}^{-1}\sum _{i\in D_n}{Z}_{i,n},$$

to prove the theorem, it suffices to show that ${\sigma }_{Y,n}^{-1}{\sum }_{i\in D_n}{Y}_{i,n}\Rightarrow N(0,1)$. Therefore, it proves convenient to switch notation from the text and to define

$$S_n=\sum _{i\in D_n}{Y}_{i,n},{\sigma }_n^2=\mathit{Var}(S_n).$$

That is, in the following, S_n denotes Σ_i∈DnY_i,n rather than Σ_i∈DnZ_i,n, and ${\sigma }_n^2$ denotes the variance of Σ_i∈DnY_i,n rather than of Σ_i∈DnZ_i,n. We now establish moment and mixing conditions for Y_i,n from the assumptions of the theorem. Observe that by definition of M_n

$$\mathbf{1}(\mid Y_{i,n}\mid >k)=\mathbf{1}(\mid Z_{i,n}/M_n\mid >k)\le \mathbf{1}(\mid Z_{i,n}/c_{i,n}\mid >k),$$

and hence

$$E[{\mid Y_{i,n}\mid }^{2+\delta }\mathbf{1}(\mid Y_{i,n}\mid >k)]\le E[{\mid Z_{i,n}/c_{i,n}\mid }^{2+\delta }\mathbf{1}(\mid Z_{i,n}/c_{i,n}\mid >k)]$$

so that Assumption 4(a) implies that

$$\underset{k\to \infty }{lim}\underset{n,i\in D_n}{sup}E[{\mid Y_{i,n}\mid }^{2+\delta }\mathbf{1}(\mid Y_{i,n}\mid >k)]=0.$$ (A.5)

Hence, Y_i,n is also uniformly L_2+δ bounded. Let ||Y||_2+δ = sup_n,i∈Dn||Y_i,n||_2+δ. Further, note that

$$\begin{array}{l}{\left|\right|Y_{i,n}-E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\left|\right|}_2=M_n^{-1}{\left|\right|Z_{i,n}-E(Z_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\left|\right|}_2\\ \le c_{i,n}^{-1}{d}_{i,n}\psi (s)\le C\psi (s)\end{array}$$ (A.6)

since ${sup}_{n,i\in D_n}{c}_{i,n}^{-1}{d}_{i,n}\le C<\infty$, by assumption. Thus, Y_i,n is uniformly L₂-NED on ε with the NED coefficients ψ(m). Finally, observe that by Assumption 4(b):

$$\underset{n}{inf}{\mid D_n\mid }^{-1}{\sigma }_n^2>0.$$ (A.7)

Hence, there exists 0 < B < ∞ such that for all n

$$B\mid D_n\mid \le {\sigma }_n^2.$$ (A.8)

2. Decomposition of Y_i,n

For any fixed s > 0, decompose X_i,n as

$$Y_{i,n}={\xi }_{i,n}^s+{\eta }_{i,n}^s$$

where ${\xi }_{i,n}^s=E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s)),{\eta }_{i,n}^s=Y_{i,n}-{\xi }_{i,n}^s$. Let

$$\begin{array}{l}{S}_{n,s}=\sum _{i\in D_n}{\xi }_{i,n}^s;{\stackrel{\sim }{S}}_{n,s}=\sum _{i\in D_n}{\eta }_{i,n}^s\\ {\sigma }_{n,s}^2=\mathit{Var}[S_{n,s}],{\stackrel{\sim }{\sigma }}_{n,s}^2=\mathit{Var}\left[{\stackrel{\sim }{S}}_{n,s}\right]\end{array}$$

Repeated use of the Minkowski inequality yields:

$$\mid {\sigma }_n-{\sigma }_{n,s}\mid \le {\stackrel{\sim }{\sigma }}_{n,s},\mid {\sigma }_n-{\stackrel{\sim }{\sigma }}_{n,s}\mid \le {\sigma }_{n,s}.$$ (A.9)

