Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances

Abstract

This study develops a methodology of inference for a widely used Cliff-Ord type spatial model containing spatial lags in the dependent variable, exogenous variables, and the disturbance terms, while allowing for unknown heteroskedasticity in the innovations. We first generalize the GMM estimator suggested in Kelejian and Prucha (1998,1999) for the spatial autoregressive parameter in the disturbance process. We also define IV estimators for the regression parameters of the model and give results concerning the joint asymptotic distribution of those estimators and the GMM estimator. Much of the theory is kept general to cover a wide range of settings.

1 Introduction¹

In recent years the economics literature has seen an increasing number of theoretical and applied econometric studies involving spatial issues.² While this increase in interest in spatial models in economics is relatively recent, spatial models have a long history in the regional science and geography literature.³ One of the most widely referenced model of spatial interactions is one that was put forth by Cliff and Ord (1973, 1981). This model is a variant of the model considered by Whittle (1954). In its simplest (and original) form the model only considers spatial spillovers in the dependent variable, and specifies the endogenous variable corresponding to a cross sectional unit in terms of a weighted average of endogenous variables corresponding to other cross sectional units, plus a disturbance term. This model is typically referred to as a spatial autoregressive model, the weighted average is typically referred to as a spatial lag, the corresponding parameter as the autoregressive parameter, and the matrix containing the weights as the spatial weights matrix. A generalized version of this model also allows for the dependent variable to depend on a set of exogenous variables and spatial lags thereof. A further generalization allows for the disturbances to be generated by a spatial autoregressive process. Consistent with the terminology developed by Anselin and Florax (1995) we refer to the combined model as a spatial autoregressive model with autoregressive disturbances of order (1, 1), for short SARAR(1, 1). We note that this model is fairly general in that it allows for spatial spillovers in the endogenous variables, exogenous variables and disturbances.

Somewhat surprisingly, even though the SARAR(1, 1) model has been a modeling tool for many years, until recently there has been a lack of formal results concerning estimation methods for this model. One method that has been employed to estimate this model is the (quasi) maximum likelihood (ML) procedure, where the likelihood function corresponds to the normal distribution. Formal results concerning the asymptotic properties of the ML estimator have been established only recently in an important contribution by Lee (2004). Given that the likelihood function involves the determinant of a matrix whose dimensions depend on the sample size and an unknown parameter, there can be significant difficulties in the practical computation of this estimator especially if the sample size is large, as it might be if the spatial units relate to counties, single family houses, etc. In part because of this Kelejian and Prucha (1999) introduced a generalized moments (GM) estimator for the autoregressive parameter of the disturbance process that is simple to compute and remains computationally feasible even for large sample sizes. In Kelejian and Prucha (1998) we used that GM estimator to introduce a generalized spatial two stage least squares estimator (GS2SLS) for the regression parameters of the spatial SARAR(1, 1) model that is again simple to compute, and demonstrated its consistency and asymptotic normality.⁴

All of the above estimators for the SARAR(1, 1) model were introduced and their asymptotic properties were derived under the assumption that the innovations in the disturbance process are homoskedastic. The lack of an estimation theory that allows for heteroskedasticity, and the lack of corresponding joint hypothesis tests for the presence of spatial dependencies in the endogenous variables, exogenous variables and/or disturbances, is a serious shortcoming. Spatial units are often heterogeneous in important characteristics, e.g., size, and hence the homoskedasticity assumption may not hold in many situations (conditionally and unconditionally). It is readily seen that if the innovations are heteroskedastic, the ML estimator considered in Lee (2004) is inconsistent, and the asymptotic distribution given in Kelejian and Prucha (1998) for the GS2SLS estimator is not appropriate. One important goal of this study is therefore to develop a methodology of inference for the SARAR(1, 1) model that allows for heteroskedastic innovations. In developing this theory we will adopt a modular approach such that much of the theory not only applies to the SARAR(1, 1) model, but can also be utilized in different settings in future research.

In more detail, in this paper we introduce a new class of GM estimators for the autoregressive parameter of a spatially autoregressive disturbance process that allows for heteroskedastic innovations. Our GM estimators are again computationally simple even in large samples. We determine their consistency; unlike in our earlier paper we also determine, under reasonably general conditions, their asymptotic distribution. Loosely speaking, in deriving those results we essentially only maintain that the disturbances are n^1/2-consistently estimated (where n is the sample size) and that the estimator of the model parameters employed in estimating the disturbances is asymptotically linear in the innovations. As a result the methodology developed in this paper covers a wide range of (linear and nonlinear) models and estimators, in addition to the SARAR(1, 1) model and estimators specific for that model. We furthermore derive results concerning the joint distribution of the GM estimators and estimators of the regression parameters to facilitate joint tests. While the results are presented for the case of two step estimation procedures where the spatial autoregressive parameter and the regression parameters are estimated in separate steps, the analysis can be readily adapted to one step procedures where all parameters are estimated in a single (but numerically more involved) step.

The general theory is then applied to develop inference methodology for the SARAR(1, 1) model. In particular, we use the GM estimator in constructing a GS2SLS estimator for the regression parameters of the SARAR(1, 1) model and demonstrate the consistency and asymptotic normality of this estimator. We also provide results concerning the joint distribution of the GM estimator and the GS2SLS estimator, which permits, among other things, testing the joint hypothesis of the absence of spatial spillovers stemming from the endogenous variables, exogenous variables or disturbances.

Another concern with the existing literature on Cliff-Ord type models, including in the above cited literature on the SARAR(1, 1) models, is the specification of the parameter space for spatial autoregressive parameters. In virtually all of the literature it is assumed that the parameter space for autoregressive parameters is the interval (−1, 1), or a subset thereof. One may conjecture that this traditional specification of the parameter space received its motivation from the time series literature. However, as discussed in detail below, choosing the interval (−1, 1) as the parameter space for the autoregressive parameter of a spatial model is not natural in the sense that the spatial autoregressive parameter always appears in those models in product form with the spatial weights matrix. Hence equivalent model formulations are obtained by applying an (arbitrary) scale factor to the autoregressive parameter and its inverse to the weights matrix. Of course, applying a scale factor to the autoregressive parameter leads to a corresponding re-scaling of its parameter space. In this paper we therefore allow for a more general specification of the parameter space. Even if a scale factor is used that results in the parameter space being the interval (−1, 1), this scale factor and correspondingly the autoregressive parameter will then typically depend on the sample size. In contrast to the existing literature we thus allow for the parameters to depend on the sample size. Our discussion of the parameter space and possible normalizations of the spatial weights matrix also points out potential pitfalls with the frequently used approach of row-normalizing the spatial weights matrix.

The paper is organized as follows: The generalized SARAR(1, 1) model is specified and interpreted in Section 2. This section also contains a discussion of the parameter space of the autoregressive parameter. In Section 3 we define and establish the large sample properties of our suggested GM estimators for the autoregressive parameter of a spatially autoregressive disturbance process. In this section we also provide results concerning the joint large sample distribution of the GM estimators and a wide class of estimator of the regression parameters. We also develop HAC type estimators for the large sample variance-covariance matrix of the suggested estimators. Section 4 contains results relating to the suggested instrumental variable estimators of the regression parameters of the SARAR(1, 1) model and their joint large sample distribution with GM estimators. Concluding remarks are given in the Section 5. Technical details are relegated to the appendices.

It proves helpful to introduce the following notation: Let A_n with n ∈ N be some matrix; we then denote the (i, j)-th element of A_n as a_ij,n. Similarly, if v_n is a vector, then v_i,n denotes the i-th element of v_n. An analogous convention is adopted for matrices and vectors that do not depend on the index n, in which case the index n is suppressed on the elements. If A_n is a square matrix, then ${\mathbf{A}}_n^{-1}$ denotes the inverse of A_n. If A_n is singular, then ${\mathbf{A}}_n^{-1}$ should be interpreted as the generalized inverse of A_n. At times it will also be helpful to denote the generalized inverse more explicitly as ${\mathbf{A}}_n^{+}$. With a_i.,n and a_.i,n we denote the i-th row and column of A_n, respectively, and with ${\mathbf{a}}_n^{i.}$ and ${\mathbf{a}}_n^{.i}$ those of ${\mathbf{A}}_n^{-1}$. If A_n is a square symmetric nonnegative matrix, then ${\mathbf{A}}_n^{1/2}$ denotes the unique symmetric and nonnegative square root of A_n. If A_n is nonsingular, then ${\mathbf{A}}_n^{-1/2}$ denotes ${({\mathbf{A}}_n^{-1})}^{1/2}$. Further, we say the row and column sums of the (sequence of) matrices A_n are bounded uniformly in absolute value if there exists a constant c_A < ∞ (that does not dependent of n) such that

$$\underset{1\le i\le n}{\text{max}}{\displaystyle \sum _{j=1}^n|a_{ij,n}|\le c_A\text{\hspace{1em}and\hspace{1em}}}\underset{1\le j\le n}{\text{max}}{\displaystyle \sum _{i=1}^n|a_{ij,n}|\le c_A\text{for all}n\in \mathbf{N}}$$

holds. As a point of interest, we note that the above condition is identical to the condition that the sequences of the maximum column sum matrix norms and maximum row sum matrix norms of A_n are bounded; cp. Horn and Johnson (1985, pp.294–5). For definiteness, let A be some vector or matrix, then ||A|| = [Tr(A′A)]^1/2. We note that this norm is submultiplicative, i.e., ||AB|| ≤ ||A|| ||B||.

2 Model

In this section we specify the generalized SARAR(1, 1) model and discuss the underlying assumptions.

2.1 Specification

Suppose a cross section of n spatial units is observed, and the interactions between those spatial units can be described by the following model:

$$\begin{array}{ccc}{\mathbf{y}}_n\hfill & =\hfill & {\mathbf{X}}_n{\beta }_n+{\mathrm{\lambda }}_n{\mathbf{W}}_n{\mathbf{y}}_n+{\mathbf{u}}_n\hfill \\ \hfill & =\hfill & {\mathbf{Z}}_n{\mathbf{\delta }}_n+{\mathbf{u}}_n\hfill \end{array}$$ (1)

and

$${\mathbf{u}}_n={\rho }_n{\mathbf{M}}_n{\mathbf{u}}_n+{\epsilon }_n,$$ (2)

with Z_n = [X_n, W_ny] and ${\mathbf{\delta }}_n=[{\beta }_n^{\prime },{\mathrm{\lambda }}_n]\prime$. Here y_n denotes the n × 1 vector of observations of the dependent variable, X_n denotes the n × k matrix of non-stochastic (exogenous) regressors, W_n and M_n are n × n non-stochastic matrices, u_n denotes the n × 1 vector of regression disturbances, ε_n is an n × 1 vector of innovations, λ_n and ρ_n are unknown scalar parameters, and β_n is a k × 1 vector of unknown parameters. The matrices W_n and M_n are typically referred to as spatial weights matrices, and λ_n and ρ_n are typically called spatial autoregressive parameters. The analysis allows for W_n = M_n, which will frequently be the case in applications. All quantities are allowed to depend on the sample size.

The vectors ȳ_n = W_ny_n and ū_n = M_nu_n are typically referred to as spatial lags of y_n and u_n, respectively. We note that all quantities are allowed to depend on the sample size and so some of the exogenous regressors may be spatial lags of exogenous variables. Thus the model is fairly general in that it allows for spatial spillovers in the endogenous variables, exogenous variables and disturbances.

The spatial weights matrices and the autoregressive parameters are assumed to satisfy the following assumption.

Assumption 1

(a) All diagonal elements of W_n and M_n are zero. (b) ${\mathrm{\lambda }}_n\in (-{\underset{¯}{a}}_n^{\mathrm{\lambda }},{\overline{a}}_n^{\mathrm{\lambda }}),{\rho }_n\in (-{\underset{¯}{a}}_n^{\rho },{\overline{a}}_n^{\rho })$ with $0<{\underset{¯}{a}}_n^{\mathrm{\lambda }},{\overline{a}}_n^{\mathrm{\lambda }}\le a^{\mathrm{\lambda }}<\mathrm{\infty }$ and $0<{\underset{¯}{a}}_n^{\rho },{\overline{a}}_n^{\rho }\le a^{\rho }<\mathrm{\infty }$. (c) The matrices I_n – λW_n and I_n – ρM_n are nonsingular for all $\mathrm{\lambda }\in (-{\underset{¯}{a}}_n^{\mathrm{\lambda }},{\overline{a}}_n^{\mathrm{\lambda }})$, and $\rho \in (-{\underset{¯}{a}}_n^{\rho },{\overline{a}}_n^{\rho })$.

Assumption 1(a) is clearly a normalization rule. Assumption 1(b) concerning the parameter space of λ_n and ρ_n will be discussed in the next subsection. Assumption 1(c) ensures that y_n and u_n are uniquely defined by (1) and (2) as

$$\begin{array}{ccc}{\mathbf{y}}_n\hfill & =\hfill & {({\mathbf{I}}_n-{\mathrm{\lambda }}_n{\mathbf{W}}_n)}^{-1}{\mathbf{X}}_n{\beta }_n+{({\mathbf{I}}_n-{\mathrm{\lambda }}_n{\mathbf{W}}_n)}^{-1}{\mathbf{u}}_n,\hfill \\ {\mathbf{u}}_n\hfill & =\hfill & {({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n)}^{-1}{\epsilon }_n.\hfill \end{array}$$ (3)

As remarked in the Introduction, spatial units are often heterogeneous in important characteristics, e.g., size. For that reason it is important to develop an estimation theory that allows for the innovations to be heteroskedastic. Therefore, we maintain the following set of assumptions with respect to the innovations.

Assumption 2

The innovations {ε_i,n : 1 ≤ i ≤ n, n ≥ 1} satisfy Eε_i,n = 0, $E({\epsilon }_{i,n}^2)={\sigma }_{i,n}^2$ with $0<{\underset{¯}{a}}^{\sigma }\le {\sigma }_{i,n}^2\le {\overline{a}}^{\sigma }<\mathrm{\infty }$, and sup_{1≤i≤n,n≥1} E|ε_i,n|^4+η < ∞ for some η > 0. Furthermore, for each n ≥ 1 the random variables ε_1,n, …, ε_n,n are totally independent.

The above assumption also allows for the innovations to depend on the sample size n, i.e., to form triangular arrays. We note that even if the innovations do not depend on n, the elements of y_n and u_n would still depend on n in light of (3) since the elements of the inverse matrices involved would generally depend on n. We maintain the following assumption concerning the spatial weights matrices.

Assumption 3

The row and column sums of the matrices W_n, M_n, (I_n – λ_nW_n)⁻¹ and (I_n – ρ_nM_n)⁻¹ are bounded uniformly in absolute value.

Given (3), Assumption 2 implies that Eu_n = 0, and that the VC matrix of u_n is given by

$$E{\mathbf{u}}_n{\mathbf{u}}_n^{\prime }={({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n)}^{-1}{\displaystyle {\mathbf{\Sigma }}_n{({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n)}^{-1}}$$

where ${\mathbf{\Sigma }}_n=\mathit{\text{diag}}({\sigma }_{i,n}^2)$. This specification allows for fairly general patterns of autocorrelation and heteroskedasticity of the disturbances. It is readily seen that the row and column sums of products of matrices, whose row and column sums are bounded uniformly in absolute value, are again uniformly bounded in absolute value; see, e.g., Kelejian and Prucha (2004), Remark A.1. Because of this, Assumptions 2 and 3 imply that the row and column sums of the variance-covariance (VC) matrix of u_n (and similarly those of y_n) are uniformly bounded in absolute value, thus limiting the degree of correlation between, respectively, the elements of u_n (and of y_n). That is, making an analogy to the time series literature, these assumptions ensure that the disturbance process and the process for the dependent variable exhibit a “fading” memory.⁵

2.2 Parameter Space for an Autoregressive Parameter

Assumption 1(b) defines the parameter space for the autoregressive parameters. In discussing this assumption we focus on W_n and λ_n. (An analogous discussion applies to M_n and ρ_n.) In the existing literature relating to Cliff-Ord models the parameter space for the autoregressive parameter is typically taken to be the interval (−1, 1) and the autoregressive parameter is assumed not to depend on the sample size. However, in applications it is typically found that for unnormalized spatial weights matrices, I_n – λW_n is singular for some values of λ ∈ (−1, 1). To avoid this situation, many applied researchers normalize each row of their spatial weights matrices in such a way that I_n – λW_n is non-singular for all λ ∈ (−1, 1). We now discuss the implications of various normalizations of the spatial weight matrix.

Suppose c_n denotes a scalar normalization factor. Clearly, this normalization factor may depend on the sample size. For example, some of our results below relate to the case in which c_n corresponds to the maximal row or column sum of the absolute values of the elements of W_n. Given such a normalizing factor, an equivalent specification of model (1) for y_n is obtained if λ_n W_n is replaced by ${\mathrm{\lambda }}_n^{*}{\mathbf{W}}_n^{*}$ where ${\mathrm{\lambda }}_n^{*}=c_n{\mathrm{\lambda }}_n$ and ${\mathbf{W}}_n^{*}={\mathbf{W}}_n/c_n$. It is important to observe that even if λ_n and its corresponding parameter space do not depend on ${\mathrm{\lambda }}_n^{*}$ and its implied parameter space will depend on the sample size as a result of the normalization of the spatial weights matrix.⁶ It is for this reason that we allow in Assumption 1 for the elements of the spatial weights matrices, and the autoregressive parameters and the corresponding parameter spaces to depend on n. Of course, Assumption 1 also covers the case where the true data generating process corresponds to a model where autoregressive parameters do not depend on n.

Assumption 1 defines the parameter space for λ_n as an interval around zero such that I_n – λW_n is nonsingular for values λ in that interval. The following trivial lemma gives bounds for that interval.

Lemma 1

Let τ_n denote the spectral radius of W_n; i.e.,

$${\tau }_n=\text{max}\left\{|{\nu }_{1,n}|,\dots ,|{\nu }_{n,n}|\right\}$$

where ν_1,n, …, ν_n,n denote the eigenvalues of W_n. Then I_n – λW_n is nonsingular for all values of λ in the interval (−1/τ_n, 1/τ_n).⁷

Clearly, if we select (−1/τ_n, 1/τ_n) as the parameter space for λ_n, then all eigenvalues of λ_n W_n are less than one in absolute value. Thus if we interpret (1) as an equilibrium relationship, then this choice of the parameter space rules out unstable Nash equilibria. Of course, we obtain an equivalent specification of the model if instead of working with W_n we work with the normalized weights matrix ${\mathbf{W}}_n^{*}={\mathbf{W}}_n/{\tau }_n$ and select the interval (−1, 1) as the parameter space for ${\mathrm{\lambda }}_n^{*}={\mathrm{\lambda }}_n{\tau }_n$. Assumption 1 is sufficiently general to cover both cases.

