GMM Estimation of Spatial Autoregressive Models in a System of Simultaneous Equations

Transcription

1 University of Colorado, Boulder CU Scholar Economics Graduate Theses & Dissertations Economics Spring GMM Estimation of Spatial Autoregressive Models in a System of Simultaneous Equations Paulo Quinderé Saraiva University of Colorado Boulder, saraiva@coloradoedu Follow this and additional works at: Part of the Economics Commons, and the Statistical Models Commons Recommended Citation Saraiva, Paulo Quinderé, "GMM Estimation of Spatial Autoregressive Models in a System of Simultaneous Equations" (2015) Economics Graduate Theses & Dissertations 63 This Dissertation is brought to you for free and open access by Economics at CU Scholar It has been accepted for inclusion in Economics Graduate Theses & Dissertations by an authorized administrator of CU Scholar For more information, please contact cuscholaradmin@coloradoedu

2 GMM Estimation of Spatial Autoregressive Models in a System of Simultaneous Equations by Paulo Quinderé Saraiva BA, Universidade Federal do Ceará, 2003 MS, Universidade Federal do Ceará, 2005 MS, Oregon State University, 2009 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Economics 2015

3 This thesis entitled: GMM Estimation of Spatial Autoregressive Models in a System of Simultaneous Equations written by Paulo Quinderé Saraiva has been approved for the Department of Economics Xiaodong Liu Prof Robert McNown Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline

4 iii Saraiva, Paulo Quinderé (PhD, Economics) GMM Estimation of Spatial Autoregressive Models in a System of Simultaneous Equations Thesis directed by Prof Xiaodong Liu This dissertation proposes a generalized method of moments (GMM) estimation framework for the spatial autorregressive (SAR) model in a system of simultaneous equations with homoskedastic and heteroskedastic disturbances It includes two chapters based on joint work with Prof Xiaodong Liu The first chapter extends the GMM estimator in Lee (2007) to estimate SAR models with endogenous regressors and homoskedastic disturbances We propose a new set of quadratic moment equations exploring the correlation of the spatially lagged dependent variable with the disturbance term of the main regression equation and with the endogenous regressor The proposed GMM estimator is more efficient than the IV-based linear estimators in the literature, and computationally simpler than the ML estimator With carefully constructed quadratic moment equations, the GMM estimator can be asymptotically as efficient as the full information ML estimator Monte Carlo experiment shows that the proposed GMM estimator performs well in finite samples The second chapter proposes a GMM estimator for the SAR model in a system of simultaneous equations with heteroskedastic disturbances Besides linear moment conditions, the GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations Compared with the QML approach considered in Yang and Lee (2014), the GMM estimator is easier to implement and robust under heteroskedasticity of an unknown form We also derive a heteroskedasticity-robust estimator for the asymptotic covariance of the GMM estimator Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples

5 Dedication To my grandfather, Alber Garcia Quinderé

6 v Acknowledgements The path to a PhD is arduous I would not have done it without help along the way I have received both emotional and academic support throughout the years It all started with my fifth grade teacher who said I was good at math and I was foolish enough to believe her My parents, siblings and grandparents never ceased to give me support For example, my father, being an economics professor himself, taught me how to study During my bachelors he advised me to develop the habit of studying multiple sources, especially when trying to understand a difficult concept In my two masters, my advisors where respectively Ivan Castelar and Carlos Martins-Filho Both have invested immensely in my future It was because of them that I chose to study Econometrics They also guided me in obtain the required math background in my field Among the Economics department staff, Patricia Holcomb and Maria Olivares are my heroines They have gone to great lengths to help me However, I could never meet them before class, for I would always lose track of time when talking to them I am forever grateful to my advisor, Xiaodong Liu He has taught me so much, including everything I know about Spatial Econometric Models I hope to live up to all his teachings and expectations He believed in me even during the difficult times when I thought I was not going to make it That gave me the strength I needed to succeed I would also like to thank all my dissertation committee members; Jem Corcoran, Robert McNown, Scott Savage and Donald Waldman Their enthusiasm about the subject, questions and suggestions where invaluable and helped make defending my dissertation quite enjoyable