Observe that

$$E[E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\mid {\mathfrak{F}}_{i,n}(m))]=\{\begin{array}{c}E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s)),m\ge s,\\ E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(m)),m<s.\end{array}$$

and hence

$$\begin{array}{l}{\left|\right|{\eta }_{i,n}^s-E({\eta }_{i,n}^s\mid {\mathfrak{F}}_{i,n}(m))\left|\right|}_2={\left|\right|Y_{i,n}-E[Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s)]-E[Y_{i,n}\mid {\mathfrak{F}}_{i,n}(m)]+E[(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\mid {\mathfrak{F}}_{i,n}(m)]\left|\right|}_2\\ =\{\begin{array}{c}{\left|\right|Y_{i,n}-E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(m))\left|\right|}_2\le C\psi (m),\text{if}m\ge s,\\ {\left|\right|Y_{i,n}-E(Y_{i,n}\mid {\mathfrak{F}}_{i,n}(s))\left|\right|}_2\le C\psi (s)\le C\psi (m),\text{if}m<s.\end{array}\end{array}$$

since by definition the sequence ψ(m) is non-increasing. Thus, for any fixed s > 0, { ${\eta }_{i,n}^s$} is uniformly L₂-NED on ε with the same NED coefficients ψ(m) as the random field {Y_i,n}. Furthermore, as shown in the proof of Lemma A.3, { ${\eta }_{i,n}^s$} is also uniformly L_2+δ bounded.

3. Bounds for the Variances of ΣY_i;n and $\sum {\eta }_{i,n}^s$

First note that in light of Assumption 3, and observing that τ^* = δτ/(4 + 2δ) ≤ τ_* =δτ/(2 + δ) and ${\widehat{\alpha }}^{\delta /(2+\delta )}(r)\le {\widehat{\alpha }}^{\frac{\delta }{2(2+\delta )}}(r)$ we have

$$\begin{array}{l}\sum _{r=1}^{\infty }{r}^{d({\tau }_{\ast }+1)-1}{\widehat{\alpha }}^{\delta /(2+\delta )}(r)\le \sum _{r=1}^{\infty }{r}^{d({\tau }_{\ast }+1)-1}{\widehat{\alpha }}^{\frac{\delta }{2(2+\delta )}}(r)<\infty ,\\ \sum _{r=1}^{\infty }{r}^{d({\tau }^{\ast }+1)-1}{\widehat{\alpha }}^{\delta /(4+2\delta )}(r)\le \sum _{r=1}^{\infty }{r}^{d({\tau }_{\ast }+1)-1}{\widehat{\alpha }}^{\frac{\delta }{2(2+\delta )}}(r)<\infty .\end{array}$$

Using part (a) of Lemma A.3 with X_i,n = Y_i,n and recalling (A.8), we have

$$B\mid D_n\mid \le {\sigma }_n^2=\mathit{Var}(S_n)\le C\mid D_n\mid .$$

for some B > 0. Using part (b) of Lemma A.3 with $X_{i,n}={\eta }_{i,n}^s$ we have

$${\stackrel{\sim }{\sigma }}_{n,s}^2=\mathit{Var}\left({\stackrel{\sim }{S}}_{n,s}\right)\le C\mid D_n\mid {\left|\right|{\eta }_{i,n}^s\left|\right|}_2=C\mid D_n\mid \psi (s)$$ (A.10)

in light of (A.6). Hence,

$$\underset{s\to \infty }{lim}lim\underset{n\to \infty }{sup}\frac{{\stackrel{\sim }{\sigma }}_{n,s}^2}{{\sigma }_n^2}\le C\underset{s\to \infty }{lim}\psi (s)=0.$$ (A.11)

Furthermore, by (A.9) we have

$$\underset{s\to \infty }{lim}lim\underset{n\to \infty }{sup}\left|1-\frac{{\sigma }_{n,s}}{{\sigma }_n}\right|\le \underset{s\to \infty }{lim}lim\underset{n\to \infty }{sup}\frac{{\stackrel{\sim }{\sigma }}_{n,s}}{{\sigma }_n}=0$$ (A.12)

and hence for all s ≥ 1 and n ≥ 1

$$\frac{{\sigma }_{n,s}}{{\sigma }_n}\le C<\infty .$$ (A.13)

4. CLT for ${\sum }_{i\in D_n}{\xi }_{i,n}^s$

We now show that for any fixed s > 0, ${\xi }_{i,n}^s$ satisfies Theorem A.1.