For large sample sizes the computation of the eigenvalues on W_n is difficult. The following lemma gives boundaries, which are simple to compute, for a (sub)set of values of λ for which I_n – λW_n is nonsingular.

Lemma 2

Let

$${\tau }_n^{*}=\text{min}\{\underset{1\le i\le n}{\text{max}}{\displaystyle \sum _{j=1}^n|w_{ij,n}|,}\underset{1\le j\le n}{\text{max}}{\displaystyle \sum _{i=1}^n|w_{ij,n}|\}.}$$

Then ${\tau }_n\le {\tau }_n^{*}$ and consequently I_n – λW_n is nonsingular for all values of λ in the interval $(-1/{\tau }_n^{*},1/{\tau }_n^{*})$.

The above lemma suggests $(-1/{\tau }_n^{*},1/{\tau }_n^{*})$ as an alternative (although somewhat more restrictive) specification of the parameter space. Of course, we obtain an equivalent model specification if we normalize the spatial weights matrix by ${\tau }_n^{*}$ and if we choose (−1, 1) as the parameter space for the autoregressive parameter. Since the spectral radius is bound by any matrix norm, other norms in place of the maximum absolute row and column sum norms can be used, but ${\tau }_n^{*}$ is especially easy to compute.

Rather than to normalize W_n by τ_n or ${\tau }_n^{*}$ in much of the empirical literature the spatial weights matrices are normalized such that each row sums to unity. The motivation for this normalization is that if W_n is row-normalized then I_n – λW_n is nonsingular for all values of λ in the interval (−1, 1); this can be readily confirmed via Lemma 2. However, this normalization is quite different than those described above in that in row-normalizing a matrix one does not use a single normalization factor, but rather a different factor for the elements of each row. Therefore, in general, there exists no corresponding re-scaling factor for the autoregressive parameter that would lead to a specification that is equivalent to that corresponding to the un-normalized weights matrix. Consequently, unless theoretical issues suggest a row-normalized weights matrix, this approach will in general lead to a misspecified model.

The above discussion provides the motivation for our specification that the autoregressive parameters may depend on n. Furthermore, since some of the regressors may be spatial lags, we allow all of the model parameters to depend on the sample size.

3 GM Estimator for the Autoregressive Parameter ρ_n

In the following we introduce a class of GM estimators for ρ_n that can be easily computed, and prove their consistency and asymptotic normality under a set of general assumptions. We note that the discussion in this section only maintains model (2) for the disturbances u_n, but not necessarily (1) for y_n. Thus the results will also be useful in other settings such as cases where y_n is determined by a nonlinear model; see, e.g. Kelejian and Prucha (2001, p. 228). The estimators put forth below generalize the GM estimator for ρ_n introduced in Kelejian and Prucha (1999). In contrast to that earlier paper we now allow for heteroskedastic innovations ε _i,n, and optimal weighting of the moment conditions. We also do not confine the parameter space for ρ_n to be the interval (−1, 1), and allow ρ_n to depend on n. In our earlier paper we only demonstrated the consistency of the estimator. In the following we also derive the asymptotic distribution of the considered estimators.

3.1 Definition of the GM Estimator for ρ_n

In the following let ũ_n denote some predictor of u_n. Furthermore, for notational convenience let ū_n = M_nu_n and ${\overline{\overline{\mathbf{u}}}}_n={\mathbf{M}}_n{\overline{\mathbf{u}}}_n={\mathbf{M}}_n^2{\mathbf{u}}_n$, and correspondingly, ${\stackrel{\sim }{\overline{\mathbf{u}}}}_n={\mathbf{M}}_n{\stackrel{\sim }{\mathbf{u}}}_n$, and ${\stackrel{\sim }{\overline{\overline{\mathbf{u}}}}}_n={\mathbf{M}}_n^2{\stackrel{\sim }{\mathbf{u}}}_n$. Similarly, let ε̄_n = M_nε_n. It is readily seen that under Assumptions 1 and 2 we have the following moment conditions:

$$\begin{array}{ccc}{n}^{-1}E{\overline{\mathbf{\epsilon }}}_n^{\prime }{\overline{\mathbf{\epsilon }}}_n\hfill & =\hfill & n^{-1}\mathit{\text{Tr}}\{{\mathbf{M}}_n[{\mathit{\text{diag}}}_{i=1}^n(E{\epsilon }_{i,n}^2)]{\mathbf{M}}_n^{\prime }\},\hfill \\ n^{-1}E{\overline{\epsilon }}_n^{\prime }{\epsilon }_n\hfill & =\hfill & 0.\hfill \end{array}$$ (4)

It proves convenient to rewrite these conditions as

$$n^{-1}E\left[\begin{array}{c}\hfill {\mathbf{\epsilon }}_n^{\prime }{\mathbf{A}}_{1,n}{\mathbf{\epsilon }}_n\hfill \\ \hfill {\mathbf{\epsilon }}_n^{\prime }{\mathbf{A}}_{2,n}{\mathbf{\epsilon }}_n\hfill \end{array}\right]=0$$ (5)

with

$${\mathbf{A}}_{1,n}={\mathbf{M}}_n^{\prime }{\mathbf{M}}_n-dia{g}_{i=1}^n({\mathbf{m}}_{.i,n}^{\prime }{\mathbf{m}}_{.i,n}),{\mathbf{A}}_{2,n}={\mathbf{M}}_n.$$

Under Assumptions 1 and 3 it is readily seen that the diagonal elements of A_1,n and A_2,n are zero and that the row and column sums of A_1,n and A_2,n are bounded uniformly in absolute value; see, e.g., Remark A.1 in Kelejian and Prucha (2004).

Our GM estimators for ρ_n are based on these moments. Specifically, note that in light of (2) ε_n = (I_n – ρ_n M_n)u_n = u_n – ρ_n ū_n and so ${\overline{\epsilon }}_n={\overline{\mathbf{u}}}_n-{\rho }_n{\overline{\overline{\mathbf{u}}}}_n$. Substituting these expressions into (4) or (5) yields the following two equation system:

$${\gamma }_n-{\mathbf{\Gamma }}_n{\mathbf{\alpha }}_n=0$$ (6)

where ${\mathbf{\alpha }}_n=[{\rho }_n,{\rho }_n^2]\prime$ and the elements of Γ_n = [γ_rs,n]_r,s=1,2 and γ_n = [γ_1,n,γ_2,n]′ are given by

$$\begin{array}{ccc}{\gamma }_{11,n}\hfill & =\hfill & 2n^{-1}E\left\{{\overline{\overline{\mathbf{u}}}}_n^{\prime }{\overline{\mathbf{u}}}_n-\mathit{\text{Tr}}[{\mathbf{M}}_n[{\mathit{\text{diag}}}_{i=1}^n({\overline{u}}_{i,n}{u}_{i,n})]{\mathbf{M}}_n^{\prime }]\right\}\hfill \\ \hfill & =\hfill & 2n^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{M}}_n^{\prime }{\mathbf{A}}_{1,n}{\mathbf{u}}_n,\hfill \\ {\gamma }_{12,n}\hfill & =\hfill & -n^{-1}E\left\{{\overline{\overline{\mathbf{u}}}}_n^{\prime }{\overline{\overline{\mathbf{u}}}}_n+\mathit{\text{Tr}}[{\mathbf{M}}_n[{\mathit{\text{diag}}}_{i=1}^n({\overline{u}}_{i,n}^2)]{\mathbf{M}}_n^{\prime }]\right\}\hfill \\ \hfill & =\hfill & -n^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{M}}_n^{\prime }{\mathbf{A}}_{1,n}{\mathbf{M}}_n{\mathbf{u}}_n,\hfill \\ {\gamma }_{21,n}\hfill & =\hfill & n^{-1}E({\mathbf{u}}_n^{\prime }{\overline{\overline{\mathbf{u}}}}_n+{\overline{\mathbf{u}}}_n^{\prime }{\overline{\mathbf{u}}}_n)\hfill \\ \hfill & =\hfill & n^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{M}}_n^{\prime }({\mathbf{A}}_{2,n}+{\mathbf{A}}_{2,n}^{\prime }{)\mathbf{u}}_n,\hfill \\ {\gamma }_{22,n}\hfill & =\hfill & -n^{-1}E{\overline{\mathbf{u}}}_n^{\prime }{\overline{\overline{\mathbf{u}}}}_n\hfill \\ \hfill & =\hfill & -n^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{M}}_n^{\prime }{\mathbf{A}}_{2,n}{\mathbf{M}}_n{\mathbf{u}}_n,\hfill \\ {\gamma }_{1,n}\hfill & =\hfill & n^{-1}E\left\{{\overline{\mathbf{u}}}_n^{\prime }{\overline{\mathbf{u}}}_n-\mathit{\text{Tr}}[{\mathbf{M}}_n[{\mathit{\text{diag}}}_{i=1}^n(u_{i,n}^2)]{\mathbf{M}}_n^{\prime }]\right\}\hfill \\ \hfill & =\hfill & n^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{A}}_{1,n}{\mathbf{u}}_n,\hfill \\ {\gamma }_{2,n}\hfill & =\hfill & n^{-1}E{\mathbf{u}}_n^{\prime }{\overline{\mathbf{u}}}_n\hfill \\ \hfill & =\hfill & n^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{A}}_{2,n}{\mathbf{u}}_n.\hfill \end{array}$$ (7)

Now let Γ̃_n = [γ̃_rs,n]_r,s=1,2 and γ̃_n = [γ̃_1,n, γ̃_2,n]′ denote corresponding estimators for the elements of Γ_n and γ_n, which are obtained from the above expressions for the elements of Γ_n and γ_n by suppressing the expectations operator, and replacing the disturbances u_n, ū_n, and u̿_n by their predictors ũ_n, ${\stackrel{\sim }{\overline{\mathbf{u}}}}_n,\text{and}{\stackrel{\sim }{\overline{\overline{\mathbf{u}}}}}_n$, respectively. Then, the empirical analog of the relationship in (6) is

$${\stackrel{\sim }{\gamma }}_n-{\stackrel{\sim }{\mathbf{\Gamma }}}_n{\mathbf{\alpha }}_n={\mathbf{\upsilon }}_n,$$ (8)

where υ_n can be viewed as a vector of regression residuals. Our GM estimators of ρ_n, say ρ̃_n, are now defined as weighted nonlinear least squares estimators based on (8). That is, let ϒ̃_n be a 2 × 2 symmetric positive semidefinite (moments) weighting matrix; then ρ̃_n is defined as

$${\stackrel{\sim }{\rho }}_n={\stackrel{\sim }{\rho }}_n({\stackrel{\sim }{\mathrm{\Upsilon }}}_n)=\underset{\rho \in [-a^{\rho },a^{\rho }]}{\mathit{\text{argmin}}}\left\{{\left[{\stackrel{\sim }{\mathrm{\gamma }}}_n-{\stackrel{\sim }{\mathbf{\Gamma }}}_n\left[\begin{array}{c}\rho \\ {\rho }^2\end{array}\right]\right]}^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n\left[{\stackrel{\sim }{\mathrm{\gamma }}}_n-{\stackrel{\sim }{\mathbf{\Gamma }}}_n\left[\begin{array}{c}\rho \\ {\rho }^2\end{array}\right]\right]\right\}.$$ (9)

We note that the objective function for ρ̃_n remains well defined even for values of ρ_n for which I_n – ρ_nM_n is singular, which allows us to take the optimization space for ρ̃_n to be any compact interval that contains the true parameter space. For computational efficiency it is best to use the formulae for the elements of Γ̃_n and γ̃_n corresponding to the first expression on the r.h.s. of (7) and to compute ${\stackrel{\sim }{\overline{\mathbf{u}}}}_n\text{and}{\stackrel{\sim }{\overline{\overline{\mathbf{u}}}}}_n$ recursively as ${\stackrel{\sim }{\overline{\mathbf{u}}}}_n={\mathbf{M}}_n{\stackrel{\sim }{\mathbf{u}}}_n\text{and}{\stackrel{\sim }{\overline{\overline{\mathbf{u}}}}}_n={\mathbf{M}}_n{\stackrel{\sim }{\overline{\mathbf{u}}}}_n$. In this fashion one can avoid the computation of ${\mathbf{M}}_n^2$, i.e., the computation of the product of two n × n matrices.

We now relate the above estimator for the autoregressive parameter to the GM estimator introduced in Kelejian and Prucha (1999). Under homoskedasticity, ${\sigma }^2={\sigma }_{i,n}^2$ and so $E[n^{-1}{\epsilon }_n^{\prime }{\epsilon }_n]={\sigma }^2,E[n^{-1}{\overline{\epsilon }}_n^{\prime }{\overline{\epsilon }}_n]={\sigma }^2{n}^{-1}\mathit{\text{Tr}}\left\{{\mathbf{M}}_n{\mathbf{M}}_n^{\prime }\right\},\text{and}E[n^{-1}{\overline{\epsilon }}_n^{\prime }{\epsilon }_n]=0$. These three moment conditions underlie the GM estimator suggested in Kelejian and Prucha (1999). Substituting the first of these moment conditions into the second yields

$$\begin{array}{ccc}E[\frac{1}{n}{\overline{\epsilon }}_n^{\prime }{\overline{\epsilon }}_n]\hfill & =\hfill & E[n^{-1}{\epsilon }_n^{\prime }{\epsilon }_n]n^{-1}\mathit{\text{Tr}}\left\{{\mathbf{M}}_n{\mathbf{M}}_n^{\prime }\right\},\hfill \\ E[\frac{1}{n}{\overline{\epsilon }}_n^{\prime }{\epsilon }_n]\hfill & =\hfill & 0,\hfill \end{array}$$ (10)

which is clearly a special case of (4) under homoskedasticity. It is not difficult to see that the GM estimator suggested in our previous paper can be viewed as being based on the two moment conditions in (10) with ϒ̃_n = diag(υ_n, 1) and ${\upsilon }_n=1/\left[1+{[n^{-1}\mathit{\text{Tr}}\{{\mathbf{M}}_n{\mathbf{M}}_n^{\prime }\}]}^2\right]$. ⁸

3.2 Consistency of the GM Estimator for ρ_n

To establish consistency of ρ̃_n we postulate the following additional assumptions.

Assumption 4

Let ũ_i,n denote the i-th element of ũ_n. We then assume that

$${\stackrel{\sim }{u}}_{i,n}-u_{i,n}={\mathbf{d}}_{i.,n}{\mathbf{\Delta }}_n$$

where d_i.,n and Δ_n are 1 × p and p × 1 dimensional random vectors. Let d_ij,n be the j-th element of d_i.,n. Then, we assume that for some δ > 0 E|d_ij,n|^2+δ ≤ c_d < ∞ where c_d does not depend on n, and that n^1/2 ∥ Δ_n ∥ = O_p(1).

Assumption 5

(a)The smallest eigenvalue of ${\mathbf{\Gamma }}_n^{\prime }{\mathbf{\Gamma }}_n$ is uniformly bounded away from zero.⁹ (b) ϒ̃_n – ϒ_n = o_p(1), where ϒ_n are 2 × 2 non-stochastic symmetric positive definite matrices. (c) The largest eigenvalues of ϒ_n are bounded uniformly from above, and the smallest eigenvalues of ϒ_n are uniformly bounded away from zero.

Assumption 4 implies $n^{-1}{\displaystyle {\sum }_{i=1}^n{\Vert {\mathbf{d}}_{i.,n}\Vert }^{2+\delta }=O_p(1)}$, which was maintained in Kelejian and Prucha (1999), and so is slightly stronger than their assumption. Assumption 4 should be satisfied for typical linear spatial models where ũ_i is based on n^1/2-consistent estimators of regression coefficients, d_i.,n denotes the i-th row of the regressor matrix, and Δ_n denotes the difference between the parameter estimator and the true parameter values. In the next section we will actually demonstrate that Assumption 4 holds for the estimated residuals of model (1) based on an instrumental variable procedure. Assumption 4 should also be satisfied for typical non-linear models provided the response function is differentiable in the parameters, and the derivatives are (uniformly over the parameter space) bounded by some random variable with bounded 2 + δ moments; compare Kelejian and Prucha (1999).

Assumption 5 ensures that the smallest eigenvalue of ${\mathbf{\Gamma }}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Gamma }}_n$ is uniformly bounded away from zero and will be sufficient to permit us to demonstrate that ρ_n is identifiably unique w.r.t. the nonstochastic analogue of the objective function of the GM estimator. This analogue is given by the function in curly brackets on the r.h.s. of (9) with γ̃_n, Γ̃_n and ϒ̃_n replaced by γ_n, Γ_n and ϒ_n. Under homoskedasticity and ϒ̃_n = ϒ_n specified as at the end of the previous subsection this assumption is in essence equivalent to Assumption 5 in Kelejian and Prucha (1999).

Clearly Assumption 5 requires Γ_n to be nonsingular, or equivalently that $[tr({\mathbf{M}}_n^{\prime }{\mathbf{A}}_{1,n}{\mathbf{M}}_n{\mathbf{S}}_{u,n}),tr({\mathbf{M}}_n^{\prime }{\mathbf{A}}_{2,n}{\mathbf{M}}_n{\mathbf{S}}_{u,n})]\prime$ is linearly independent of $[tr({\mathbf{M}}_n^{\prime }({\mathbf{A}}_{1,n}+{\mathbf{A}}_{1,n}^{\prime }){\mathbf{S}}_{u,n}),tr({\mathbf{M}}_n^{\prime }({\mathbf{A}}_{2,n}+{\mathbf{A}}_{2,n}^{\prime }){\mathbf{S}}_{u,n})]\prime$, which is readily seen by observing that $E{\mathbf{u}}_n^{\prime }{\mathbf{M}}_n^{\prime }{\mathbf{A}}_{i,n}{\mathbf{M}}_n{\mathbf{u}}_n=tr({\mathbf{M}}_n^{\prime }{\mathbf{A}}_{i,n}{\mathbf{M}}_n{\mathbf{S}}_{u,n})\text{and}{n}^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{M}}_n^{\prime }({\mathbf{A}}_{i,n}+{\mathbf{A}}_{i,n}^{\prime }){\mathbf{u}}_n=tr({\mathbf{M}}_n^{\prime }({\mathbf{A}}_{i,n}+{\mathbf{A}}_{i,n}^{\prime }){\mathbf{S}}_{u,n})$ where S_u,n = (I_n – ρ_nM_n)⁻¹ Σ_n (I_n – ρ_nM_n)⁻¹. It is not difficult to see that this linear independence condition is an analogue to identification conditions postulated in Lee (2007b), Assumption 5(b), relating to quadratic forms under homoskedasticity.