7 vi Contents Chapter 1 Efficient GMM Estimation of SAR Models with Endogenous Regressors 1 11 Introduction 1 12 Model 3 13 GMM Estimation Estimator Identification 6 14 Asymptotic Properties Consistency and Asymptotic Normality Asymptotic Efficiency Monte Carlo Experiments Conclusion 17 Appendix 11 Likelihood Function of the SAR Model with Endogenous Regressors 18 Appendix 12 Lemmas 19 Appendix 13 Proofs 22 Appendix 14 Tables 28 2 GMM Estimation of SAR Simultaneous Equation Models with Unknown Heteroskedasticity Introduction Model and Moment Conditions 33

8 vii 23 Identification Identification of the pseudo reduced form parameters Identification of the structural parameters GMM Estimation Consistency and asymptotic normality Best moment conditions under homoskedasticity Monte Carlo Conclusion 48 Appendix 21 Lemmas 49 Appendix 22 Proofs 52 Appendix 23 Tables 65 Bibliography 70

9 viii Tables Table 11 2SLS, 3SLS and GMM Estimation (n = 245) SLS, 3SLS and GMM Estimation (n = 490) SLS, 3SLS and GMM Estimation (n = 245) SLS, 3SLS and GMM Estimation (n = 490) SLS, 3SLS, GMM and MLE Estimation (n = 98, under Heteroskedasticity) SLS, 3SLS, GMM and MLE Estimation (n = 490, under Heteroskedasticity) SLS, 3SLS, GMM and MLE Estimation (n = 98, under Homoskedasticity) SLS, 3SLS, GMM and MLE Estimation (n = 490, under Homoskedasticity) 69

10 Chapter 1 Efficient GMM Estimation of SAR Models with Endogenous Regressors 11 Introduction In recent years, spatial econometric models play a vital role in empirical research on regional and urban economics By expanding the notion of space from geographic space to economic space and social space, these models can be used to study cross-sectional interactions in much wider applications including education (eg Lin, 2010; Sacerdote, 2011; Carrell et al, 2013), crime (eg Patacchini and Zenou, 2012; Lindquist and Zenou, 2014), industrial organization (eg König et al, 2014), finance (eg Denbee et al, 2014), etc Among spatial econometric models, the spatial autoregressive (SAR) model introduced by Cliff and Ord (1973, 1981) has received the most attention In this model, the cross-sectional dependence is modeled as the weighted average outcome of neighboring units, typically referred to as the spatially lagged dependent variable As the spatially lagged dependent variable is endogenous, likelihood- and moment-based methods have been proposed to estimate the SAR model (eg Kelejian and Prucha, 1998; Kelejian and Prucha, 1999; Lee, 2004; Lee, 2007; Lee and Liu, 2010) In particular, for the SAR model with exogenous regressors, Lee (2007) proposes a generalized method of moments (GMM) estimator that combines linear moment conditions, with the (estimated) mean of the spatially lagged dependent variable as the instrumental variable (IV), and quadratic moment conditions based on the covariance structure of the spatially lagged dependent variable and the model disturbance term The GMM estimator improves estimation efficiency of IV-based linear estimators and is computational simple relative to the maximum likelihood (ML) estimator Fur-

11 2 thermore, Lin and Lee (2010) show that a sub-class of the GMM estimators is consistent in the presence of an unknown form of heteroskedasticity in model disturbances, and thus more robust relative to the ML estimator For SAR models with endogenous regressors, Liu (2012) and Liu and Lee (2013) consider, respectively, the limited information maximum likelihood (LIML) and two stage least squares (2SLS) estimators, in the presence of many potential IVs Liu and Lee (2013) also propose a criterion based on the approximate mean square error of the 2SLS estimator to select the optimal set of IVs In this paper, we extend the GMM estimator in Lee (2007) to estimate SAR models with endogenous regressors We propose a new set of quadratic moment equations exploring (i) the covariance structure of the spatially lagged dependent variable and the disturbance term of the main regression equation and (ii) the covariance structure of the spatially lagged dependent variable and the endogenous regressor The proposed GMM estimator is thus a full information estimator as it uses information across equations Compare to other full information estimators for a system of simultaneous equations with spatial interdependence, the GMM estimator is more efficient than the three stage least squares (3SLS) estimator in Kelejian and Prucha (2004), and computationally simpler than the ML estimator in Yang and Lee (2014) With carefully constructed quadratic moment equations, the GMM estimator can be asymptotically as efficient as the full information ML estimator We also conduct a limited Monte Carlo experiment to show that the proposed GMM estimator performs well in finite samples The rest of the paper is organized as follows In Section 2, we introduce the SAR model with endogenous regressors In Section 3, we define the GMM estimator and discuss the identification of model parameters In Section 4, we study the asymptotic properties of the GMM estimator and discuss the optimal moment conditions to use Section 5 reports Monte Carlo experiment results Section 6 briefly concludes The proofs are collected in the appendix Throughout the paper, we adopt the following notation For an n n matrix A = [a ij ] i,j=1,,n, let A (s) = A + A, vec D (A) = (a 11,, a nn ), and diag(a) = diag(a 11,, a nn ) The row (or column) sums of A are uniformly bounded in absolute value if max n i=1,,n j=1 a ij (or