First, since ${sup}_{n,i\in D_n}E\left[{\mid {\xi }_{i,n}^s\mid }^{2+\delta }\right]<\infty$, the process { ${\xi }_{i,n}^s$} is uniformly L_2+δ′-integrable for δ′ = δ/2, i.e.,

$$\underset{k\to \infty }{lim}\underset{n,i\in D_n}{sup}E[{\mid {\xi }_{i,n}^s\mid }^{2+\delta /2}\mathbf{1}(\mid {\xi }_{i,n}^s\mid >k)]=0.$$

Second, since ${\xi }_{i,n}^s$ is a measurable function of ε_i,n for any u, v ∈ ℕ and r > 2s

$${\overline{\alpha }}_{\xi }(u,v,r)\le \overline{\alpha }({\mathit{uMs}}^d,{\mathit{vMs}}^d,r-2s)\le C{(u+v)}^{\tau }\widehat{\alpha }(r-2s)$$

We next to show that α̂(r) = O(r^{−d(2μ+1) −γ}) for μ = max {τ, 2/δ} and some γ > 0. By assumption,

$$\sum _{r=1}^{\infty }{r}^{d({\tau }_{\ast }+1)-1}{\widehat{\alpha }}^{\frac{\delta }{2(2+\delta )}}(r)<\infty ,$$

where τ_* = δτ/(2 + δ), which implies

$$\widehat{\alpha }(r)=o(r^{-2d(2+\delta )({\tau }_{\ast }+1)/\delta })=o(r^{-d[2(\tau +2/\delta )+1]-d})=o(r^{-d[2\mu +1]-d})$$

since μ ≤ τ + 2/δ for μ = max {τ, 2/δ}. Thus, α̂(r) = O(r^{−d(2μ+1) −γ}) for γ = d.

We next show that for sufficiently large s,

$$0<lim\underset{n\to \infty }{inf}{\mid D_n\mid }^{-1}{\sigma }_{n,s}^2.$$

By (A.8),

$$B^{1/2}\le inf{\mid D_n\mid }^{-1/2}{\sigma }_n$$

Since lim_s→∞ ψ(s) = 0, there exists s_* such that in light of (A.10) for all s ≥ s_*,

$${\mid D_n\mid }^{-1/2}{\stackrel{\sim }{\sigma }}_{n,s}\le C{\psi }^{1/2}(s)\le B^{1/2}/2.$$ (A.14)

Hence by (A.9) for all s ≥ s_*, |D_n|^−1/2(σ_n− σ̃_n,s) ≤ |D_n|^−1/2σ_n,s, and thus inf_n |D_n|^−1/2σ_n,s ≥ inf_n |D_n|^−1/2σ_n − sup_n |D_n|^−1/2 σ̃_n,s. Using (A.7) and (A.14), we have

$$lim\underset{n\to \infty }{inf}{\mid D_n\mid }^{-1/2}{\sigma }_{n,s}\ge B^{1/2}-\frac{B^{1/2}}{2}=\frac{B^{1/2}}{2}>0$$

Thus, for all s ≥ s_*,

$${\sigma }_{n,s}^{-1}\sum _{i\in D_n}{\xi }_{i,n}^s\Rightarrow N(0,1)\text{as}n\to \infty .$$ (A.15)

Since the first s_* terms do not affect the analysis below we take in the following s_* = 1.