We note that while Assumption 5 should be satisfied in many settings, it does not cover situations where all elements of the spatial weights matrix converge to zero uniformly as n → ∞ - see Lee (2004) - since in this case the elements of Γ_n would tend to zero. On the other hand, Assumption 5 does not generally rule out settings where M_n is row normalized, there is an increasing number of nonzero elements in each row, and the row sums of the absolute values of the non-normalized elements are uniformly bounded.

The vector of derivatives (multiplied by minus one) of the moment conditions (6) w.r.t. ρ_n is given by J_n = Γ_n[1, 2ρ_n]′. As expected, the limiting distribution of the GM estimator ρ̃_n will be seen to depend on the inverse of ${\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n$. Assumption 5 also ensure that ${\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n$ is nonsingular.

Because of the equivalence of matrix norms it follows from Assumption 5 that the elements of ϒ_n and ${\mathrm{\Upsilon }}_n^{-1}$ are O(1).

We can now give our basic consistency result for ρ̃_n.

Theorem 1

Let ρ̃_n = ρ̃_n (ϒ̃_n) denote the GM estimator defined by (9). Then, provided the optimization space contains the parameter space, and given Assumptions 1–5,

$${\stackrel{\sim }{\rho }}_n-{\rho }_n\stackrel{p}{\to }0asn\to \mathrm{\infty }.$$

Clearly the conditions of the theorem regarding ϒ̃_n and ϒ_n are satisfied for ϒ̃_n = ϒ_n = I₂. In this case the estimator reduces to the nonlinear least squares estimator based on (8). This estimator can, e.g., be used to obtain initial consistent estimates of the autoregressive parameter. Choices of ϒ̃_n that lead to an efficient GM estimator for ρ_n (but require some consistent initial estimate of ρ_n) will be discussed below in conjunction with the asymptotic normality result.

3.3 Asymptotic Distribution of the GM Estimator for ρ_n

Let ${\mathbf{D}}_n=[{\mathbf{d}}_{1.,n}^{\prime },\dots ,{\mathbf{d}}_{n.,n}^{\prime }]\prime$ where d_i.,n is defined in Assumption 4 so that ũ_n – u_n = D_nΔ_n. To establish the asymptotic normality of ρ̃_n we need some additional assumptions.

Assumption 6

For any n × n real matrix A_n whose row and column sums are bounded uniformly in absolute value

$$n^{-1}{\mathbf{D}}_n^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n-n^{-1}E{\mathbf{D}}_n^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n=o_p(1).$$

A sufficient condition for Assumption 6 is, e.g., that the columns of D_n are of the form π_n + Π_nε_n, where the elements of π_n are bounded in absolute value and the row and column sums of Π_n are uniformly bounded in absolute value; see Lemma C.2. This will indeed be the case in many applications. In the next section we will verify that this assumption holds for the model given by (1) and (2), and where D_n equals the (negative of the) design matrix Z_n.

Assumption 7

Let Δ_n be as defined in Assumption 4. Then

$$n^{1/2}{\mathbf{\Delta }}_n=n^{-1/2}{\mathbf{T}}_n^{\prime }{\epsilon }_n+o_p(1),$$

where T_n is a n × p dimensional real nonstochastic matrix whose elements are uniformly bounded in absolute value.

As remarked above, typically Δ_n denotes the difference between the parameter estimator and the true parameter values. Assumption 7 will be satisfied by many estimators. In the next section we verify that this assumption indeed holds for the considered instrumental variable estimators for the parameters of model (1).

It may be helpful to provide some insight concerning the variance of the limiting distribution of the GM estimator n^1/2(ρ̃_n – ρ_n) given below. To that effect we note that an inspection of the derivation of this limiting distribution in Appendix C shows that it depends on the limiting distribution of the (properly normalized) vector of quadratic forms

$${\mathbf{v}}_n=n^{-1/2}\left[\begin{array}{c}\frac{1}{2}{\epsilon }_n^{\prime }({\mathbf{A}}_{1,n}+{\mathbf{A}}_{1,n}^{\prime }){\epsilon }_n+{\mathbf{a}}_{1,n}^{\prime }{\epsilon }_n\hfill \\ \frac{1}{2}{\epsilon }_n^{\prime }({\mathbf{A}}_{2,n}+{\mathbf{A}}_{2,n}^{\prime }){\epsilon }_n+{\mathbf{a}}_{2,n}^{\prime }{\epsilon }_n\hfill \end{array}\right]$$ (11)

where for r = 1, 2 the n × n matrices A_r,n are defined in (5), and where the n × 1 vectors a_r,n are defined as

$${\mathbf{a}}_{r,n}={\mathbf{T}}_n{\mathbf{\alpha }}_{r,n}$$ (12)

with

$${\mathbf{\alpha }}_{r,n}=n^{-1}E[{\mathbf{D}}_n^{\prime }({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })({\mathbf{A}}_{r,n}+{\mathbf{A}}_{r,n}^{\prime })({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n){\mathbf{u}}_n].$$

From (11) and (12) we see that, in general, the limiting distribution of n^1/2(ρ̃_n – ρ_n) will depend on the limiting distribution of n^1/2Δ_n via the matrix T_n, unless α_r,n = 0. Clearly, if D_n is not stochastic, then α_r,n = 0. Within the the context of model (1) and with D_n equal to the (negative of the) design matrix Z_n this would be the case if the model does not contain a spatial lag of the endogenous variable.

Observing further that the diagonal elements of the matrices A_r,n are zero it follows from Lemma A.1 that the VC matrix of the vector of quadratic forms in (11) is given by Ψ_n = (ψ_rs,n) where for r, s = 1, 2

$${\psi }_{rs,n}={(2n)}^{-1}tr[({\mathbf{A}}_{r,n}+{\mathbf{A}}_{r,n}^{\prime }){\mathbf{\Sigma }}_n({\mathbf{A}}_{s,n}+{\mathbf{A}}_{s,n}^{\prime }){\mathbf{\Sigma }}_n]+n^{-1}{\mathbf{a}}_{r,n}^{\prime }{\mathbf{\Sigma }}_n{\mathbf{a}}_{s,n}.$$ (13)

We now have the following result concerning the asymptotic distribution of ρ̃_n. We note that the theorem does not assume convergence of the matrices involved.

Theorem 2

(Asymptotic normality) Let ρ̃_n be the weighted nonlinear least squares estimators defined by (9). Then, provided the optimization space contains the parameter space, given Assumptions 1–7, and given that ${\mathrm{\lambda }}_{\text{min}}({\mathbf{\Psi }}_n)\ge c_{\mathrm{\Psi }}^{*}>0$, we have

$$n^{1/2}({\stackrel{\sim }{\rho }}_n-{\rho }_n)={({\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n)}^{-1}{\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Psi }}_n^{1/2}{\mathit{\xi }}_n+o_p(1)$$ (14)

where

$$\begin{array}{ccc}{\mathbf{J}}_n\hfill & =\hfill & {\mathbf{\Gamma }}_n\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2{\rho }_n\hfill \end{array}\right],\hfill \\ {\mathit{\xi }}_n\hfill & =\hfill & {\mathbf{\Psi }}_n^{-1/2}{\mathbf{v}}_n\stackrel{d}{\to }N(\mathbf{0},{\mathbf{I}}_2)\hfill \end{array}$$ (15)

. Furthermore n^1/2 (ρ̃_n – ρ_n) = O_p(1) and

$${\mathbf{\Omega }}_{{\stackrel{\sim }{\rho }}_n}({\mathrm{\Upsilon }}_n)={({\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n)}^{-1}{\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Psi }}_n{\mathrm{\Upsilon }}_n{\mathbf{J}}_n{({\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n)}^{-1}\ge \mathit{\text{const}}>0.$$ (16)

The above theorem implies that the difference between the cumulative distribution function of n^1/2(ρ̃_n – ρ_n) and that of N [0, Ω_ρ̃n] converges pointwise to zero, which justifies the use of the latter distribution as an approximation of the former.¹⁰

Remark 1

Clearly ${\mathbf{\Omega }}_{{\stackrel{\sim }{\rho }}_n}({\mathbf{\Psi }}_n^{-1})={({\mathbf{J}}_n^{\prime }{\mathbf{\Psi }}_n^{-1}{\mathbf{J}}_n)}^{-1}\text{and}{\mathbf{\Omega }}_{{\stackrel{\sim }{\rho }}_n}({\mathrm{\Upsilon }}_n)-{\mathbf{\Omega }}_{{\stackrel{\sim }{\rho }}_n}({\mathbf{\Psi }}_n^{-1})$ is positive semi-definite. Thus choosing ϒ̃_n as a consistent estimator for ${\mathbf{\Psi }}_n^{-1}$ leads to the efficient GM estimator. Such a consistent estimator will be developed in the next subsection. As discussed in the proof of the above theorem, the elements of Ψ_n are uniformly bounded in absolute value and hence ${\mathrm{\lambda }}_{\text{max}}({\mathbf{\Psi }}_n)\le c_{\mathbf{\Psi }}^{**}$ for some $c_{\mathbf{\Psi }}^{**}<\mathrm{\infty }$. Since by assumption also $0<c_{\mathbf{\Psi }}^{*}\le {\mathrm{\lambda }}_{\text{min}}({\mathbf{\Psi }}_n)$ it follows that the conditions on the eigenvalues of γ_n postulated in Assumption 5 are automatically satisfied by ${\mathbf{\Psi }}_n^{-1}$ We note that Ψ_n is, in general, only identical to the VC matrix of the moment vector in (5) if a_r,n = 0. The terms involving a_r,n reflect the fact that the GM estimator is based on estimators of the disturbances u_n and not on the true disturbances. As noted above, J_n equals the vector of derivatives (multiplied by minus one) of the moment conditions (6) w.r.t. ρ_n, and thus Ω_ρ̃n (ϒ_n) has the usual structure, except that here Ψ_n is not identical to the VC matrix of the moment vector.

Remark 2

From (11), (12), (14) and (15) we see that n^1/2(ρ̃_n – ρ_n) depends linearly on a vector of linear quadratic forms in the innovations ε_n plus a term of order o_p(1). This result is helpful in establishing the joint distribution of ρ̃_n with that of estimators of some of the other model parameters of interest. In particular, it may be of interest to derive the joint limiting distribution of n^1/2 Δ_n and n^1/2 (ρ̃_n – ρ_n). By Assumption 7, n^1/2 Δ_n is asymptotically linear in ε_n and hence the joint limiting distribution can be readily derived using the CLT for linear quadratic forms given in Appendix A. We will illustrate this below within the context of IV estimators for model (1).

We next introduce a consistent estimator for Ω_ρ̃n For this purpose let

$${\stackrel{\sim }{\mathbf{J}}}_n={\stackrel{\sim }{\mathbf{\Gamma }}}_n\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2{\stackrel{\sim }{\rho }}_n\hfill \end{array}\right]$$ (17)

where Γ̃_n is defined above by (7) and the discussion after that equation. We next define a HAC type estimator for Ψ_n whose elements are defined by (13). For this purpose let

$${\stackrel{\sim }{\mathbf{\Sigma }}}_n={\mathit{\text{diag}}}_{i=1,\dots ,n}({\stackrel{\sim }{\epsilon }}_{i,n}^2)$$

with ε̃_n = (I_n – ρ̃_n M_n) ũ_n. We furthermore need to specify an estimator for a_r,n = T_nα_r,n. The matrix T_n introduced in Assumption 7 will in many applications be of the form

$${\mathbf{T}}_n={\mathbf{F}}_n{\mathbf{P}}_n\text{\hspace{1em}with\hspace{1em}}{\mathbf{F}}_n={\mathbf{H}}_n\text{\hspace{1em}or\hspace{1em}}{\mathbf{F}}_n={({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })}^{-1}{\mathbf{H}}_n,$$ (18)

where H_n is a real nonstochastic n × p_* matrix of instruments, and P_n is a real nonstochastic p_* × p matrix, with p as in Assumption 7. In the next section we will consider instrumental variable estimators for the parameters of model (1) and (2). In that section we will see that if Δ_n corresponds to these instrumental variable estimators, then the matrix T_n will indeed have the above structure, and where P_n can be estimated consistently by some estimator P̃_n. We now define our estimator for T_n as¹¹

$${\stackrel{\sim }{\mathbf{T}}}_n={\stackrel{\sim }{\mathbf{F}}}_n{\stackrel{\sim }{\mathbf{P}}}_n\text{\hspace{1em}with\hspace{1em}}{\stackrel{\sim }{\mathbf{F}}}_n={\mathbf{H}}_n\text{\hspace{1em}or\hspace{1em}}{\stackrel{\sim }{\mathbf{F}}}_n={({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n^{\prime })}^{+}{\mathbf{H}}_n.$$ (19)

In light of (12) it now seems natural to estimate a_r,n by

$${\stackrel{\sim }{\mathbf{a}}}_{r,n}={\stackrel{\sim }{\mathbf{T}}}_n{\stackrel{\sim }{\mathbf{\alpha }}}_{r,n.}$$ (20)

with

$${\stackrel{\sim }{\mathbf{\alpha }}}_{r,n}=n^{-1}[{\mathbf{D}}_n^{\prime }({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n^{\prime })({\mathbf{A}}_{r,n}+{\mathbf{A}}_{r,n}^{\prime })({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n){\stackrel{\sim }{\mathbf{u}}}_n].$$

Given the above we now introduce the following HAC type estimator Ψ̃_n = (ψ̃_{rs, n}) where for r, s = 1, 2

$${\stackrel{\sim }{\psi }}_{rs,n}={(2n)}^{-1}tr\left[({\mathbf{A}}_{r,n}+{\mathbf{A}}_{r,n}^{\prime }){\stackrel{\sim }{\mathbf{\Sigma }}}_n({\mathbf{A}}_{s,n}+{\mathbf{A}}_{s,n}^{\prime }){\stackrel{\sim }{\mathbf{\Sigma }}}_n\right]+n^{-1}{\stackrel{\sim }{\mathbf{a}}}_{r,n}^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\stackrel{\sim }{\mathbf{a}}}_{s,n}.$$ (21)

Furthermore, based on Ψ̃_n we define the following estimator for Ω_ρ̃n:

$${\stackrel{\sim }{\mathbf{\Omega }}}_{{\stackrel{\sim }{\rho }}_n}={({\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n)}^{+}{\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{\Psi }}}_n{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n{({\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n)}^{+}.$$ (22)

The next theorem establishes the consistency of Ψ̃_n and Ω̃_ρ̃n.

Theorem 3

(VC matrix estimation) Suppose all of the assumptions of Theorem 2, apart from Assumption 5, hold and that additionally all of the fourth moments of the elements of D_n are uniformly bounded. Suppose furthermore that (a) that the elements of the nonstochastic matrices H_n are uniformly bounded in absolute value,(b) sup_n |ρ_n| < 1 and the row and column sums of M_n are bounded uniformly in absolute value by, respectively, one and some finite constant (possibly after a renormalization of the weights matrix and parameter space as discussed in Section 2.2), and (c) P̃_n – P_n = o_p(1) with P_n = O(1). Then

$${\stackrel{\sim }{\mathbf{\Psi }}}_n-{\mathbf{\Psi }}_n=o_p(1),{\stackrel{\sim }{\mathbf{\Psi }}}_n^{-1}-{\mathbf{\Psi }}_n^{-1}=o_p(1).$$

If furthermore Assumption 5 holds, then also

$${\stackrel{\sim }{\mathbf{\Omega }}}_{{\stackrel{\sim }{\rho }}_n}-{\mathbf{\Omega }}_{{\stackrel{\sim }{\rho }}_n}=o_p(1).$$

The hypothesis of zero spatial correlation in the disturbances, i.e., H₀ : ρ_n = 0, can now be tested in terms of N [0,Ω̃_ρ̃n].

Remark 3

We note that the above theorem also holds if ρ̃_n is replaced by any other estimator ${\stackrel{\sim }{\stackrel{\sim }{\rho }}}_n\text{with}{n}^{1/2}({\stackrel{\sim }{\stackrel{\sim }{\rho }}}_n-{\rho }_n)=O_p(1)$. In case F_n = H_n condition (b) can be dropped. The consistency result for ${\stackrel{\sim }{\mathbf{\Psi }}}_n^{-1}$ verifies that this estimator for ${\mathbf{\Psi }}_n^{-1}$ can indeed be used in the formulation of an efficient GM estimator, as discussed after Theorem 2.

Remark 4

The modules underlying the derivation of Theorems 2 and 3 can be readily extended to cover a wider class of estimators. A crucial underlying ingredient is the CLT for vectors of linear quadratic forms given in Appendix A, which was used to establish the limiting distribution of the vector of linear quadratic forms (11); that CLT is based on Kelejian and Prucha (2001). We emphasize that while in (5) we consider two moment conditions, all of the above results generalize trivially to the case where the GM estimator for ρ_n corresponds to m moment conditions

$$n^{-1}E[E{\epsilon }_n^{\prime }{\mathbf{A}}_{1,n}{\epsilon }_n,\dots ,{\epsilon }_n^{\prime }{\mathbf{A}}_{m,n}{\epsilon }_n]\prime =0,$$ (23)

where the diagonal elements of A_r,n are zero and the row and column sums of A_r,n are bounded uniformly in absolute value. The focus of this paper is on two-step estimation procedures, which is motivated by their computational simplicity, generality of the first step (where residuals may come from nonlinear models) and, since at least under homoskedasticity, Monte Carlo experiments suggest that very little efficiency is lost, see, e.g., Das, Kelejian and Prucha (2003). Given instruments H_n, one-step GMM estimators for all parameters, i.e., ρ_n, λ_n, β_n, of the SARAR(1, 1) model could be defined by augmenting those moment conditions (23) by the conditions

$$n^{-1}E{\mathbf{H}}_n^{\prime }{\epsilon }_n=n^{-1}E[{\mathbf{h}}_{.1,n}^{\prime }{\epsilon }_n,\dots ,{\mathbf{h}}_{.p*,n}^{\prime }{\epsilon }_n]\prime =0.$$ (24)

with

$${\epsilon }_n=({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n)({\mathbf{y}}_n-{\mathbf{X}}_n{\beta }_n-{\mathrm{\lambda }}_n{\mathbf{W}}_n{\mathbf{y}}_n).$$

The limiting distribution of the stacked moment vector follows immediately from the CLT for vectors of linear quadratic forms. Theorem 3 establishes consistent estimation of the VC matrix of the vector of (normalized) linear quadratic forms (11). Estimation of the VC matrix of the vector of (normalized) linear quadratic forms corresponding to the stacked moment conditions (23) and (24), is analogous. In fact, in this case estimation simplifies in that the components of the vector are either quadratic or linear, and the elements of the linear terms h_r,n are observed.¹²

3.4 Joint Asymptotic Distribution of GM Estimator for ρ_n and Estimators of Other Model Parameters

In the following we discuss how the above results can be extended to obtain the joint asymptotic distribution of the GM estimator for ρ_n and of other estimators that are asymptotically linear in the innovations ε_n, i.e., that are of the form considered in Assumption 7. As remarked above, the IV estimators for the regression parameters of model (1) and (2) considered in the next section will be of this form. Based on the joint distribution it will then be possible to test joint hypothesis concerning ρ_n and other model parameters.