12 3 max j=1,,n n i=1 a ij ) is bounded 12 Model Consider a SAR model with an endogenous regressor 1 given by y 1 = λ 0 Wy 1 + φ 0 y 2 + X 1 β 0 + u 1, (11) where y 1 is an n 1 vector of observations on the dependent variable, W is an n n nonstochastic spatial weights matrix with a zero diagonal, y 2 is an n 1 vector of observations on an endogenous regressor, X 1 is an n K 1 matrix of observations on K 1 nonstochastic exogenous regressors, and u 1 is an n 1 vector of iid innovations 2 Wy 1 is usually referred to as the spatially lagged dependent variable Let X = [X 1, X 2 ], where X 2 is an n K 2 matrix of observations on K 2 excluded nonstochastic exogenous variables The reduced form of the endogenous regressor y 2 is assumed to be y 2 = Xγ 0 + u 2, (12) where u 2 is an n 1 vector of iid innovations Let θ 0 = (δ 0, γ 0 ), with δ 0 = (λ 0, φ 0, β 0), denote the vector of true parameter values in the data generating process (DGP) The following regularity conditions are common in the literature of SAR models (see, eg, Lee, 2007; Kelejian and Prucha, 2010) Assumption 11 Let u 1,i and u 2,i denote, respectively, the i-th elements of u 1 and u 2 (i) (u 1,i, u 2,i ) is iid(0, Σ), where Σ = σ2 1 σ 12 σ 12 σ2 2 (ii) E u k,i u l,i u r,i u s,i 1+η is bounded for k, l, r, s = 1, 2 and some small constant η > 0 1 In this paper, we focus on the model with a single endogenous regressor for exposition purpose The model and proposed estimator can be easily generalized to accommodate any fixed number of endogenous regressors 2 y 1, y 2, u 1, u 2, X, W are allowed to depend on the sample size n, ie, to formulate triangular arrays as in Kelejian and Prucha (2010) Nevertheless, we suppress the subscript n to simplify the notation

13 Assumption 12 (i) The elements of X are uniformly bounded constants (ii) X has full column rank K X = K 1 + K 2 (iii) lim n n 1 X X exists and is nonsingular 4 Assumption 13 (i) All diagonal elements of the spatial weights matrix W are zero (ii) λ 0 ( λ, λ) with 0 < λ, λ c λ < (iii) S(λ) = I n λw is nonsingular for all λ ( λ, λ) (iv) The row and column sums of W and S(λ 0 ) 1 are uniformly bounded in absolute value Assumption 14 θ 0 is in the interior of a compact and convex parameter space Θ 13 GMM Estimation 131 Estimator Let S = S(λ 0 ) = I n λ 0 W and G = WS 1 Under Assumption 13, model (11) has a reduced form y 1 = S 1 X 1 β 0 + φ 0 S 1 Xγ 0 + S 1 u 1 + φ 0 S 1 u 2, (13) which implies that Wy 1 = GX 1 β 0 + φ 0 GXγ 0 + Gu 1 + φ 0 Gu 2 (14) As Wy 1 and y 2 are endogenous, consistent estimation of (11) requires IVs for Wy 1 and y 2 From (28), the deterministic part of Wy 1 is a linear combination of the columns in GX = [GX 1, GX 2 ] Therefore, GX can be used as an IV matrix for Wy 1 3 From (12), X can be used as an IV matrix for y 2 In general, let Q be an n K Q matrix of IVs such that E(Q u 1 ) = E(Q u 2 ) = 0 Let u 1 (δ) = S(λ)y 1 φy 2 X 1 β and u 2 (γ) = y 2 Xγ, where δ = (λ, φ, β ) The linear moment function for the GMM estimation is given by g 1 (θ) = (I 2 Q) u(θ), where denotes the Kronecker product, u(θ) = [u 1 (δ), u 2 (γ) ], and θ = (δ, γ ) 4 3 The IV matrix GX is not feasible as G involves the unknown parameter λ 0 Under Assumption 13, GX = WX + λ 0W 2 X + λ 2 0W 3 X + Therefore, we can use the leading order terms WX, W 2 X, W 3 X of the series expansion as feasible IVs for Wy 4 In practice, we could use two different IV matrices Q 1 and Q 2 to construct linear moment functions Q 1u 1(δ) and Q 2u 2(δ) The GMM estimator with g 1(θ) is (asymptotically) no less efficient than that with Q 1u 1(δ) and Q 2u 2(δ) if Q includes all linearly indepedent columns of Q 1 and Q 2