5. CLT for ${\sigma }_n^{-1}{\sum }_{i\in D_n}{Y}_{i,n}$

Finally, using Lemma A.1 we now show that, given the maintained NED assumption, the just established CLT in (A.15) for the approximators ${\xi }_{i,n}^s$ can be carried over to the the Y_i,n. Define

$$W_n={\sigma }_n^{-1}\sum _{i\in D_n}{Y}_{i,n},V_{ns}={\sigma }_n^{-1}\sum _{i\in D_n}{\xi }_{i,n}^s,W_n-V_{ns}={\sigma }_n^{-1}\sum _{i\in D_n}{\eta }_{i,n}^s$$

so that we can exploit Lemma A.1 to prove that

$$W_n={\sigma }_n^{-1}\sum _{i\in D_n}{Y}_{i,n}\Rightarrow V~N(0,1).$$

We first verify condition (iii) of Lemma A.1. By Markov’s inequality and (A.11), for every ε > 0 we have

$$\begin{array}{l}\underset{s\to \infty }{lim}lim\underset{n\to \infty }{sup}P(\mid W_n-V_{ns}\mid >\epsilon )=\underset{s\to \infty }{lim}lim\underset{n\to \infty }{sup}P(\left|{\sigma }_n^{-1}\sum _{i\in D_n}{\eta }_{i,n}^s\right|>\epsilon )\\ \le \underset{s\to \infty }{lim}lim\underset{n\to \infty }{sup}\frac{{\stackrel{\sim }{\sigma }}_{n,s}^2}{{\epsilon }^2{\sigma }_n^2}=0.\end{array}$$

Next observe that $V_{ns}=\frac{{\sigma }_{n,s}}{{\sigma }_n}[{\sigma }_{n,s}^{-1}{\sum }_{i\in D_n}{\xi }_{i,n}^s]$. We proceed to show W_n ⇒ V by contradiction. For that purpose let be the set of all probability measures on (ℝ; ), and observe that we can metrize by, e.g., the Prokhorov distance d(., .). Let μ_n and μ be the probability measures corresponding to W_n and V, respectively, then W_n ⇒ V, or μ_n ⇒ μ, iff d(μ_n, μ) → 0 as n → ∞. Now suppose μ_n does not converge to μ. Then for some ε > 0 there exists a subsequence {n(m)} such that d(μ_n(m), μ) > ε for all n(m). By (A.13), we have 0 ≤ σ_n,s/σ_n ≤ C < ∞ for all s, n ≥ 1. Hence, 0 ≤ σ_n(m),s/σ_n(m) ≤ C < ∞ for all n(m). Consequently, for s = 1 there exists a subsubsequence {n(m(l₁))} such that σ_n(m(l1)),1/σ_n(m(l1)) → p(1) as l₁ → ∞. For s = 2, there exists a subsubsubsequence {n(m(l₁(l₂)))} such that σ_{n(m(l1(l2))),2}/σ_{n(m(l1 (l2)))} → p(2) as l₂ → ∞. The argument can be repeated for s = 3, 4…. Now construct a subsequence {n_l} such that n₁ corresponds to the first element of {n(m(l₁))}, n₂ corresponds to the second element of {n(m(l₁(l₂)))}, and so on, then

$$\underset{l\to \infty }{lim}\frac{{\sigma }_{n_l,s}}{{\sigma }_{n_l}}=p(s)$$ (A.16)

for s = 1, 2, … Given (A.15), it follows that as l → ∞

$$V_{n_{l}s}\Rightarrow V_s~N(0,p^2(s)).$$

Then, it follows from (A.12) that

$$\underset{s\to \infty }{lim}\mid p(s)-1\mid \le \underset{s\to \infty }{lim}\underset{l\to \infty }{lim}\left|p(s)-\frac{{\sigma }_{n_l,s}}{{\sigma }_{n_l}}\right|+\underset{s\to \infty }{lim}\underset{n\ge 1}{sup}\left|\frac{{\sigma }_{n,s}}{{\sigma }_n}-1\right|=0.$$

Thus V_s ⇒ V and thus by Lemma A.1 W_nl ⇒ V ~ N(0, 1) as l → ∞. Since {n_l} ⊆ {n(m)} this contradicts the assumption that d(μ_n(m), μ) > ε for all n(m). This completes the proof of the CLT.

Proof of Corollary 1

The proof is available online on the authors’ webpages.