In the following we will give results concerning the joint asymptotic distribution of ρ̃_n – ρ_n and Δ_n as considered in Assumptions 4 and 7 in conjunction with the estimation of the disturbances u_n. Clearly, in general it will be of interest to have available results not only concerning the joint asymptotic distribution of ρ̃_n – ρ_n and Δ_n, but also concerning other estimators, say, ${\mathbf{\Delta }}_n^{*}$ that are of the general form considered in Assumption 7. To avoid having to introduce further notation we give our result in terms of ρ̃_n – ρ_n and Δ_n, but then comment on what changes would be needed in the formulae to accommodate other estimators ${\mathbf{\Delta }}_n^{*}$ in place of Δ_n. The discussion assumes that T_n = F_nP_n and T̃_n = F̃_nP̃_n as defined in the previous subsection.

In light of the above discussion we expect that the joint limiting distribution of n^1/2(ρ̃_n – ρ_n) and n^1/2 Δ_n will depend on the limiting distribution of $[n^{-1/2}{\mathbf{F}}_n^{\prime }{\epsilon }_n,{\mathbf{v}}_n^{\prime }]\prime$. Observing again that the diagonal elements of the matrices A_r,n are zero it follows from Lemma A.1 that the VC matrix of this vector of linear and linear quadratic form is given by

$${\mathbf{\Psi }}_{\mathrm{o},n}=\left[\begin{array}{cc}{\mathbf{\Psi }}_{\mathrm{\Delta }\mathrm{\Delta },n}\hfill & {\mathbf{\Psi }}_{\mathrm{\Delta }\rho ,n}\hfill \\ {\mathbf{\Psi }}_{\mathrm{\Delta }\rho ,n}^{\prime }\hfill & \hfill {\mathbf{\Psi }}_n\hfill \end{array}\right]$$ (25)

with ${\mathbf{\Psi }}_{\mathrm{\Delta }\mathrm{\Delta },n}=n^{-1}{\mathbf{F}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{F}}_n,{\mathbf{\Psi }}_{\mathrm{\Delta }\rho ,n}=n^{-1}{\mathbf{F}}_n^{\prime }{\mathbf{\Sigma }}_n[{\mathbf{a}}_{1,n},{\mathbf{a}}_{2,n}]$ and where Ψ_n is defined by (13). We shall also employ the following estimator for Ψ_o,n:

$${\stackrel{\sim }{\mathbf{\Psi }}}_{\mathrm{o},n}=\left[\begin{array}{cc}{\stackrel{\sim }{\mathbf{\Psi }}}_{\mathrm{\Delta }\mathrm{\Delta },n}\hfill & {\stackrel{\sim }{\mathbf{\Psi }}}_{\mathrm{\Delta }\rho ,n}\hfill \\ {\stackrel{\sim }{\mathbf{\Psi }}}_{\mathrm{\Delta }\rho ,n}^{\prime }\hfill & \hfill {\stackrel{\sim }{\mathbf{\Psi }}}_n\hfill \end{array}\right]$$ (26)

with ${\stackrel{\sim }{\mathbf{\Psi }}}_{\mathrm{\Delta }\mathrm{\Delta },n}=n^{-1}{\stackrel{\sim }{\mathbf{F}}}_n^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\stackrel{\sim }{\mathbf{F}}}_n,{\stackrel{\sim }{\mathbf{\Psi }}}_{\mathrm{\Delta }\rho ,n}=n^{-1}{\stackrel{\sim }{\mathbf{F}}}_n^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n[{\stackrel{\sim }{\mathbf{a}}}_{1,n},{\stackrel{\sim }{\mathbf{a}}}_{2,n}]$ and where Ψ̃_n is defined by (21).

We now have the following result concerning the joint limiting distribution of n^1/2(ρ̃_n – ρ_n) and n^1/2 Δ_n.

Theorem 4

Suppose all of the assumptions of Theorem 3 hold, and ${\mathrm{\lambda }}_{\text{min}}({\mathbf{\Psi }}_{\mathrm{o},n})\ge c_{{\mathbf{\Psi }}_{\mathrm{o}}}^{*}>0$. Then

$$\begin{array}{ccc}\left[\begin{array}{c}\hfill n^{1/2}{\mathbf{\Delta }}_n\hfill \\ \hfill n^{1/2}({\stackrel{\sim }{\rho }}_n-{\rho }_n)\hfill \end{array}\right]\hfill & =\hfill & \left[\begin{array}{cc}\hfill {\mathbf{P}}_n^{\prime }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {({\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n)}^{-1}{\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n\hfill \end{array}\right]{\mathbf{\Psi }}_{\mathrm{o},n}^{1/2}{\mathit{\xi }}_{\mathrm{o},n}+o_p(1),\hfill \\ \hfill {\mathit{\xi }}_{\mathrm{o},n}& =\hfill & {\mathbf{\Psi }}_{\mathrm{o},n}^{-1/2}\left[n^{-1/2}{\mathbf{F}}_n^{\prime }{\epsilon }_n,{\mathbf{v}}_n^{\prime }\right]\prime \stackrel{d}{\to }N(\mathbf{0},{\mathbf{I}}_{p_{*}+2}).\hfill \end{array}$$ (27)

Furthermore let

$${\mathbf{\Omega }}_{\mathrm{o},n}=\left[\begin{array}{cc}\hfill {\mathbf{P}}_n^{\prime }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {({\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n)}^{-1}{\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n\hfill \end{array}\right]{\mathbf{\Psi }}_{\mathrm{o},n}\left[\begin{array}{cc}\hfill {\mathbf{P}}_n\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathrm{\Upsilon }}_n{\mathbf{J}}_n{({\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n)}^{-1}\hfill \end{array}\right]$$ (28)

$${\stackrel{\sim }{\mathbf{\Omega }}}_{\mathrm{o},n}=\left[\begin{array}{cc}\hfill {\stackrel{\sim }{\mathbf{P}}}_n^{\prime }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ({\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n)+{\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n\hfill \end{array}\right]{\stackrel{\sim }{\mathbf{\Psi }}}_{\mathrm{o},n}\left[\begin{array}{cc}\hfill {\stackrel{\sim }{\mathbf{P}}}_n\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n({\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n)+\hfill \end{array}\right]$$ (29)

then Ψ̃_o,n – Ψ_o,n = o_p(1), Ω̃_o,n – Ω_o,n = o_p(1), and Ψ_o,n = O(1), Ω_o,n = O(1).

The above theorem implies that the difference between the joint cumulative distribution function of $n^{-1/2}[{\mathbf{\Delta }}_n^{\prime },({\stackrel{\sim }{\rho }}_n-{\rho }_n)]\prime$ and that of N [0, Ω_o,n] converges pointwise to zero, which justifies the use of the latter distribution as an approximation of the former. The theorem also states that Ω̃_o,n is a consistent estimator for Ω_o,n.

Remark 5

The above result generalizes readily to cases where we are interested in the joint distribution between ρ̃_n – ρ_n and some other estimator, say, ${\mathbf{\Delta }}_n^{*}$ where $n^{1/2}{\mathbf{\Delta }}_n^{*}=n^{-1/2}{\mathbf{T}}_n^{*\prime }{\epsilon }_n+o_p(1),{\mathbf{T}}_n^{*}={\mathbf{F}}_n^{*}{\mathbf{P}}_n^{*}\text{and}{\stackrel{\sim }{\mathbf{T}}}_n={\stackrel{\sim }{\mathbf{F}}}_n^{*}{\stackrel{\sim }{\mathbf{P}}}_n^{*}$, assuming that analogous assumptions are maintained for this estimator. In particular, the results remain valid, but with ${\mathbf{\Psi }}_{\mathrm{\Delta }\mathrm{\Delta },n}=n^{-1}{\mathbf{F}}_n^{*\prime }{\mathbf{\Sigma }}_n{\mathbf{F}}_n^{*},{\mathbf{\Psi }}_{\mathrm{\Delta }\rho ,n}=n^{-1}{\mathbf{F}}_n^{*\prime }{\mathbf{\Sigma }}_n[{\mathbf{a}}_{1,n},{\mathbf{a}}_{2,n}],{\stackrel{\sim }{\mathbf{\Psi }}}_{\mathrm{\Delta }\mathrm{\Delta },n}=n^{-1}{\stackrel{\sim }{\mathbf{F}}}_n^{*\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\stackrel{\sim }{\mathbf{F}}}_n^{*},{\stackrel{\sim }{\mathbf{\Psi }}}_{\mathbf{\Delta }\rho ,n}=n^{-1}{\stackrel{\sim }{\mathbf{F}}}_n^{*\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n[{\stackrel{\sim }{\mathbf{a}}}_{1,n},{\stackrel{\sim }{\mathbf{a}}}_{2,n}]$, and P_n, P̃_n replaced by ${\mathbf{P}}_n^{*},{\stackrel{\sim }{\mathbf{P}}}_n^{*}$.

4 Instrumental Variable Estimator for δ_n

As remarked, the consistency and asymptotic normality results developed in an important paper by Lee (2004) for the quasi-ML estimator for the SARAR(1,1) model defined by (1) and (2) under the assumption of homoskedastic innovations do not carry over to the case where the innovations are heteroskedastic. In fact, under heteroskedasticity the limiting objective function of the quasi-ML estimator would generally not be maximized at the true parameter values, and therefore the quasi-ML estimator would be inconsistent. Also, the asymptotic normality results developed by Kelejian and Prucha (1998), Kelejian, Prucha and Yuzefovich (2004) and Lee (2003) for instrumental variable (IV) estimators of the SARAR(1,1) model do not carry over to the case where the innovations are heteroskedastic. In the following we provide results concerning the asymptotic distribution of IV estimators allowing the innovations to be heteroskedastic. More specifically, we will show that the considered IV estimators satisfy certain conditions such that their asymptotic distribution can be readily obtained via Theorem 4. We also allow for a more general definition of the parameter space of the spatial autoregressive parameters to avoid certain pitfalls discussed in Section 2.

4.1 Instruments

It is evident from (3) that in general W_ny_n will be correlated with the disturbances u_n, which motivates the use of IV estimation procedures. We maintain the following assumptions w.r.t. the n × k regressor matrices X_n, and the n × _p* instrument matrices H_n.

Assumption 8

The regressor matrices X_n have full column rank (for n large enough). Furthermore, the elements of the matrices X_n are uniformly bounded in absolute value.

Assumption 9

The instrument matrices H_n have full column rank p_* ≥ k+1 (for all n large enough). Furthermore, the elements of the matrices H_n are uniformly bounded in absolute value. Additionally H_n is assumed to, at least, contain the linearly independent columns of (X_n, M_n X_n).

Assumption 10

The instruments H_n satisfy furthermore:

1. ${\mathbf{Q}}_{\mathbf{\text{HH}}}={\text{lim}}_{n\to \mathrm{\infty }}{n}^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{H}}_n$ is finite, and nonsingular.

2. ${\mathbf{Q}}_{\mathbf{\text{HZ}}}=p{\text{lim}}_{n\to \mathrm{\infty }}{n}^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{Z}}_n$ and ${\mathbf{Q}}_{\mathbf{\text{HMZ}}}=p{\text{lim}}_{n\to \mathrm{\infty }}{n}^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{M}}_n{\mathbf{Z}}_n$ are finite and have full column rank. Furthermore, let Q_HZ*(ρ_n) = Q_HZ – ρ_n Q_HMZ, then the smallest eigenvalue of ${\mathbf{Q}}_{\mathbf{\text{HZ}}*}^{\prime }({\rho }_n){\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{\mathbf{\text{HZ}}*}({\rho }_n)$ is bounded away from zero uniformly in n.

3. ${\mathbf{Q}}_{\mathbf{H}\mathbf{\Sigma }\mathbf{H}}={\text{lim}}_{n\to \mathrm{\infty }}{n}^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{H}}_n$ is finite and nonsingular.

The above assumptions are similar to those maintained in Kelejian and Prucha (1998, 2004), and Lee (2003), and so a discussion which is quite similar to those given in those papers also applies here. Regarding the specification of the instruments H_n observe first that

$$E({\mathbf{W}}_n{\mathbf{y}}_n)={\mathbf{W}}_n{({\mathbf{I}}_n-{\mathrm{\lambda }}_n{\mathbf{W}}_n)}^{-1}{\mathbf{X}}_n{\beta }_n={\displaystyle \sum _{i=0}^{\mathrm{\infty }}{\mathrm{\lambda }}_n^i{\mathbf{W}}_n^{i+1}{\mathbf{X}}_n{\beta }_n}$$ (30)

provided that the characteristic roots of λ_n W_n are less than one in absolute value; compare Lemma 1 and 2 concerning the choice of the parameter space for λ_n. The instrument matrices H_n will be used to instrument Z_n = (X_n, W_ny_n) and M_n Z_n = (M_n X_n, M_n W_ny_n) in terms of their predicted values from a least squares regression on H_n, i.e., Ẑ_n = P_Hn Z_n and $\widehat{{\mathbf{M}}_n{\mathbf{Z}}_n}={\mathbf{P}}_{{\mathbf{H}}_n}{\mathbf{M}}_n{\mathbf{Z}}_n$ with ${\mathbf{P}}_{{\mathbf{H}}_n}={\mathbf{H}}_n{({\mathbf{H}}_n^{\prime }{\mathbf{H}}_n)}^{-1}{\mathbf{H}}_n^{\prime }$. Towards approximating the ideal instruments E(Z_n) = (X_n, W_nE(y_n)) and E(M_n Z_n) = (M_n X_n, M_n W_n E(y_n)) it seems reasonable, in light of (30), to take H_n to be a subset of the linearly independent columns of

$$({\mathbf{X}}_n,{\mathbf{W}}_n{\mathbf{X}}_n,{\mathbf{W}}_n^2{\mathbf{X}}_n,\dots ,{\mathbf{W}}_n^q{\mathbf{X}}_n,{\mathbf{M}}_n{\mathbf{X}}_n,{\mathbf{M}}_n{\mathbf{W}}_n{\mathbf{X}}_n,,\dots ,{\mathbf{M}}_n{\mathbf{W}}_n^q{\mathbf{X}}_n)$$ (31)

where q is a pre-selected finite constant.¹³ We note that if H_n is selected as in (31) it follows from Assumptions 3 and 8 that its elements will be bounded in absolute value as postulated in Assumption 9. Assumption 9 ensures that X_n and M_n X_n are instrumented by themselves. Finally we note that the assumption that H_n has full column rank could be relaxed at the expense of working with generalized inverses.

4.2 Definition, Consistency and Asymptotic Normality

Towards estimating the model (1) and (2) we propose a three step procedure. In the first step the model is estimated by two stage least squares (2SLS) using the instruments H_n. In the second step the autoregressive parameter, ρ_n, is estimated using the generalized moments estimation approach from Section 3 based on the 2SLS residuals obtained via the first step. In the third step, the regression model in (1) is re-estimated by 2SLS after transforming the model via a Cochrane-Orcutt-type transformation to account for the spatial correlation.

More specifically, the first step 2SLS estimator is defined as:

$${\stackrel{\sim }{\mathbf{\delta }}}_n={({\widehat{\mathbf{Z}}}_n^{\prime }{\mathbf{Z}}_n)}^{-1}{\widehat{\mathbf{Z}}}_n^{\prime }{\mathbf{y}}_n,$$ (32)

where ${\widehat{\mathbf{Z}}}_n={\mathbf{P}}_{{\mathbf{H}}_n}{\mathbf{Z}}_n=({\mathbf{X}}_n,\widehat{{\mathbf{W}}_n{\mathbf{y}}_n})\text{and}\widehat{{\mathbf{W}}_n{\mathbf{y}}_n}={\mathbf{P}}_{{\mathbf{H}}_n}{\mathbf{W}}_n{\mathbf{y}}_n$. In the second step we estimate ρ_n by the GM procedure defined by (9) based on the 2SLS residuals ũ_n = y_n – Z_n δ̃_n. We denote the GM estimator again as ρ̃_n.

The next lemma shows that various assumptions maintained in Section 3 w.r.t. the estimator of the regression parameters and estimated residuals are automatically satisfied by the 2SLS estimator δ̃_n and the corresponding residuals.

Lemma 3

Suppose Assumptions 1–3 and 8–10 hold, and sup_n ||β_n|| < ∞. Let D_n = −Z_n, then the fourth moments of the elements of D_n are uniformly bounded, Assumption 6 holds, and:

1. $n^{1/2}({\stackrel{\sim }{\mathbf{\delta }}}_n-{\mathbf{\delta }}_n)=n^{-1/2}{\mathbf{T}}_n^{\prime }{\mathbf{\epsilon }}_n+o_p(1)$ with T_n = F_n P_n and where

   $$\begin{array}{ccc}{\mathbf{P}}_n\hfill & =\hfill & {\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{\mathbf{\text{HZ}}}{[{\mathbf{Q}}_{\mathbf{\text{HZ}}}^{\prime }{\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{\mathbf{\text{HZ}}}]}^{-1},\hfill \\ {\mathbf{F}}_n\hfill & =\hfill & {({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })}^{-1}{\mathbf{H}}_n.\hfill \end{array}$$

2. $n^{-1/2}{\mathbf{T}}_n^{\prime }{\mathit{\epsilon }}_n=O_p(1)$.