B Appendix: Proofs for Section 4

Proof of Theorem 3

We show that

$$\underset{\theta \in \mathrm{\Theta }}{sup}\mid Q_n(\theta )-{\overline{Q}}_n(\theta )\mid \stackrel{p}{\to }0$$ (B.1)

as n → ∞. As discussed in the text, given that the θ_0n are identifiably unique it then follows immediately from, e.g., Pötscher and Prucha (1997), Lemma 3.1, that $\nu ({\widehat{\theta }}_n,{\theta }_{0n})\stackrel{p}{\to }0$ as claimed.

We start by proving that

$${\mid D_n\mid }^{-1}\sum _{i\in D_n}[q_{i,n}(Z_{i,n},\theta )-{Eq}_{i,n}(S_{i,n},\theta )]\stackrel{p}{\to }0$$ (B.2)

for each θ ∈ Θ, by applying the LLN given as Theorem 1 in the text to q_i,n(Z_i,n, θ). By Assumption 6(a), we have sup_n,i∈DnE |q_i,n(Z_i,n, θ)|^p < ∞ for each θ ∈ Θ and p = 2, which verifies Assumption 2(a) for q_i,n(Z_i,n, θ) with c_i,n = 1. By Assumption 6(b), the q_i,n(Z_i,n, θ) are uniformly L₁-NED on ε, and hence w.o.l.g. we can take d_i,n = 1. Furthermore, by Assumption 6(b) the input process ε is α-mixing, and the α-mixing coefficients satisfy Assumption 2(b). Consequently (B.2) follows directly from Theorem 1 applied to q_i,n(Z_i,n, θ).

Next, by Proposition 1 of Jenish and Prucha (2009), Assumption 6(c) implies that q_i,n is L₀ stochastically equicontinuous on Θ, i.e., for every ε > 0

$$lim\underset{n\to \infty }{sup}\frac{1}{\mid D_n\mid }\sum _{i\in D_n}P\left(\underset{\nu (\theta ,{\theta }^{•})\le \delta }{sup}\mid q_{i,n}(Z_{i,n},\theta )-q_{i,n}(Z_{i,n},{\theta }^{•})\mid >\epsilon \right)\to 0\text{as}\delta \to 0.$$

Furthermore, in light of Assumption 6(a) the q_i,n(Z_i,n, θ) 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 verified the pointwise LLN in (B.2) it now follows directly from that theorem that

$$\underset{\theta \in \mathrm{\Theta }}{sup}\mid R_n(\theta )-{ER}_n(\theta )\mid \stackrel{p}{\to }0$$ (B.3)

with R_n(θ) = |D_n|⁻¹Σ_i∈Dnq_i,n(Z_i,n, θ), and that the ER_n(θ) are uniformly equicontinuous on Θ in the sense that

$$lim\underset{n\to \infty }{sup}\underset{{\theta }^{•}\in \mathrm{\Theta }}{sup}\underset{\nu (\theta ,{\theta }^{•})\le \delta }{sup}\mid {ER}_n(\theta )-{ER}_n({\theta }^{•})\mid \to 0\text{as}\delta \to 0.$$

To prove (B.1), observe that

$$\begin{array}{l}\underset{\theta \in \mathrm{\Theta }}{sup}\mid Q_n(\theta )-{\overline{Q}}_n(\theta )\mid \le \underset{\theta \in \mathrm{\Theta }}{sup}\mid R_n{(\theta )}^{\prime }{PR}_n(\theta )-{ER}_n(\theta ){\mathit{PER}}_n(\theta )\mid +\underset{\theta \in \mathrm{\Theta }}{sup}\mid R_n{(\theta )}^{\prime }(P_n-P)R_n(\theta )\mid \\ \le \underset{\theta \in \mathrm{\Theta }}{sup}\mid R_n{(\theta )}^{\prime }{PR}_n(\theta )-{ER}_n(\theta ){\mathit{PER}}_n(\theta )\mid +2\underset{\theta \in \mathrm{\Theta }}{sup}{\mid R_n(\theta )\mid }^2\mid P_n-P\mid .\end{array}$$ (B.4)

Furthermore observe that Assumption 6(a) we have E [sup_θ∈Θ|q_i,n(Z_i,n, θ)|] ≤ K and E [sup_θ∈Θ|q_i,n(Z_i,n, θ)|]² ≤ K for some finite constant K. Thus

$$\underset{\theta \in \mathrm{\Theta }}{sup}E\mid R_n(\theta )\mid \le E\underset{\theta \in \mathrm{\Theta }}{sup}\mid R_n(\theta )\mid \le {\mid D_n\mid }^{-1}\sum _{i\in D_n}E\underset{\theta \in \mathrm{\Theta }}{sup}\mid q_{i,n}(Z_{i,n},\theta )\mid \le K$$ (B.5)

$$\begin{array}{l}E\underset{\theta \in \mathrm{\Theta }}{sup}{\mid R_n(\theta )\mid }^2\le {\mid D_n\mid }^{-2}\sum _{i,j\in D_n}E\left[\underset{\theta \in \mathrm{\Theta }}{sup}\mid q_{i,n}(Z_{i,n},\theta )\mid \underset{\theta \in \mathrm{\Theta }}{sup}\mid q_{j,n}(Z_{j,n},\theta )\mid \right]\\ \le {\mid D_n\mid }^{-2}\sum _{i,j\in D_n}{\left[E{\left(\underset{\theta \in \mathrm{\Theta }}{sup}\mid q_{i,n}(Z_{i,n},\theta )\mid \right)}^2\right]}^{1/2}{\left[E{\left(\underset{\theta \in \mathrm{\Theta }}{sup}\mid q_{j,n}(Z_{j,n},\theta )\mid \right)}^2\right]}^{1/2}\le K.\end{array}$$ (B.6)

Now consider the first terms on the r.h.s. of the last inequality of (B.4). From (B.5) we see that E |R_n(θ)| takes on its values in a compact set. Given (B.3) it now follows immediately from part (a) of Lemma 3.3 of Pötscher and Prucha (1997) that

$$\underset{\theta \in \mathrm{\Theta }}{sup}\mid R_n{(\theta )}^{\prime }{PR}_n(\theta )-{ER}_n(\theta ){\mathit{PER}}_n(\theta )\mid \stackrel{P}{\to }0.$$ (B.7)

Next we show that also the second term on the r.h.s. of the last inequality of (B.4) converges in probability to zero. To see that this is indeed the case observe that sup_θ∈Θ |R_n(θ)|² = O_p(1) in light of (B.6) and $\mid P_n-P\mid \stackrel{P}{\to }0$ by assumption. This completes the proof of (B.1).

Having established that ER_n(θ) are uniformly equicontinuous on Θ, the uniform equicontinuity of Q̄_n(θ) on Θ follows immediately from Lemma 3.3(b) of Pötscher and Prucha (1997).

Proof of Theorem 4

Clearly by Theorem 3 we have θ̂_n − θ_0n = o_p(1).

Step 1

The estimators θ̂_n corresponding to the objective function (13) satisfy the following first order conditions:

$${\nabla }_{\theta }{R}_n{({\widehat{\theta }}_n)}^{\prime }{P}_n\left[\mid D_n{\mid }^{1/2}{R}_n({\widehat{\theta }}_n)\right]=o_p(1).$$ (B.8)

The o_p(1) term on the r.h.s. reflects that the first order conditions may not hold if θ̂_n falls onto the boundary of Θ, and that the probability of that event goes to zero as n → ∞, since the θ_0n are uniformly in the in the interior of Θ by Assumption 7(a). If θ̂_n is in the interior of Θ, then the l.h.s. of (B.8) is zero.

Taking the mean value expansion of R_n(θ̂_n) about θ_0n yields

$$R_n({\widehat{\theta }}_n)=R_n({\theta }_{0n})+{\nabla }_{\theta }{R}_n({\stackrel{\sim }{\theta }}_n)({\widehat{\theta }}_n-{\theta }_{0n})$$ (B.9)

where θ̃_n ∈ Θ is between θ̃_n and θ_0n (component-by-component). Let

$${\widehat{A}}_n={\nabla }_{\theta }{R}_n{({\widehat{\theta }}_n)}^{\prime }{P}_n{\nabla }_{\theta }{R}_n({\stackrel{\sim }{\theta }}_n)\text{and}{\widehat{B}}_n={\nabla }_{\theta }{R}_n{({\widehat{\theta }}_n)}^{\prime }{P}_n{\left[{\mid D_n\mid }^{-1}{\mathrm{\sum }}_n\right]}^{1/2},$$