3. P_n = O_p(1) and P̃_n − P_n = o_p(1) for

   $$\begin{array}{ccc}{\stackrel{\sim }{\mathbf{P}}}_n\hfill & =\hfill & {(n^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{H}}_n)}^{-1}(n^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{Z}}_n)\times \hfill \\ \hfill & \hfill & {[(n^{-1}{\mathbf{Z}}_n^{\prime }{\mathbf{H}}_n){(n^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{H}}_n)}^{-1}(n^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{Z}}_n)]}^{-1},\hfill \end{array}$$

The condition sup_n ||β_n|| < ∞ is trivially satisfied if β_n = β. Of course, parts (a) and (b) together imply that δ̃_n is n^1/2-consistent for δ_n.

Clearly ũ_n = u_n + D_nΔ_n with D_n = −Z_n and Δ_n = δ̃_n – δ_n. Lemma 3 shows that in essence under Assumptions 1–3 and 8–10 the 2SLS residuals automatically satisfy the conditions postulated in Assumptions 4, 6 and 7 with D_n = −Z_n, Δ_n = δ̃_n – δ_n and T_n as specified in the lemma. Consequently the results concerning consistency and asymptotic normality of the GM estimator for ρ_n in Theorems 1 and 2 apply in particular to the GM estimator ρ̃_n based on 2SLS residuals. Lemma 3 also establishes that the fourth moments of the elements of D_n = −Z_n are uniformly bounded. The lemma also gives explicit expression for P_n and P̃_n and verifies the conditions postulated w.r.t. to those matrices in Theorems 3 and 4. Hence the results of those two theorems also cover the GM estimator ρ̃_n and the 2SLS estimator δ̃_n. In particular, Theorem 4 gives the joint limiting distribution of n^1/2(ρ̃_n – ρ_n) and n^1/2 Δ_n = n^1/2(δ̃_n – δ_n) where D_n = −Z_n, the matrices P_n, F_n, P̃_n are as in Lemma 3, and ${\stackrel{\sim }{\mathbf{F}}}_n={({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n^{\prime })}^{+}{\mathbf{H}}_n$¹⁴

We now turn to the third step. Applying a Cochrane-Orcutt-type transformation to (1) yields

$${\mathbf{y}}_{n*}({\rho }_n)={\mathbf{Z}}_{n*}({\rho }_n){\mathbf{\delta }}_n+{\mathbf{\epsilon }}_n,$$ (33)

where y_n*(ρ_n) = y_n – ρ_n M_ny_n and Z_n*(ρ_n) = Z_n – ρ_nM_nZ_n. Our estimator δ̂_n for δ_n in this third step is now defined as the 2SLS procedure applied to the transformed model (33) after replacing ρ_n by ρ̃_n. That is,

$${\widehat{\delta }}_n={[{\widehat{\mathbf{Z}}}_{n*}({\stackrel{\sim }{\rho }}_n)\prime {\mathbf{Z}}_{n*}({\stackrel{\sim }{\rho }}_n)]}^{-1}{\widehat{\mathbf{Z}}}_{n*}({\stackrel{\sim }{\rho }}_n)\prime {\mathbf{y}}_{n*}({\stackrel{\sim }{\rho }}_n)$$ (34)

where Ẑ_n* (ρ̃_n) = P_HnZ_n* (ρ̃_n). To express the dependence of δ̂_n on ρ̃_n we also write δ̂_n = δ̂_n (ρ̃_n).

The next lemma shows again that various assumptions maintained in Section 3 w.r.t. the estimator of the regression parameters and estimated residuals are automatically satisfied by the generalized spatial 2SLS estimator δ̂_n and corresponding residuals.

Lemma 4

Suppose the assumptions of Lemma 3, and let δ̂_n (ρ̂_n) be as defined by (34), where ρ̂_n is any n^1/2-consistent estimator for ρ_n (such as the GM estimator ρ̃_n based on 2SLS residuals). Then

1. $n^{1/2}[{\widehat{\mathbf{\delta }}}_n({\widehat{\rho }}_n)-{\mathbf{\delta }}_n]=n^{-1/2}{\mathbf{T}}_n^{*\prime }{\mathbf{\epsilon }}_n+o_p(1)$ with ${\mathbf{T}}_n^{*}={\mathbf{F}}_n^{*}{\mathbf{P}}_n^{*}$ and where

   $$\begin{array}{ccc}{\mathbf{P}}_n^{*}\hfill & =\hfill & {\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{\mathbf{\text{HZ}}*}({\rho }_n){[{\mathbf{Q}}_{\mathbf{\text{HZ}}*}^{\prime }({\rho }_n){\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{\mathbf{\text{HZ}}*}({\rho }_n)]}^{-1}\hfill \\ {\mathbf{F}}_n^{*}\hfill & =\hfill & {\mathbf{H}}_n.\hfill \end{array}$$

2. $n^{-1/2}{\mathbf{T}}_n^{*\prime }{\epsilon }_n=O_p(1)$.

3. ${\mathbf{P}}_n^{*}=O_p(1)\mathit{\text{and}}{\stackrel{\sim }{\mathbf{P}}}_n^{*}-{\mathbf{P}}_n^{*}=o_p(1)$ for

   $$\begin{array}{ccc}{\stackrel{\sim }{\mathbf{P}}}_n^{*}\hfill & =\hfill & {(n^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{H}}_n)}^{-1}(n^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{Z}}_{n*}({\widehat{\rho }}_n))\times \hfill \\ \hfill & \hfill & {[(n^{-1}{\mathbf{Z}}_{n*}^{\prime }({\widehat{\rho }}_n){\mathbf{H}}_n){(n^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{H}}_n)}^{-1}(n^{-1}{\mathbf{H}}_n^{\prime }{\mathbf{Z}}_{n*}({\widehat{\rho }}_n))]}^{-1}.\hfill \end{array}$$

Frequently we will be interested in the joint distribution of the generalized spatial 2SLS estimator δ̂_n(ρ̂_n) and the GM estimator ρ̃_n In light of Lemma 3 and 4 the limiting distribution of n^1/2[(δ̂_n – δ_n)′, (ρ̃_n – ρ_n)]′ now follows immediately from Theorem 4 and Remark 5 after that theorem, with ${\mathbf{\Delta }}_n^{*}={\widehat{\mathbf{\delta }}}_n-{\mathbf{\delta }}_n$. The asymptotic variance covariance matrix and its corresponding estimator are given by (28) and (29) with modifications as described in Remark 5 and D_n = −Z_n. The expressions for the matrices P_n, F_n, P̃_n are as in Lemma 3, and ${\stackrel{\sim }{\mathbf{F}}}_n={({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n^{\prime })}^{+}{\mathbf{H}}_n$ The expressions for the matrices ${\mathbf{P}}_n^{*},{\mathbf{F}}_n^{*},{\stackrel{\sim }{\mathbf{P}}}_n^{*}$ are as in Lemma 4, and ${\stackrel{\sim }{\mathbf{F}}}_n^{*}={\mathbf{H}}_n$. The joint distribution can then be used to test in particular the joint hypothesis H₀ : λ_n = ρ_n = 0 in the usual manner.

Finally, consider the estimated residuals corresponding to δ̂_n i.e., ${\stackrel{\sim }{\mathbf{u}}}_n^{*}={\mathbf{y}}_n-{\mathbf{Z}}_n{\widehat{\mathbf{\delta }}}_n={\mathbf{u}}_n+{\mathbf{D}}_n{\mathbf{\Delta }}_n^{*}$ Clearly, in light of Lemma 4 we could use those residuals to define a corresponding GM estimator based on those residuals, and a discussion analogous to that after Lemma 3 would also apply applies here. Of course, further iterative procedures are possible, and their asymptotic properties would again be covered by the asymptotic theory developed above.

Remark 6

The estimation theory of Section 3 was developed under a set of fairly general assumptions. The results given in this section are illustrative as to how Assumptions 4, 6, 7 can be implied for the 2SLS and GS2SLS estimator of the SARAR(1,1) model from the more primitive Assumptions 8 – 10. The assumption that X_n is nonstochastic rules out the presence of endogenous regressors (apart from the spatial lag W_ny_n). Now suppose that X_n = [X̅_n, Y_n] and correspondingly D_n = −Z_n = – [X̅_n, Y_n, W_nY_n], where X̅_n satisfies Assumptions 8–10, with X_n replaced by X̅_n (including in the formulation of the instruments), and where Y_n is a matrix of endogenous variables. Then given the fourth moments of the elements of D_n are uniformly bounded, and Assumption 6 holds, parts (a),(b) and (c) of Lemmata 3 and 4 still hold but with Ẑ_n = P_Hn Z_n = [X̅_n, P_Hn Y_n, P_Hn W_nY_n] and

$$\begin{array}{ccc}{\widehat{\mathbf{Z}}}_{n*}({\stackrel{\sim }{\rho }}_n)\hfill & =\hfill & {\mathbf{P}}_{{\mathbf{H}}_n}{\mathbf{Z}}_{n*}({\stackrel{\sim }{\rho }}_n)\hfill \\ \hfill & =\hfill & [({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n){\overline{\mathbf{X}}}_n,{\mathbf{P}}_{{\mathbf{H}}_n}({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n^{\prime }){\mathbf{Y}}_n,{\mathbf{P}}_{{\mathbf{H}}_n}({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n){\mathbf{W}}_n{\mathbf{y}}_n)].\hfill \end{array}$$

In specifying a full system of equations analogous to Kelejian and Prucha (2004) one could furthermore develop more primitive conditions that ensure the moment condition for the elements of D_n as well as Assumption 6.

5 Some Monte Carlo Results

In order to obtain some insights relating to the small sample properties of our suggested estimators, as well as the quasi-maximum likelihood estimator, we undertook a limited Monte Carlo study. Because of space constraints we do not report the details of that study here; however, those details can be found in Kelejian and Prucha (2007). In essence the results of that Monte Carlo study were quite consistent with the theoretical results presented above. Specifically, in our considered cases with heteroskedastic innovations the generalized spatial 2SLS/GM estimators had very little bias even in small samples. In contrast, the ML estimators were significantly biased. Furthermore, the ML biases did not decline with the sample size. We also note that the rejection rates of tests corresponding to the generalized spatial 2SLS/GM estimators were close to the nominal value of 0.05, suggesting that the derived large sample distribution and the estimators of the asymptotic variance covariance matrix determined above provide indeed useful small sample appoximations. On the other hand, rejection rates for the ML estimators exibited serious biases ranging from 0.176 to 1.000.

In the cases we considered with homoskedastic innovations both the generalized spatial 2SLS/GM estimators and ML estimators showed very little bias. In addition, the loss of efficiency of the generalized spatial 2SLS/GM estimators relative to the ML estimators was generally modest.

6 Summary and Suggestions for Further Research

In this paper we introduce a new class of GM estimators for the autoregressive parameter of a spatially autoregressive disturbance process allowing for innovations with unknown heteroskedasticity. The estimation theory for the GM estimators is developed in a modular fashion under a fairly general set of assumptions, and should cover many (linear and nonlinear) models. The general theory is then utilized to establish the consistency and asymptotic normality of IV estimators for the regression parameters for an important class of spatial models, frequently referred to as SARAR(1, 1) models. The paper provides results concerning the joint asymptotic distribution of the GM estimator for the autoregressive parameter of the disturbance process and IV estimators for the model regression parameters. Among other things, the results allow for a joint test that the autoregressive parameters corresponding to the spatial lags of the dependent variable and disturbance term are both zero We also provide a discussion of the specification of the parameter space for SARAR(1, 1) models. We demonstrate that a computationally simple re-scaling of the weights matrix leads to an equivalent model containing a (re-scaled) autoregressive parameter which has a user-friendly parameter space. Unfortunately, previous studies in the literature have often re-scaled their weights matrix in such a way that the “before and after” scaled models are not equivalent.

One suggestion for further research is to extend the results to panel data. A further suggestion for future research would be a comprehensive Monte Carlo study which focuses on the small sample properties of the considered estimators. Such a study may also consider comparisons with the quasi-ML estimator (specified under homoskedasticity).

A Appendix: CLT for Vectors of Linear Quadratic Forms

In the following we state, for the convenience of the reader, a central limit theorem (CLT) for vectors of linear quadratic forms with heteroskedastic innovations. This CLT is based on a CLT given in Kelejian and Prucha (2001) for the scalar case. We first state a lemma that collects useful results on the mean and VC matrix between (and as a special case the variance of) linear quadratic forms.

Lemma A.1

Let ζ = (ζ₁, …, ζ_n)′ be a random vector with zero mean and positive definite variance covariance matrix Ξ, let A = (a_ij) and B= (b_ij) be n × n nonstochastic symmetric matrices, and let a and b be n × 1 nonstochastic vectors. Consider the decomposition Ξ = SS′, let A_* = (a_ij,*) = S′AS and B_* = (b_ij,*) = S′BS, and let a_* = S′a and let b_* = S′b. Furthermore let η = (η₁, …, η_n)′ = S⁻¹ζ. Then assuming that the elements of η are independently distributed with zero mean, unit variance, and finite third and fourth moments $E({\eta }_i^3)={\mu }_{{\eta }_i}^{(3)}\text{and}E({\eta }_i^4)={\mu }_{{\eta }_i}^{(4)}$ we have

$$\begin{array}{c}E(\mathbf{\zeta }\prime \mathbf{A}\mathbf{\zeta }+\mathbf{a}\prime \mathbf{\zeta })=tr({\mathbf{A}}_{*})=tr(\mathbf{A}\mathbf{\Xi }),\hfill \\ \mathit{\text{cov}}(\mathbf{\zeta }\prime \mathbf{A}\mathbf{\zeta }+\mathbf{a}\prime \mathbf{\zeta },\mathbf{\zeta }\prime \mathbf{B}\mathbf{\zeta }+\mathbf{b}\prime \mathbf{\zeta })=2tr(\mathbf{A}\mathbf{\Xi }\mathbf{B}\mathbf{\Xi })+\mathbf{a}\prime \mathbf{\Xi }\mathbf{b}\hfill \\ +{\displaystyle \sum _{i=1}^n{a}_{ii,*}{b}_{ii,*}\left[{\mu }_{{\eta }_i}^{(4)}-3\right]+{\displaystyle \sum _{i=1}^n(a_{i,*}{b}_{ii{,}_{*}}+b_{i,*}{a}_{ii,*}){\mu }_{{\eta }_i}^{(3)}}.}\hfill \end{array}$$

Remark A.1

The above expression for the covariance exploits the independence of the elements of η. A convenient way to obtain these expressions is to re-write the linear quadratic form as a sum of martingale differences as, e.g., in Kelejian and Prucha (2001, Appendix A). If the diagonal elements of A_* and B_* are zero the last two terms drop out from the expression for the covariance, and so one need only assume the existence of second moments. The last two terms also drop out from the expression for the covariance in the case where ζ – or equivalently η – is normally distributed, since in this case ${\mu }_{{\eta }_i}^{(3)}=0\text{and}{\mu }_{{\eta }_i}^{(4)}=3$. Of course, the variance of a linear quadratic form is obtained as the special case where A = B and a = b. Obviously, in case A and B are not symmetric the above formulae apply with A and B replaced by (A + A′)/2 and (B + B′)/2.

Consider the linear quadratic forms (r = 1, …, m)

$${\mathbf{Q}}_{r,n}={\mathit{\xi }}_n^{\prime }{\mathbf{A}}_{r,n}{\mathit{\xi }}_n+{\mathbf{a}}_{r,n}^{\prime }{\mathit{\xi }}_n$$

where ξ_n = (ξ_1,n, …, ξ_n,n)′ is an n × 1 random vector, A_r,n = (a_ij,r,n)_i,j=1,…,n is an n × n nonstochastic real matrix, and a_r,n = (a_1,r,n, …, a_n,r,n)′ is an n × 1 nonstochastic real vector. We maintain the following assumptions:

Assumption A.1

The real valued random variables of the array {ξ_i,n: 1 ≤ i ≤ n, n ≥ 1} satisfy Eξ_i,n = 0. Furthermore, for each n ≥ 1 the random variables ξ_1,n, …, ξ_n,n are totally independent.

Assumption A.2

For r = 1,…, m the elements of the array of real numbers {a_ij,r,n : 1 ≤ i, j ≤ n, n ≥ 1} satisfy a_ij,r,n = a_ji,r,n and ${\text{sup}}_{1\le j\le n,n\ge 1}{\displaystyle {\sum }_{i=1}^n\left|a_{ij,r,n}\right|<}\mathrm{\infty }$. The elements of the array of real numbers {a_i,r,n : 1 ≤ i ≤ n, n ≥ 1} satisfy ${\text{sup}}_n{n}^{-1}{\displaystyle {\sum }_{i=1}^n|a_{i,r,n}{|}^{2+{\eta }_1}<}\mathrm{\infty }$ for some η₁ > 0.

Assumption A.3

For r = 1, …, m one of the following two conditions holds:

1. sup_{1≤i≤n,n≤1} E|ξ_i,n|^2+η₂ < ∞ for some η₂ > 0 and a_ii,r,n = 0.

2. sup_{1≤i≤n,n≤1} E|ξ_i,n|^4+η₂ < ∞ for some η₂ > 0 (but possibly a_ii,r,n ≠ 0).

Let μ_Qr,n and σ_Qrs,n denote the mean of Q_r,n and the covariance between Q_r,n and Q_s,n, respectively, for r, s = 1, …, m. Then it follows immediately from Lemma A.1 that under Assumptions A.1 and A.3

$$\begin{array}{ccc}{\mu }_{{\mathbf{Q}}_{r,n}}\hfill & =\hfill & {\displaystyle \sum _{i=1}^n{a}_{ii,r,n}{\sigma }_{i,n}^2},\hfill \\ {\sigma }_{{\mathbf{Q}}_{rs,n}}\hfill & =\hfill & 2{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{a}_{ij,r,n}{a}_{ij,s,n}{\sigma }_{i,n}^2{\sigma }_{j,n}^2+{\displaystyle \sum _{i=1}^n{a}_{i,r,n}{a}_{i,s,n}{\sigma }_{i,n}^2}}}\hfill \\ \hfill & \hfill & +{\displaystyle \sum _{i=1}^n{a}_{ii,r,n}{a}_{ii,s,n}\left[{\mu }_{i,n}^{(4)}-3{\sigma }_{i,n}^4\right]+{\displaystyle \sum _{i=1}^n(a_{i,r,n}{a}_{ii,s,n}+a_{i,s,n}{a}_{ii,r,n}){\mu }_{i,n}^{(3)}}}\hfill \end{array}$$

where ${\sigma }_{i,n}^2=E{\xi }_{i,n}^2,{\mu }_{i,n}^{(3)}=E{\xi }_{i,n}^3\text{and}{\mu }_{i,n}^{(4)}=E{\xi }_{i,n}^4$. In case Assumption A.3(a) holds, the mean of Q_r,n is zero and the last two terms drop out from the expression for the variance.