then combining (B.8) and (B.9) gives

$$\begin{array}{l}{\mid D_n\mid }^{1/2}\left({\widehat{\theta }}_n-{\theta }_{0n}\right)=\left[I-{\widehat{A}}_n^{+}{\widehat{A}}_n\right]{\mid D_n\mid }^{1/2}\left({\widehat{\theta }}_n-{\theta }_{0n}\right)-{\widehat{A}}_n^{+}{\nabla }_{\theta }{R}_n{({\widehat{\theta }}_n)}^{\prime }{P}_n\left[{\mid D_n\mid }^{1/2}{R}_n({\theta }_{on})\right]+{\widehat{A}}_n^{+}{o}_p(1)\\ =\left[I-{\widehat{A}}_n^{+}{\widehat{A}}_n\right]{\mid D_n\mid }^{1/2}\left({\widehat{\theta }}_n-{\theta }_{0n}\right)-{\widehat{A}}_n^{+}{\widehat{B}}_n\left[{\mathrm{\sum }}_n^{-1/2}\mid D_n\mid R_n({\theta }_{0n})\right]+{\widehat{A}}_n^{+}{o}_p(1),\end{array}$$ (B.10)

where ${\widehat{A}}_n^{+}$ denotes the generalized inverse of Â_n.

Step 2

By Assumptions 7(c) the q_i,n(Z_i,n, θ_0n) are uniformly L₂-NED and uniformly L_2+δ-integrable with c_i,n = 1. Given Assumptions 7(d),(g) it is now readily seen that the process {q_i,n(Z_i,n, θ_0n), i ∈ D_n} satisfies all assumptions of the CLT for vector-valued NED processes, given as Corollary 1 in the text, with c_in = 1. (Note that Assumption 4(d) is satisfied automatically since the q_i,n(Z_i,n, θ_0n) are uniformly L₂-NED.) Hence,

$${\mathrm{\sum }}_n^{-1/2}\mid D_n\mid R_n({\theta }_{0n})={\mathrm{\sum }}_n^{-1/2}\sum _{i\in D_n}{q}_{i,n}(Z_{i,n},{\theta }_{0n})\Rightarrow N(0,I_{p_q}),$$ (B.11)

with Σ_n = Var [Σ_i∈Dnq_i,n(Z_i,n, θ_0n)] and sup_n λ_max [|D_n|⁻¹ Σ_n]< ∞.

Step 3

By Assumptions 7(c),(d),(e) the functions ∇_θq_i,n(Z_i,n, θ) satisfy for each θ ∈ Θ the LLN given as Theorem 1 in the text with c_i,n = 1, observing that Assumption 2(b) is implied by 3. By argumentation analogous as used in the proof of consistency we have

$${\mid D_n\mid }^{-1}\sum _{i\in D_n}({\nabla }_{\theta }{q}_{i,n}(Z_{i,n},\theta )-E{\nabla }_{\theta }{q}_{i,n}(Z_{i,n},\theta ))\stackrel{p}{\to }0.$$

By Proposition 1 of Jenish and Prucha (2009), Assumption 7(f) implies that the ∇_θq_i,n(Z_i,n; θ) are uniformly L₀-equicontinuous on Θ. Given L₀-equicontinuity and Assumption 7(e), we have by the ULLN of Jenish and Prucha (2009):

$$\underset{\theta \in \mathrm{\Theta }}{sup}\mid {\nabla }_{\theta }{R}_n(\theta )-E{\nabla }_{\theta }{R}_n(\theta )\mid \stackrel{p}{\to }0.$$ (B.12)

and furthermore, the E∇_θR_n(θ) are uniformly equicontinuous on Θ in the sense:

$$lim\underset{n\to \infty }{sup}\underset{{\theta }^{\prime }\in \mathrm{\Theta }}{sup}\underset{\mid \theta -{\theta }^{\prime }\mid <\delta }{sup}\mid E{\nabla }_{\theta }{R}_n(\theta )-E{\nabla }_{\theta }{R}_n(\theta )\mid \to 0$$ (B.13)

as δ → 0. In light of (B.12) and (B.13), and given that θ̂_n − θ_0n = o_p(1) and hence θ̃_n − θ_0n = o_p(1), if follows further that

$${\nabla }_{\theta }{R}_n({\widehat{\theta }}_n)-E{\nabla }_{\theta }{R}_n({\theta }_{0n})\stackrel{p}{\to }0,\text{and}{\nabla }_{\theta }{R}_n({\stackrel{\sim }{\theta }}_n)-E{\nabla }_{\theta }{R}_n({\theta }_{0n})\stackrel{p}{\to }0.$$

Hence,

$${\widehat{A}}_n-A_n\stackrel{p}{\to }0\text{and}{\widehat{B}}_n-B_n\stackrel{p}{\to }0,$$ (B.14)

where Â_n and B̂_n are as defined above, and

$$A_n={[E{\nabla }_{\theta }{R}_n({\theta }_{0n})]}^{\prime }P[E{\nabla }_{\theta }{R}_n({\theta }_{0n})]\text{and}{B}_n={[E{\nabla }_{\theta }{R}_n({\theta }_{0n})]}^{\prime }P{\left[{\mid D_n\mid }^{-1}{\mathrm{\sum }}_n\right]}^{1/2}.$$

Step 4

Given Assumptions 7(e),(f), and since P is positive definite, we have |A_n| = O(1) and $\mid A_n^{-1}\mid =O(1)$, respectively. Hence by, e.g., Lemma F1 in Pötscher and Prucha (1997) we have Â_n = O_p(1), ${\widehat{A}}_n^{+}=O_p(1)$, Â_n is nonsingular with probability tending to one, and ${\widehat{A}}_n^{+}-A_n^{-1}\stackrel{p}{\to }0$. In light of the above it follows from (B.10) that

$$\begin{array}{l}{\mid D_n\mid }^{1/2}\left({\widehat{\theta }}_n-{\theta }_{0n}\right)=-{\widehat{A}}_n^{+}{\widehat{B}}_n\left[{\mathrm{\sum }}_n^{-1/2}\mid D_n\mid R_n({\theta }_{0n})\right]+o_p(1)\\ =-A_n^{-1}{B}_n\left[{\mathrm{\sum }}_n^{-1/2}\mid D_n\mid R_n({\theta }_{0n})\right]+o_p(1)\end{array}$$

Recalling that sup_n λ_max [|D_n|⁻¹ Σ_n]< ∞, Assumptions 7(e) implies that |B_n| = O_p(1). In light of Assumption 7(g),(h) $B_n{B}_n^{\prime }$ is invertible and furthermore ${(B_n{B}_n^{\prime })}^{-1}=O(1)$. Thus $\left|{(A_n^{-1}{B}_n{B}_n^{\prime }{A_n^{-1}}^{\prime })}^{-1}\right|\le {\mid A_n\mid }^2\mid {(B_n{B}_n^{\prime })}^{-1}\mid =O(1)$ and therefore

$${(A_n^{-1}{B}_n{B}_n^{\prime }{A_n^{-1}}^{\prime })}^{-1/2}{\mid D_n\mid }^{1/2}\left({\widehat{\theta }}_n-{\theta }_{0n}\right)=-{(A_n^{-1}{B}_n{B}_n^{\prime }{A_n^{-1}}^{\prime })}^{-1/2}{A}_n^{-1}{B}_n\left[{\mathrm{\sum }}_n^{-1/2}\mid D_n\mid R_n({\theta }_{0n})\right]+o_p(1).$$

The claim that ${(A_n^{-1}{B}_n{B}_n^{\prime }{A_n^{-1}}^{\prime })}^{-1/2}{\mid D_n\mid }^{1/2}\left({\widehat{\theta }}_n-{\theta }_{0n}\right)\Rightarrow N(0,I_k)$ now follows, e.g., from Corollary F4(b) in Pötscher and Prucha (1997).