In the following we give a CLT for the m × 1 vector of linear quadratic forms

$${\mathbf{V}}_n={\left[{\mathbf{Q}}_{1,n},\dots ,{\mathbf{Q}}_{m,n}\right]}^{\prime }.$$

Let μ_Vn = EV_n = [μ_Q1,n,…, μ_Qm,n]′ and Σ_Vn = [σ_Qrs,n]r,s=1,…,m denote the mean and VC matrix of V_n, respectively. We then have the following theorem.

Theorem A.1

Suppose Assumptions A.1–A.3 hold and n⁻¹λ_min(Σ_Vn) ≥ c for some c > 0. Let ${\mathrm{\Sigma }}_{V_n}=\left({\mathbf{\Sigma }}_{V_n}^{1/2}\right){\left({\mathbf{\Sigma }}_{V_n}^{1/2}\right)}^{\prime }$, then

$${\mathbf{\Sigma }}_{V_n}^{-1/2}({\mathbf{V}}_n-{\mathit{\mu }}_{V_n})\stackrel{d}{\to }N(\mathbf{0},{\mathbf{I}}_m).$$

Remark A.2

Since the diagonal elements of a positive definite matrix are greater than or equal to the smallest eigenvalue of that matrix it follows that n⁻¹λ_min(ΣV_n) ≥ c implies that n−¹ σ_Qn,ii ≥ c. Therefore this assumption automatically implies the assumption maintained in Theorem 1 of Kelejian and Prucha (2001) w.r.t. the variance of a linear quadratic form for each of the linear quadratic forms Q_r,n. Of course, the theorem remains valid, if all assumptions are assumed to hold for n > n₀ where n₀ is finite.

Proof of Theorem A.1

Let α be some arbitrary m × 1 vector with ||α|| = 1, let

$${\mathbf{\pi }}_n=({\pi }_{1,n},\dots ,{\pi }_{m,n})=n^{1/2}\alpha \prime {\mathbf{\Sigma }}_{V_n}^{-1/2},$$

and define

$${\mathbf{Q}}_n=n^{1/2}\alpha \prime {\mathbf{\Sigma }}_{V_n}^{-1/2}{\mathbf{V}}_n={\displaystyle \sum _{r=1}^m{\pi }_{r,n}{\mathit{\xi }}_n^{\prime }{\mathbf{A}}_{r,n}{\mathit{\xi }}_n+{\displaystyle \sum _{r=1}^m{\pi }_{r,n}{\mathbf{a}}_{r,n}^{\prime }{\mathit{\xi }}_n={\mathit{\xi }}_n^{\prime }{\mathbf{C}}_n{\mathit{\xi }}_n+{\mathbf{d}}_n^{\prime }{\mathit{\xi }}_n},}$$

where ${\mathbf{C}}_n={\displaystyle {\sum }_{r=1}^m{\pi }_{r,n}{\mathbf{A}}_{r,n}}\text{and}{\mathbf{d}}_n={\displaystyle {\sum }_{r=1}^m{\pi }_{r,n}{\mathbf{a}}_{r,n}}$. Observe that

$$\begin{array}{ccc}{\mu }_{{\mathbf{Q}}_n}\hfill & =\hfill & E{\mathbf{Q}}_n=n^{1/2}\alpha \prime {\mathbf{\Sigma }}_{V_n}^{-1/2}{\mathit{\mu }}_{V_n}={\displaystyle \sum _{r=1}^m{\pi }_{r,n}{\mu }_{{\mathbf{Q}}_{r,n}}},\hfill \\ {\sigma }_{{\mathbf{Q}}_n}^2\hfill & =\hfill & \mathit{\text{var}}({\mathbf{Q}}_n)=n\alpha \prime {\mathbf{\Sigma }}_{V_n}^{-1/2}{\mathbf{\Sigma }}_{V_n}{\mathbf{\Sigma }}_{V_n}^{-1/2\prime }\alpha =n.\hfill \end{array}$$

To prove the theorem, using the Cramer Wold device, it suffices to show that

$$\alpha \prime {\mathbf{\Sigma }}_{V_n}^{-1/2}({\mathbf{V}}_n-{\mathit{\mu }}_{V,n})=\frac{{\mathbf{Q}}_n-{\mu }_{{\mathbf{Q}}_n}}{{\sigma }_{{\mathbf{Q}}_n}}\stackrel{d}{\to }N(0,1).$$ (A.1)

To show that (A.1) holds we verify that ξ_n and C_n and d_n satisfy the conditions of Theorem 1 in Kelejian and Prucha (2001). Assumptions A.1 and A.3 are identical to Assumptions 1 and 3 in Kelejian and Prucha (2001). Furthermore, $n^{-1}{\sigma }_{{\mathbf{Q}}_n}^2=1$. Thus it suffices to verify that Assumption 2 in Kelejian and Prucha (2001) holds for the elements of C_n and d_n.

Clearly ${\mathrm{\lambda }}_{\text{max}}(n{\mathbf{\Sigma }}_{V_n}^{-1})=1/{\mathrm{\lambda }}_{\text{min}}(n^{-1}{\mathbf{\Sigma }}_{V_n})\le 1/c$. By Proposition 43 in Dhrymes (1978, p. 470) the matrices $n{\mathbf{\Sigma }}_{V_n}^{-1}=n{\mathbf{\Sigma }}_{V_n}^{-1/2\prime }{\mathbf{\Sigma }}_{V_n}^{-1/2}\text{and}n{\mathbf{\Sigma }}_{V_n}^{-1/2}{\mathbf{\Sigma }}_{V_n}^{-1/2\prime }$ have the same characteristic roots. Thus ${\left|\right|{\pi }_n\left|\right|}^2=\alpha \prime (n{\mathbf{\Sigma }}_{V_n}^{-1/2}{\mathbf{\Sigma }}_{V_n}^{-1/2\prime })\alpha \le {\mathrm{\lambda }}_{\text{max}}(n{\mathbf{\Sigma }}_{V_n}^{-1/2}{\mathbf{\Sigma }}_{V_n}^{-1/2\prime }){\left|\right|\alpha \left|\right|}^2={\mathrm{\lambda }}_{\text{max}}(n{\mathbf{\Sigma }}_{V_n}^{-1})\le 1/c<\mathrm{\infty }$, and hence |π_r,n| ≤ c_π where $c_{\pi }=1/\sqrt{c}$

Observe that

$$|c_{ij,n}|={\displaystyle \sum _{r=1}^m|{\pi }_{r,n}||a_{ij,r,n}|}\le c_{\pi }{\displaystyle \sum _{r=1}^m|a_{ij,r,n}|}$$

and hence

$$\underset{1\le j\le n,n\ge 1}{\text{sup}}{\displaystyle \sum _{i=1}^n|c_{ij,n}|\le c_{\pi }{\displaystyle \sum _{r=1}^m\underset{1\le j\le n,n\ge 1}{\text{sup}}}{\displaystyle \sum _{i=1}^n|a_{ij,r,n}|<\mathrm{\infty }}}$$

in light of Assumption A.2. This shows that the elements of C_n satisfy Assumption 2 in Kelejian and Prucha (2001).

Next observe that

$$\begin{array}{ccc}{|d_{i,n}|}^{2+{\eta }_1}\hfill & \le \hfill & {\left[{\displaystyle \sum _{r=1}^m|{\pi }_{r,n}||a_{i,r,n}|}\right]}^{2+{\eta }_1}\le c_{\pi }^{2+{\eta }_1}{\left[{\displaystyle \sum _{r=1}^m|a_{i,r,n}|}\right]}^{2+{\eta }_1}\hfill \\ \hfill & \le \hfill & c_{\pi }^{2+{\eta }_1}{m}^{2+{\eta }_1}{\displaystyle \sum _{r=1}^m{|a_{i,r,n}|}^{2+{\eta }_1}}\hfill \end{array}$$

and hence

$$\underset{n}{\text{sup}}{n}^{-1}{\displaystyle \sum _{i=1}^n{|d_{i,n}|}^{2+{\eta }_1}\le c_{\pi }^{2+{\eta }_1}{m}^{2+{\eta }_1}{\displaystyle \sum _{r=1}^m\underset{n}{\text{sup}}{n}^{-1}{\displaystyle \sum _{i=1}^n{|a_{i,r,n}|}^{2+{\eta }_1}<\mathrm{\infty }}}}$$

in light of Assumption A.2. Therefore the elements of d_n satisfy Assumption 2 in Kelejian and Prucha (2001).

B Appendix: Proofs for Section 2

Proof of Lemma 1

Clearly I_n – λW_n is non-singular for λ = 0. For λ ≠ 0 we have det(I_n – λW_n) = (−λ)ⁿ det(W_n – λ⁻¹I_n). Consequently I_n – λW_n is non-singular for values of λ⁻¹ ∉ {ν_1,n, …, ν_n,n}. In particular, I_n – λW_n is non-singular for |λ⁻¹| > τ_n. Rewriting this last inequality as $|\mathrm{\lambda }|<{\tau }_n^{-1}$ completes the proof.

Proof of Lemma 2

As an immediate consequence of Geršgorin’s Theorem – see, e.g., Horn and Johnson (1985), pp. 344–346 – we have ${\tau }_n=\text{max}\{|{\nu }_{1,n}|,\dots ,|{\nu }_{n,n}|\}\le {\tau }_n^{*}$. The claim now follows from Lemma 1.

C Appendix: Proofs for Section 3

Remark C.1

Suppose the row and column sums of the np × np matrices A_n = (a_ij,n) are bounded uniformly in absolute value by some finite constants c_A, then ${\displaystyle {\sum }_{i=1}^{np}{|a_{ij,n}|}^q\le c_A^q}$ for q < 1 This result is trivially seen to hold since ${\displaystyle {\sum }_{i=1}^{np}{|a_{ij,n}|}^q=c_A^{q-1}}{\displaystyle {\sum }_{i=1}^{np}|a_{ij,n}|{|a_{ij,n}/c_A|}^{q-1}\le c_A^{q-1}}{\displaystyle {\sum }_{i=1}^{np}|a_{ij,n}|\le c_A^q}$.

In this appendix we shall make use of several lemmata. Because of space constraints we do not provide explicit proofs for these lemmata. However, explicit proofs are given in the underlying working paper, Kelejian and Prucha (2007b), which can be obtained from the authors.

Lemma C.1

Suppose the row and column sums of the real nonstochastic n × n matrices A_n are uniformly bounded in absolute value. Let u_n be defined by (2) and let ũ_n denote a predictor for u_n. Suppose Assumptions 1 – 4 hold then:

1. $n^{-1}E|{\mathbf{u}}_n^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n|=O(1),\mathit{\text{var}}(n^{-1}{\mathbf{u}}_n^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n)=o(1)$ and

   $$n^{-1}{\stackrel{\sim }{\mathbf{u}}}_n^{\prime }{\mathbf{A}}_n{\stackrel{\sim }{\mathbf{u}}}_n-n^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n=o_p(1).$$
2. $n^{-1}E|{\mathbf{d}}_{.s,n}^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n|=O(1)$, s = 1, …, p, where d_.s,n denotes the s-th column of D_n, and

   $$n^{-1}{\mathbf{D}}_n^{\prime }{\mathbf{A}}_n{\stackrel{\sim }{\mathbf{u}}}_n-n^{-1}E{\mathbf{D}}_n^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n=o_p(1).$$
3. If furthermore Assumption 6 holds, then

   $$n^{-1/2}{\stackrel{\sim }{\mathbf{u}}}_n^{\prime }{\mathbf{A}}_n{\stackrel{\sim }{\mathbf{u}}}_n=n^{-1/2}{\mathbf{u}}_n^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n+{\mathbf{\alpha }}_n^{\prime }{n}^{1/2}{\mathbf{\Delta }}_n+o_p(1)$$

   with ${\mathbf{\alpha }}_n=n^{-1}E{\mathbf{D}}_n^{\prime }({\mathbf{A}}_n+{\mathbf{A}}_n^{\prime }){\mathbf{u}}_n$. (Of course, in light of (b) we have α_n = O(1) and $n^{-1}{\mathbf{D}}_n^{\prime }({\mathbf{A}}_n+{\mathbf{A}}_n^{\prime }){\stackrel{\sim }{\mathbf{u}}}_n-{\mathbf{\alpha }}_n=o_p(1)$.)

Proof

A proof of the lemma is given in Kelejian and Prucha (2007b).

Proof of Theorem 1

The existence and measurability of ρ̃_n is assured by, e.g., Lemma 3.4 in Pötscher and Prucha (1997). The objective function of the weighted nonlinear least squares estimator and its corresponding non-stochastic counterpart are given by, respectively,

$$\begin{array}{ccc}{R}_n(\omega ,\rho )\hfill & =\hfill & \left[{\stackrel{\sim }{\mathbf{\Gamma }}}_n(\rho ,{\rho }^2)\prime -{\stackrel{\sim }{\gamma }}_n\right]\prime {\stackrel{\sim }{\mathrm{\Upsilon }}}_n\left[{\stackrel{\sim }{\mathbf{\Gamma }}}_n(\rho ,{\rho }^2)\prime -{\stackrel{\sim }{\gamma }}_n\right]\hfill \\ \hfill {\overline{R}}_n(\rho )& =\hfill & \left[{\mathbf{\Gamma }}_n(\rho ,{\rho }^2)\prime -{\gamma }_n\right]\prime {\mathrm{\Upsilon }}_n\left[{\mathbf{\Gamma }}_n(\rho ,{\rho }^2)\prime -{\gamma }_n\right].\hfill \end{array}$$

To prove the consistency of ρ̃_n we show that the conditions of, e.g., Lemma 3.1 in Pötscher and Prucha (1997) are satisfied for the problem at hand. We first show that ρ_n is an identifiably unique sequence of minimizers of R̅_n Observe that R̅_n(ρ) ≥ 0 and that R̅_n(ρ_n) = 0. From (6) we have ${\gamma }_n={\mathbf{\Gamma }}_n[{\rho }_n,{\rho }_n^2]\prime$. Utilizing Assumption 5 we then get

$$\begin{array}{ccc}{\overline{R}}_n(\rho )-{\overline{R}}_n({\rho }_n)\hfill & =\hfill & {\overline{R}}_n(\rho )\hfill \\ \hfill & =\hfill & [\rho -{\rho }_n,{\rho }^2-{\rho }_n^2]{\mathbf{\Gamma }}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Gamma }}_n[\rho -{\rho }_n,{\rho }^2-{\rho }_n^2]\prime \hfill \\ \hfill & \ge \hfill & {\mathrm{\lambda }}_{\text{min}}({\mathrm{\Upsilon }}_n){\mathrm{\lambda }}_{\text{min}}({\mathbf{\Gamma }\prime }_n{\mathbf{\Gamma }}_n)[\rho -{\rho }_n,{\rho }^2-{\rho }_n^2][\rho -{\rho }_n,{\rho }^2-{\rho }_n^2]\prime \hfill \\ \hfill & \ge \hfill & {\mathrm{\lambda }}_{*}{[\rho -{\rho }_n]}^2\hfill \end{array}$$

for some λ_* > 0. Hence for every ε > 0 and n we have:

$$\begin{array}{cc}\hfill & \underset{\{\rho \in [-a^{\rho },a^{\rho }]:\Vert \rho -{\rho }_n\Vert \ge \epsilon \}}{\text{inf}}[{\overline{R}}_n(\rho )-{\overline{R}}_n({\rho }_n)]\hfill \\ \ge \hfill & \underset{\{\rho \in [-a^{\rho },a^{\rho }]:\Vert \rho -{\rho }_n\Vert \ge \epsilon \}}{\text{inf}}{\mathrm{\lambda }}_{*}^2{[\rho -{\rho }_n]}^2={\mathrm{\lambda }}_{*}{\epsilon }^2>0,\hfill \end{array}$$

which proves that ρ_n is identifiably unique. Next let Φ_n = [Γ_n, –γ_n] and Φ̃_n = [Γ̃_n, − γ̂_n], then

$$\begin{array}{ccc}|R_n(\omega ,\rho )-{\overline{R}}_n(\rho )|\hfill & =\hfill & \left|[\rho ,{\rho }^2,1]\left[{\stackrel{\sim }{\mathbf{\Phi }}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{\Phi }}}_n-{\mathbf{\Phi }}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Phi }}_n\right][\rho ,{\rho }^2,1]\prime \right|\hfill \\ \hfill & \le \hfill & \Vert {\stackrel{\sim }{\mathbf{\Phi }}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{\Phi }}}_n-{\mathbf{\Phi }}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Phi }}_n\Vert ||\rho ,{\rho }^2,1|{|}^2\hfill \\ \hfill & \le \hfill & \Vert {\stackrel{\sim }{\mathbf{\Phi }}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{\Phi }}}_n-{\mathbf{\Phi }}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Phi }}_n\Vert [1+{(a^{\rho })}^2+{(a^{\rho })}^4].\hfill \end{array}$$

As is readily seen from the respective second expressions on the r.h.s. of (7), the elements of Φ_n and Φ̃_n are all of the form $n^{-1}E{\mathbf{u}}_n^{\prime }{\mathbf{A}}_n{\mathbf{u}}_n\text{and}{n}^{-1}{\stackrel{\sim }{\mathbf{u}}}_n^{\prime }{\mathbf{A}}_n{\stackrel{\sim }{\mathbf{u}}}_n$ where the row and column sums of A_n are bounded uniformly in absolute value. It now follows immediately from Lemma C.1 that ${\stackrel{\sim }{\mathbf{\Phi }}}_n-{\mathbf{\Phi }}_n\stackrel{p}{\to }0$, and that the elements of Φ̃_n and Φ_n are, respectively, O_p(1) and O(1). The analogous properties are seen to hold for the elements of ϒ̃_n and ϒ_n in light of Assumption 5. Given this it follows from the above inequality that R_n(ω, ρ) – R̅_n(ρ) converges to zero uniformly over the optimization space [−a^ρ, a^ρ], i.e.,

$$\underset{\rho \in [-a^{\rho },a^{\rho }]}{\text{sup}}|R_n(\omega ,\rho )-{\overline{R}}_n(\rho )|\le \Vert {\stackrel{\sim }{\mathbf{\Phi }}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{\Phi }}}_n-{\mathbf{\Phi }}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Phi }}_n\Vert [1+{(a^{\rho })}^2+{(a^{\rho })}^4]\stackrel{p}{\to }0$$

as n → ∞. The consistency of ρ̃_n now follows directly from Lemma 3.1 in Pötscher and Prucha (1997).

We make use of the following lemma.

Lemma C.2

Let u_n be defined by (2) and let ${\mathbf{D}}_n=[{\mathbf{d}}_{1.,n}^{\prime },\dots ,{\mathbf{d}}_{n.,n}^{\prime }]\prime$, where d_i.,n is defined in Assumption 4. Suppose Assumptions 1 – 3 hold, and suppose furthermore that the columns of D_n are of the form π_n + Π_nε_n, where the elements of π_n are bounded in absolute value and the row and column sums of Π_n are uniformly bounded in absolute value. Then (a) $E{d}_{ij,n}^4\le \mathit{\text{const}}<\mathrm{\infty }$, and thus the moment condition in Assumption 4 is automatically implied, and (b) Assumption 6 is automatically implied.

Proof

A proof of the lemma is given in Kelejian and Prucha (2007b).

Proof of Theorem 2

Define

$$q_n({\rho }_n,{\mathbf{\Delta }}_n)={\stackrel{\sim }{\gamma }}_n-{\stackrel{\sim }{\mathbf{\Gamma }}}_n\left[\begin{array}{c}\hfill {\rho }_n\hfill \\ \hfill {\rho }_n^2\hfill \end{array}\right]=\left[\begin{array}{c}{n}^{-1}{\stackrel{\sim }{\mathbf{u}}}_n^{\prime }{\mathbf{C}}_{1,n}{\stackrel{\sim }{\mathbf{u}}}_n\hfill \\ n^{-1}{\stackrel{\sim }{\mathbf{u}}}_n^{\prime }{\mathbf{C}}_{2,n}{\stackrel{\sim }{\mathbf{u}}}_n\hfill \end{array}\right]$$ (C.1)

with ${\mathbf{C}}_{r,n}=(1/2)({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })({\mathbf{A}}_{r,n}+{\mathbf{A}}_{r,n}^{\prime })({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n)$, and where the matrices A_r,n are defined by (5), r = 1, 2. The second equality in (C.1) is seen to hold in light of the discussion surrounding equations (4)–(9). For later use we observe that the rows and column sums of C_r,n are uniformly bounded in absolute value; see, e.g., Remark A.1 in Kelejian and Prucha (2004).

We have shown in Theorem 1 that the GM estimator ρ̃_n defined in (9) is consistent. Apart on a set whose probability tends to zero the estimator satisfies the following first order condition

$$q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)\prime {\stackrel{\sim }{\mathrm{\Upsilon }}}_n\frac{\partial q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho }=0.$$

Substituting the mean value theorem expression

$$q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)=q_n({\rho }_n,{\mathbf{\Delta }}_n)+\frac{\partial q_n({\overline{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho }({\stackrel{\sim }{\rho }}_n-{\rho }_n)$$

into the first order condition yields

$$\frac{\partial q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho \prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n\frac{\partial q_n({\overline{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho }{n}^{1/2}({\stackrel{\sim }{\rho }}_n-{\rho }_n)=-\frac{\partial q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho \prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{n}^{1/2}{q}_n({\rho }_n,{\mathbf{\Delta }}_n)$$ (C.2)

where ρ̄_n is some between value. Observe that

$$\frac{\partial q_n(\rho ,{\mathbf{\Delta }}_n)}{\partial \rho }=-{\stackrel{\sim }{\mathbf{\Gamma }}}_n\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\rho \hfill \end{array}\right]$$ (C.3)

and consider the nonnegative scalars

$$\begin{array}{ccc}{\stackrel{\sim }{\mathbf{\Xi }}}_n\hfill & =\hfill & \frac{\partial q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho \prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n\frac{\partial q_n({\overline{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho }={\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2{\stackrel{\sim }{\rho }}_n\hfill \end{array}\right]}^{\prime }{\stackrel{\sim }{\mathbf{\Gamma }}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{\Gamma }}}_n\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2{\overline{\rho }}_n\hfill \end{array}\right],\hfill \\ {\mathbf{\Xi }}_n\hfill & =\hfill & {\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2{\rho }_n\hfill \end{array}\right]}^{\prime }{\mathbf{\Gamma }}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Gamma }}_n\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2{\rho }_n\hfill \end{array}\right].\hfill \end{array}$$ (C.4)

In proving Theorem 1 we have demonstrated that ${\stackrel{\sim }{\mathbf{\Gamma }}}_n-{\mathbf{\Gamma }}_n\stackrel{p}{\to }0$ and that the elements of Γ̃_n and Γ_n are O_p(1) and O(1), respectively. By Assumption 5 we have ϒ̃_n – ϒ_n = o_p(1) and also that the elements of ϒ̃_n and ϒ_n are O_p(1) and O(1), respectively. Since ρ̃_n and ρ̄_n are consistent and bounded in absolute value, clearly

$${\stackrel{\sim }{\mathbf{\Xi }}}_n-{\mathbf{\Xi }}_n\stackrel{p}{\to }0$$ (C.5)

as n → ∞, and furthermore Ξ̃_n = O_p(1) and Ξ_n = O(1). In particular ${\mathbf{\Xi }}_n\le {\mathrm{\lambda }}_{\mathrm{\Xi }}^{**}$ where ${\mathrm{\lambda }}_{\mathrm{\Xi }}^{**}$ is some finite constant. In light of Assumption 5 we have ${\mathbf{\Xi }}_n\ge {\mathrm{\lambda }}_{\text{min}}({\mathrm{\Upsilon }}_n){\mathrm{\lambda }}_{\text{min}}({\mathbf{\Gamma }}_n^{\prime }{\mathbf{\Gamma }}_n)(1+4{\rho }_n^2)\ge {\mathrm{\lambda }}_{\mathrm{\Xi }}^{*}$ for some ${\mathrm{\lambda }}_{\mathrm{\Xi }}^{*}>0$. Hence $0<{\mathbf{\Xi }}_n^{-1}\le 1/{\mathrm{\lambda }}_{\mathrm{\Xi }}^{*}<\mathrm{\infty }$, and thus we also have ${\mathbf{\Xi }}_n^{-1}=O(1)$. Let ${\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}$ denote the generalized inverse of Ξ̃_n. It then follows as a special case of Lemma F1 in Pötscher and Prucha (1997) that Ξ̃_n is nonsingular eventually with probability tending to one, that ${\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}=O_p(1)$, and that

$${\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}-{\mathbf{\Xi }}_n^{-1}\stackrel{p}{\to }0$$ (C.6)

as n → ∞.

Premultiplying (C.2) by ${\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}$ and rearranging terms yields

$$n^{1/2}({\stackrel{\sim }{\rho }}_n-{\rho }_n)=\left[1-{\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}{\stackrel{\sim }{\mathbf{\Xi }}}_n\right]n^{1/2}({\stackrel{\sim }{\rho }}_n-{\rho }_n)-{\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}\frac{\partial q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho \prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{n}^{1/2}{q}_n({\rho }_n,{\mathbf{\Delta }}_n).$$

In light of the above discussion the the first term on the r.h.s. is zero on ω-sets of probability tending to one. This yields

$$n^{1/2}({\stackrel{\sim }{\rho }}_n-{\rho }_n)=-{\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}\frac{\partial q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho \prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{n}^{1/2}{q}_n({\rho }_n,{\mathbf{\Delta }}_n)+o_p(1).$$ (C.7)

Observe that

$${\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}\frac{\partial q_n({\stackrel{\sim }{\rho }}_n,{\mathbf{\Delta }}_n)}{\partial \rho \prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n-{\mathbf{\Xi }}_n^{-1}{\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2{\rho }_n\hfill \end{array}\right]}^{\prime }{\mathbf{\Gamma }}_n^{\prime }{\mathrm{\Upsilon }}_n=o_p(1).$$ (C.8)

In light of (C.1) and Lemma C.1 the elements of n^1/2q_n (ρ_n, Δ_n) can be expressed as (r = 1, 2)

$$n^{-1/2}{\stackrel{\sim }{\mathbf{u}}}_n^{\prime }{\mathbf{C}}_{r,n}{\stackrel{\sim }{\mathbf{u}}}_n=n^{-1/2}{\mathbf{u}}_n^{\prime }{\mathbf{C}}_{r,n}{\mathbf{u}}_n+{\mathbf{\alpha }}_{r,n}^{\prime }{n}^{1/2}{\mathbf{\Delta }}_n+o_p(1)$$

where

$${\mathbf{\alpha }}_{r,n}=2n^{-1}E{\mathbf{D}}_n^{\prime }{\mathbf{C}}_{r,n}{\mathbf{u}}_n.$$

Furthermore, the lemma implies that the elements of α_r,n are uniformly bounded in absolute value. Utilizing u_n = (I_n – ρ_n M_n)⁻¹ ε_n and Assumption 7 we have

$$n^{1/2}{q}_n({\rho }_n,{\mathbf{\Delta }}_n)=n^{-1/2}\left[\begin{array}{c}\frac{1}{2}{\mathbf{\epsilon }}_n^{\prime }({\mathbf{A}}_{1,n}+{\mathbf{A}}_{1,n}^{\prime }){\mathbf{\epsilon }}_n+{\mathbf{a}}_{1,n}^{\prime }{\mathbf{\epsilon }}_n\hfill \\ \frac{1}{2}{\epsilon }_n^{\prime }({\mathbf{A}}_{2,n}+{\mathbf{A}}_{2,n}^{\prime }){\mathbf{\epsilon }}_n+{\mathbf{a}}_{2,n}^{\prime }{\mathbf{\epsilon }}_n\hfill \end{array}\right]+o_p(1)$$ (C.9)

where a_r,n = T_nα_r,n, r = 1, 2. Observe that the elements of a_r,n are uniformly bounded in absolute value. As discussed before Theorem 2 in the text, the VC matrix of the vector of quadratic forms on the r.h.s. of (C.9) is given by Ψ_n where the elements of that matrix are given in (13). Those elements can be written more explicitly as

$${\psi }_{rs,n}=\frac{1}{2n}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n(a_{ij,r,n}+a_{ji,r,n})(a_{ij,s,n}+a_{ji,s,n}){\sigma }_{i,n}^2{\sigma }_{j,n}^2+\frac{1}{n}{\displaystyle \sum _{i=1}^n{a}_{i,r,n}{a}_{i,s,n}{\sigma }_{i,n}^2}}}$$ (C.10)

By assumption λ_min(Ψ_n) ≥ const > 0. Since the matrices A_r,n, the vectors a_r,n, and the innovations ε_n satisfy all of the remaining assumptions of the central limit theorem for vectors of linear quadratic forms given above as Theorem A.1 it follows that

$${\mathit{\xi }}_n=-{\mathbf{\Psi }}_n^{-1/2}{n}^{-1/2}\left[\begin{array}{c}\frac{1}{2}{\mathbf{\epsilon }}_n^{\prime }({\mathbf{A}}_{1,n}+{\mathbf{A}}_{1,n}^{\prime }){\mathbf{\epsilon }}_n+{\mathbf{a}}_{1,n}^{\prime }{\mathbf{\epsilon }}_n\hfill \\ \frac{1}{2}{\mathbf{\epsilon }}_n^{\prime }({\mathbf{A}}_{2,n}+{\mathbf{A}}_{2,n}^{\prime }){\mathbf{\epsilon }}_n+{\mathbf{a}}_{2,n}^{\prime }{\mathbf{\epsilon }}_n\hfill \end{array}\right]\stackrel{d}{\to }N(\mathbf{0},{\mathbf{I}}_2).$$ (C.11)

Since the row and column sums of the matrices A_r,n are uniformly bounded in absolute value, and since the elements of a_r,n and the variances are uniformly bounded by finite constants it is readily seen from (C.10) that the elements of Ψ_n, and hence those of ${\mathbf{\Psi }}_n^{1/2}$ are uniformly bounded. It now follows from (C.7), (C.8) and (C.11) that

$$n^{1/2}({\stackrel{\sim }{\rho }}_n-{\rho }_n)={\mathbf{\Xi }}_n^{-1}{\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2{\rho }_n\hfill \end{array}\right]}^{\prime }{\mathbf{\Gamma }}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Psi }}_n^{1/2}{\mathit{\xi }}_n+o_p(1).$$ (C.12)

Observing that ${\mathbf{\Xi }}_n={\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n$, where J_n = Γ_n[1, 2ρ_n]′, this establishes (14). Since all of the nonstochastic terms on the r.h.s. of (C.12) are O(1) it follows that n^1/2(ρ̃_n – ρ_n) = O_p(1). Next recall that $0<{\mathrm{\lambda }}_{\mathrm{\Xi }}^{*}\le {\mathbf{\Xi }}_n\le {\mathrm{\lambda }}_{\mathrm{\Xi }}^{**}<\mathrm{\infty }$. Hence

$${\mathbf{\Xi }}_n^{-1}{\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{\Psi }}_n{\mathrm{\Upsilon }}_n{\mathbf{J}}_n{\mathbf{\Xi }}_n^{-1}\ge {\mathrm{\lambda }}_{\text{min}}({\mathbf{\Psi }}_n){[{\mathrm{\lambda }}_{\text{min}}({\mathrm{\Upsilon }}_n)]}^2{\mathrm{\lambda }}_{\text{min}}({\mathbf{\Gamma }}_n^{\prime }{\mathbf{\Gamma }}_n)(1+4{\rho }_n^2)/{({\mathrm{\lambda }}_{\mathrm{\Xi }}^{**})}^2\ge \mathit{\text{const}}>0.$$

This establishes the last claim of the theorem.

As part of proving Theorem 3 will be to show that Ψ̃_n – Ψ_n = o_p(1). Observe that the elements can be written as

$${\stackrel{\sim }{\psi }}_{rs,n}={\stackrel{\sim }{\psi }}_{rs,n}^{*}+{\stackrel{\sim }{\psi }}_{rs,n,}^{**},\text{\hspace{1em}and\hspace{1em}}{\psi }_{rs,n}={\psi }_{rs,n}^{*}+{\psi }_{rs,n,}^{**}$$ (C.13)

with

$$\begin{array}{cc}{\stackrel{\sim }{\psi }}_{rs,n}^{*}={(2n)}^{-1}{\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{j=1}^n{a}_{ij,n}{\stackrel{\sim }{\epsilon }}_{i,n}^2{\stackrel{\sim }{\epsilon }}_{j,n}^2,}}\hfill & {\psi }_{rs,n}^{*}={(2n)}^{-1}{\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{j=1}^n{a}_{ij,n}{\sigma }_{i,n}^2{\sigma }_{j,n}^2,}}\hfill \\ {\stackrel{\sim }{\psi }}_{rs,n}^{*}=n^{-1}{\stackrel{\sim }{\mathbf{\alpha }}}_{r,n}^{\prime }{\stackrel{\sim }{\mathbf{P}}}_n^{\prime }{\stackrel{\sim }{\mathbf{F}}}_n^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\stackrel{\sim }{\mathbf{F}}}_n{\stackrel{\sim }{\mathbf{P}}}_n{\stackrel{\sim }{\mathbf{\alpha }}}_{s,n},\hfill & {\psi }_{rs,n}^{*}=n^{-1}{\mathbf{\alpha }}_{r,n}^{\prime }{\mathbf{P}}_n^{\prime }{\mathbf{F}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{F}}_n{\mathbf{P}}_n{\mathbf{\alpha }}_{s,n},\hfill \end{array}$$ (C.14)

where a_ij,n = (a_ij,r,n + a_ji,r,n)(a_ij,s,n + a_ji,s,n).

The next two lemmata will be used to show that ${\stackrel{\sim }{\psi }}_{rs,n}^{*}-{\stackrel{\sim }{\psi }}_{rs,n}^{*}=o_p(1)$.

Lemma C.3

Suppose Assumptions 1 – 3 hold. Let ${\mathrm{\Lambda }}_n=n^{-1}({\mathbf{\sigma }}_n^2)\prime {\mathbf{A}}_n{\mathbf{\sigma }}_n^2$ and ${\overline{\mathrm{\Lambda }}}_n=n^{-1}({\mathbf{\epsilon }}_n^2)\prime {\mathbf{A}}_n{\mathbf{\epsilon }}_n^2\mathit{\text{with}}{\mathbf{\sigma }}_n^2=({\sigma }_{1,n}^2,\dots ,{\sigma }_{n,n}^2)\prime \text{and}{\mathbf{\epsilon }}_n^2=({\epsilon }_{1,n}^2,\dots ,{\epsilon }_{n,n}^2)\prime$ and where the n × n matrices A_n are real nonstochastic and symmetric. Suppose further that the diagonal elements of the matrices A_n are zero and that their row and column sums are uniformly bounded in absolute value. Then EΛ̅_n = Λ_n = O(1) and var (Λ̅_n = o(1), and hence ${\overline{\mathrm{\Lambda }}}_n-{\mathrm{\Lambda }}_n\stackrel{p}{\to }0$ as n → ∞, and Λ̅_n = O_p(1).

Proof

A proof of the lemma is given in Kelejian and Prucha (2007b).

Lemma C.4

Suppose Assumptions 1 – 4 hold. Let ε_n = (I_n – ρ_n M_n) u_n, and let ε̃_n = (I_n – ρ̃_n M_n) ũ_n with ũ_n = u_n + D_n Δ_n and ${\mathbf{D}}_n=[{\mathbf{d}}_{1.,n}^{\prime },\dots ,{\mathbf{d}}_{n.,n}^{\prime }]\prime$, and where ρ̃_n can be any estimator that satisfies n^1/2 (ρ̃_n – ρ_n) = O_p(1). Define ${\stackrel{\sim }{\mathrm{\Lambda }}}_n=n^{-1}({\stackrel{\sim }{\mathbf{\epsilon }}}_n^2)\prime {\mathbf{A}}_n({\stackrel{\sim }{\mathbf{\epsilon }}}_n^2),{\overline{\mathrm{\Lambda }}}_n=n^{-1}({\mathbf{\epsilon }}_n^2)\prime {\mathbf{A}}_n{\mathbf{\epsilon }}_n^2\mathit{\text{with}}{\stackrel{\sim }{\mathbf{\epsilon }}}_n^2=({\stackrel{\sim }{\epsilon }}_{1,n}^2,\dots ,{\stackrel{\sim }{\epsilon }}_{n,n}^2)\prime ,{\mathbf{\epsilon }}_n^2=({\epsilon }_{1,n}^2,\dots ,{\epsilon }_{n,n}^2)\prime$ and where the n × n matrices A_n are real nonstochastic and symmetric. Suppose further that the diagonal elements of the matrices A_n are zero and that their row and column sums are uniformly bounded in absolute value, and that $E{d}_{ij,n}^4\le K_d<\mathrm{\infty }$. Then, ${\stackrel{\sim }{\mathrm{\Lambda }}}_n-{\overline{\mathrm{\Lambda }}}_n\stackrel{p}{\to }0$ as n → ∞, and Λ̃_n = O_p(1).

Proof

A proof of the lemma is given in Kelejian and Prucha (2007b).

The next two lemmata will be used to show that ${\stackrel{\sim }{\psi }}_{rs,n}^{**}-{\psi }_{rs,n}^{**}=o_p(1)$, where ${\stackrel{\sim }{\psi }}_{rs,n}^{**}\text{and}{\psi }_{rs,n}^{**}$ are defined in (C.13) and (C.14).

Lemma C.5

Suppose Assumptions 1 – 4 hold. Let ε_n = (I_n – ρ_n M_n) u_n, and let ε̃_n = (I_n – ρ̃_n M_n) ũ_n with ũ_n = u_n + D_nΔ_n and ${\mathbf{D}}_n=[{\mathbf{d}}_{1.,n}^{\prime },\dots ,{\mathbf{d}}_{n.,n}^{\prime }]\prime$, and where ρ̃_n be any estimator that satisfies ρ̃_n – ρ_n = o_p(1). Let a_n and b_n be n × 1 vectors whose elements are uniformly bounded in absolute value by c < ∞ and let ${\mathbf{\Sigma }}_n={\mathit{\text{diag}}}_{i=1,\dots ,n}({\sigma }_i^2)\mathit{\text{and}}{\stackrel{\sim }{\mathbf{\Sigma }}}_n={\mathit{\text{diag}}}_{i=1,\dots ,n}({\stackrel{\sim }{\epsilon }}_{i,n}^2)$. Then:

1. $n^{-1}{\mathbf{a}}_n^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\mathbf{b}}_n–n^{-1}{\mathbf{a}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{b}}_n=o_p(1)$ and $n^{-1}{\mathbf{a}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{b}}_n=O(1)$.

2. There exist random variables ς_n that do not depend on a_n and b_n such that

   $$\left|n^{-1}{\mathbf{a}}_n^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\mathbf{b}}_n-n^{-1}{\mathbf{a}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{b}}_n\right|\le K(c)(1+{\varsigma }_n)$$

   with ς_n = o_p(1) and where K(c) < ∞ is a constant that depends monotonically on c (as well as on some other bounds maintained in the assumptions).

Proof

A proof of the lemma is given in Kelejian and Prucha (2007b).

Lemma C.6

Suppose Assumptions 1 – 4 hold. Furthermore assume that sup_n|ρ_n| < 1, and the row sums and column sums of M_n are bounded uniformly in absolute value by, respectively, one and some finite constant. Let ε_n = (I_n – ρ_n M_n) u_n, and let ε̃_n = (I_n – ρ̃_n M_n) ũ_n with ũ_n = u_n + D_n Δ_n and ${\mathbf{D}}_n=[{\mathbf{d}}_{1.,n}^{\prime },\dots ,{\mathbf{d}}_{n.,n}^{\prime }]\prime$, and where ρ̃_n can be any estimator that satisfies ρ̃_n – ρ_n = o_p(1). Let

$${\mathbf{F}}_n={\stackrel{\sim }{\mathbf{F}}}_n={\mathbf{H}}_n$$

or

$${\mathbf{F}}_n={({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n)}^{-1}{\mathbf{H}}_n\mathit{\text{\hspace{1em}and\hspace{1em}}}{\stackrel{\sim }{\mathbf{F}}}_n={({\mathbf{I}}_n-{\stackrel{\sim }{\rho }}_n{\mathbf{M}}_n)}^{+}{\mathbf{H}}_n,$$

where H_n is an n × p_* matrix whose elements are uniformly bounded in absolute value, let ${\mathbf{\Sigma }}_n={\mathit{\text{diag}}}_{i=1,\dots ,n}({\sigma }_i^2)\mathit{\text{and}}{\stackrel{\sim }{\mathbf{\Sigma }}}_n={\mathit{\text{diag}}}_{i=1,\dots ,n}({\stackrel{\sim }{\epsilon }}_{i,n}^2)$, then $n^{-1}{\stackrel{\sim }{\mathbf{F}}}_n^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\stackrel{\sim }{\mathbf{F}}}_n-n^{-1}{\mathbf{F}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{F}}_n=o_p(1)\mathit{\text{and}}{n}^{-1}{\mathbf{F}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{F}}_n=O(1)$.¹⁵

Proof

A proof of the lemma is given in Kelejian and Prucha (2007b).

Proof of Theorem 3

We first demonstrate that Ψ̃_n – Ψ_n = o_p(1), utilizing the expressions for the elements of Ψ̃_n and Ψ_n in (C.13) and (C.14). Observe that under the maintained assumption the row and column sums of the absolute elements of the matrices A_r,n and A_s,n are uniformly bounded, and thus clearly are those of the matrices A_n = (a_ij,n) with a_ij,n = (a_ij,r,n + a_ji,r,n)(a_ij,s,n + a_ji,s,n). It then follows directly from Lemmata C.3 and C.4 that ${\stackrel{\sim }{\psi }}_{rs,n}^{*}-{\psi }_{rs,n}^{*}=o_p(1),{\psi }_{rs,n}^{*}=O(1)\text{and}{\stackrel{\sim }{\psi }}_{rs,n}^{*}=O_p(1)$.

Observing that the row and column sums of $({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })({\mathbf{A}}_{r,n}+{\mathbf{A}}_{r,n}^{\prime })({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n)$ are bounded uniformly in absolute value it follows from Lemma C.1 that α̃_r,n – α_r,n = o_p(1), α_r,n = O(1) and hence α̃_r,n = O_p(1). By assumption P̃_n – P_n = o_p(1), P_n = O(1), and hence P̃_n = O_p(1). Observe that by Lemma C.6 we have $n^{-1}{\stackrel{\sim }{\mathbf{F}}}_n^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\stackrel{\sim }{\mathbf{F}}}_n-n^{-1}{\mathbf{F}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{F}}_n=o_p(1),n^{-1}{\mathbf{F}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{F}}_n=O(1)$, and hence $n^{-1}{\stackrel{\sim }{\mathbf{F}}}_n^{\prime }{\stackrel{\sim }{\mathbf{\Sigma }}}_n{\stackrel{\sim }{\mathbf{F}}}_n=O_p(1)$. Thus ${\stackrel{\sim }{\psi }}_{rs,n}^{**}-{\psi }_{rs,n}^{**}=o_p(1),{\psi }_{rs,n}^{**}=O(1),\text{and}{\stackrel{\sim }{\psi }}_{rs,n}^{**}=O_p(1)$. Hence Ψ̃_n – Ψ_n = o_p(1), Ψ_n = O(1) and Ψ̃_n = O_p(1).

By Assumption 5 we have ϒ̃_n – ϒ_n = o_p(1), ϒ_n = O(1) and thus ϒ̃_n = O_p(1). Let ${\mathbf{\Xi }}_n={\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n\text{and}{\stackrel{\sim }{\mathbf{\Xi }}}_n={\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n$, then, as demonstrated in the proof of Theorem 2, ${\stackrel{\sim }{\mathbf{J}}}_n=O_p(1),{\mathbf{J}}_n=O(1),{\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}=O_p(1)\text{and}{\mathbf{\Xi }}_n^{-1}=O_p(1)$, and furthermore ${\stackrel{\sim }{\mathbf{J}}}_n-{\mathbf{J}}_n\stackrel{p}{\to }0\text{and}{\stackrel{\sim }{\mathbf{\Xi }}}_n^{+}-{\mathbf{\Xi }}_n^{-1}\stackrel{p}{\to }0$ as n → ∞. (There is a slight difference in the definition of Ξ̃_n here and in the proof of Theorem 2, which does not affect the claim.) Given the above results it is now obvious that Ω̃_ρ̄n – Ω_ρ̄n = o_p(1).

Proof of Theorem 4

We concentrate again on the case ${\mathbf{F}}_n={({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })}^{-1}{\mathbf{H}}_n$. The first line in (27) is seen to hold in light of Assumption 7 and Theorem 2. We next verify that ${\mathit{\xi }}_{\mathrm{o},n}\stackrel{d}{\to }N(\mathbf{0},{\mathbf{I}}_{p_{*}+2})$ utilizing the central limit theorem for vectors of linear quadratic forms given above as Theorem A.1. By assumption ${\mathrm{\lambda }}_{\text{min}}({\mathbf{\Psi }}_{\mathrm{o},n})\ge c_{{\mathrm{\Psi }}_{\mathrm{o}}}^{*}>0$. In proving Theorem 2 we verified that the innovations ε_n and the elements of a_r,n and A_r,n appearing in v_n all satisfy the conditions of the central limit theorem. Since the row and column sums of ${({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })}^{-1}$ and the elements of H_n are uniformly bounded in absolute value it follows that the elements of F_n are also uniformly bounded in absolute value. Thus all conditions of Theorem A.1 are satisfied, which verifies the claim.

In proving Theorems 2 and 3 we have shown that Ψ̃_n – Ψ_n = o_p(1), Ψ_n = O(1) and thus Ψ̃_n = O_p(1). By analogous argumentation it is readily seen that the other sub-matrices of Ψ̃_o,n and Ψ_o,n have the same properties, and thus Ψ̃_o,n – Ψ_o,n = o_p(1) and Ψ_o,n = O(1). By assumption P̃_n – P_n = o_p(1), P_n = O(1) and thus P̃_n = O_p(1), and furthermore ϒ̃_n – ϒ_n = o_p(1), ϒ_n = O(1) and thus ϒ̃_n = O_p(1). In proving Theorem 2 we have verified that ${\stackrel{\sim }{\mathbf{J}}}_n-{\mathbf{J}}_n\stackrel{p}{\to }0,{\mathbf{J}}_n=O(1)$ and J̃_n = O_p(1), and furthermore that ${({\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n)}^{+}-{({\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n)}^{-1}=o_p(1),{({\stackrel{\sim }{\mathbf{J}}}_n^{\prime }{\stackrel{\sim }{\mathrm{\Upsilon }}}_n{\stackrel{\sim }{\mathbf{J}}}_n)}^{+}=O_p(1)\text{and}{({\mathbf{J}}_n^{\prime }{\mathrm{\Upsilon }}_n{\mathbf{J}}_n)}^{-1}=O(1)$ The claim that Ω̃_o,n – Ω_o,n(ϒ_n) = o_p(1) and Ω_o,n = O(1) is now obvious.

D Appendix: Proofs for Section 4

Proof of Lemma 3

Utilizing (3) it follows from Assumptions 3, 8 and sup_n ||β_n|| < ∞ that all columns of Z_n = [X_n, W_ny_n] are of the form ϑ_n = π_n + Π_nε_n, where the elements of the n × 1 vector π_n are bounded in absolute value and the row and column sums of the n × n matrix Π_n are uniformly bounded in absolute value by some finite constant; compare, e.g., Remark A.1. in Kelejian and Prucha (2004). The claims that the fourth moments of D_n = −Z_n are uniformly bounded by a finite constant, and that Assumption 6 holds, now follows directly from Lemma C.2.

Clearly

$$n^{1/2}({\stackrel{\sim }{\mathbf{\delta }}}_n-{\mathbf{\delta }}_n)={\stackrel{\sim }{\mathbf{P}}}_n^{\prime }{n}^{-1/2}{\mathbf{F}}_n^{\prime }{\mathbf{\epsilon }}_n$$

where P̃_n is defined in the lemma and ${\mathbf{F}}_n={({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })}^{-1}{\mathbf{H}}_n$. Given Assumption 10 clearly P̃_n = P_n + o_p(1) and P_n = O(1), with P_n defined in the lemma. Since by Assumption 3 the row and column sums of (I_n – ρ_n M_n)⁻¹ are uniformly bounded in absolute value, and since by Assumption 9 the elements of H_n are uniformly bounded in absolute value, it follows that the elements of F_n are uniformly bounded in absolute value. By Assumption 2, E(ε_n) = 0 and its diagonal VC matrix, Σ_n, has uniformly bounded elements. Therefore, $E{n}^{-1/2}{\mathbf{F}}_n^{\prime }{\epsilon }_n=0$ and the elements of $VC(n^{-1/2}{\mathbf{F}}_n^{\prime }{\mathbf{\epsilon }}_n)=n^{-1}{\mathbf{F}}_n^{\prime }{\mathbf{\Sigma }}_n{\mathbf{F}}_n$ are also uniformly bounded in absolute value. Thus, by Chebyshev’s inequality $n^{-1/2}{\mathbf{F}}_n^{\prime }{\mathbf{\epsilon }}_n=O_p(1)$, and consequently $n^{1/2}({\stackrel{\sim }{\mathbf{\delta }}}_n-{\mathbf{\delta }}_n)={\mathbf{P}}_n^{\prime }{n}^{-1/2}{\mathbf{F}}_n^{\prime }{\mathbf{\epsilon }}_n+o_p(1)\text{and}{\mathbf{P}}_n^{\prime }{n}^{-1/2}{\mathbf{F}}_n^{\prime }{\mathbf{\epsilon }}_n=O_p(1)$. This completes the proof recalling that T_n = F_n P_n.

Proof of Lemma 4

Note from (1) and (2) that

$${\mathbf{y}}_{n*}({\widehat{\rho }}_n)={\mathbf{Z}}_{n*}({\widehat{\rho }}_n){\mathbf{\delta }}_n+{\epsilon }_n-({\widehat{\rho }}_n-{\rho }_n){\mathbf{M}}_n{\mathbf{u}}_n$$

and hence

$$\begin{array}{cc}\hfill & n^{1/2}[{\widehat{\mathbf{\delta }}}_n({\widehat{\rho }}_n)-{\mathbf{\delta }}_n]\hfill \\ =\hfill & {\left[n^{-1}{\widehat{\mathbf{Z}}}_{n*}^{\prime }({\widehat{\rho }}_n){\mathbf{Z}}_{n*}({\widehat{\rho }}_n)\right]}^{-1}{n}^{-1/2}{\widehat{\mathbf{Z}}}_{n*}^{\prime }({\widehat{\rho }}_n)[{\epsilon }_n-({\widehat{\rho }}_n-{\rho }_n){\mathbf{M}}_n{\mathbf{u}}_n]\hfill \\ =\hfill & {\stackrel{\sim }{\mathbf{P}}}_n^{*\prime }\left[n^{-1/2}{\mathbf{F}}_n^{*\prime }{\epsilon }_n-({\widehat{\rho }}_n-{\rho }_n)n^{-1/2}{\mathbf{F}}_n^{**\prime }{\epsilon }_n\right],\hfill \end{array}$$

where ${\stackrel{\sim }{\mathbf{P}}}_n^{*}$ is defined in the lemma, and with ${\mathbf{F}}_n^{*}={\mathbf{H}}_n\text{and}{\mathbf{F}}_n^{**}={({\mathbf{I}}_n-{\rho }_n{\mathbf{M}}_n^{\prime })}^{-1}{\mathbf{M}}_n^{\prime }{\mathbf{H}}_n$. In light of Assumption 10, and since ρ̂ is n^1/2-consistent, it follows that

$$n^{-1}{\widehat{\mathbf{Z}}}_{n*}^{\prime }({\widehat{\rho }}_n){\mathbf{Z}}_{n*}({\widehat{\rho }}_n)-{\mathbf{Q}}_{\mathbf{\text{HZ}}*}^{\prime }({\rho }_n){\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{\mathbf{\text{HZ}}*}({\rho }_n)=o_p(1).$$

Since by Assumption 10 we have ${\mathbf{Q}}_{\mathbf{\text{HZ}}*}^{\prime }({\rho }_n){\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{{\mathbf{\text{HZ}}}_{*}}({\rho }_n)=O(1)\text{and}{[{\mathbf{Q}}_{\mathbf{\text{HZ}}*}^{\prime }({\rho }_n){\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{\mathbf{\text{HZ}}*}({\rho }_n)]}^{-1}=O(1)$ it follows that

$${[n^{-1}{\widehat{\mathbf{Z}}}_{n*}^{\prime }({\widehat{\rho }}_n){\mathbf{Z}}_{n*}({\widehat{\rho }}_n)]}^{-1}-{[{\mathbf{Q}}_{\mathbf{\text{HZ}}*}^{\prime }({\rho }_n){\mathbf{Q}}_{\mathbf{\text{HH}}}^{-1}{\mathbf{Q}}_{\mathbf{\text{HZ}}*}({\rho }_n)]}^{-1}=o_p(1);$$

compare, e.g., Pötscher and Prucha (1997), Lemma F1. In light of this it follows further that ${\stackrel{\sim }{\mathbf{P}}}_n^{*}-{\mathbf{P}}_n^{*}=o_p(1)\text{and}{\mathbf{P}}_n^{*}=O(1)$, with ${\mathbf{P}}_n^{*}$ defined in the lemma. By argumentation analogous to that in the proof of Lemma 3 it is readily seen that $n^{-1/2}{\mathbf{F}}_n^{*\prime }{\epsilon }_n=O_p(1)\text{and}{n}^{-1/2}{\mathbf{F}}_n^{**\prime }{\epsilon }_n=O_p(1)$. Consequently $n^{1/2}[{\widehat{\mathbf{\delta }}}_n({\widehat{\rho }}_n)-{\mathbf{\delta }}_n]={\mathbf{P}}_n^{*\prime }{n}^{-1/2}{\mathbf{F}}_n^{*\prime }{\epsilon }_n+o_p(1)\text{and}{\mathbf{P}}_n^{*\prime }{n}^{-1/2}{\mathbf{F}}_n^{*\prime }{\epsilon }_n=O_p(1)$, observing again that ρ̂_n – ρ_n = o_p(1). This completes the proof recalling that ${\mathbf{T}}_n^{*}={\mathbf{F}}_n^{*}{\mathbf{P}}_n^{*}$.