14 5 Besides the linear moment functions, Lee (2007) proposes to use quadratic moment functions based on the covariance structure of the spatially lagged dependent variable and model disturbances to improve estimation efficiency We generalize this idea to SAR models with endogenous regressors Substitution of (12) into (11) leads to a pseudo reduced form y 1 = λ 0 Wy 1 + φ 0 Xγ 0 + X 1 β 0 + u 1 + φ 0 u 2 (15) By exploring the covariance structure of the spatially lagged dependent variable Wy 1 and the disturbances of (15), we propose the following quadratic moment functions g 2 (θ) = [g 2,11 (δ), g 2,12 (θ), g 2,21 (θ), g 2,22 (γ) ] with g 2,11 (δ) = [Ξ 1u 1 (δ),, Ξ mu 1 (δ)] u 1 (δ) g 2,12 (θ) = [Ξ 1u 1 (δ),, Ξ mu 1 (δ)] u 2 (γ) g 2,21 (θ) = [Ξ 1u 2 (γ),, Ξ mu 2 (γ)] u 1 (δ) g 2,22 (γ) = [Ξ 1u 2 (γ),, Ξ mu 2 (γ)] u 2 (γ) where Ξ j is an n n constant matrix with tr(ξ j ) = 0 for j = 1,, m 5 Possible candidates for Ξ j are W, W 2 n 1 E(W 2 )I n, etc 6 These quadratic moment functions are based on the moment conditions that E(u 1 Ξ ju 1 ) = E(u 1 Ξ ju 2 ) = E(u 2 Ξ ju 1 ) = E(u 2 Ξ ju 2 ) = 0 for j = 1,, m Let g(θ) = [g 1 (θ), g 2 (θ) ], (16) and Ω = Var[g(θ 0 )] The following assumption is from Lee (2007) Assumption 15 (i) The elements of Q are uniformly bounded constants (ii) Ξ j is an n n constant matrix with tr(ξ j ) = 0 for j = 1,, m The row and column sums of Ξ j are uniformly bounded in absolute value (iii) lim n n 1 Ω exists and is nonsingular 5 In practice, we could use different sets of weighting matrices {Ξ 11,j} m 11 j=1, {Ξ12,j}m 12 j=1, {Ξ21,j}m 21 j=1 and {Ξ22,j}m 22 j=1 for the quadratic moment functions g 2,11(θ), g 2,12(θ), g 2,21(θ) and g 2,22(θ) respectively The quadratic moment functions g 2(θ) are (asymptotically) no less efficient than that with {Ξ 11,j} m 11 j=1, {Ξ12,j}m 12 j=1, {Ξ21,j}m 21 j=1 and {Ξ22,j}m 22 j=1 if {Ξ 1,, Ξ m} = {Ξ 11,j} m 11 j=1 {Ξ12,j}m 12 j=1 {Ξ21,j}m 21 j=1 {Ξ22,j}m 22 j=1 6 We discuss the optimal Q and Ξ in Section 142

15 Combining both linear and quadratic moment functions, the GMM estimator of θ 0 is given 6 by θ gmm = arg min θ Θ g(θ) F Fg(θ), (17) for some matrix F such that lim n F exists and has full row rank greater than or equal to dim(θ) In the GMM literature, F F is known as the GMM weighting matrix For instance, one can use the identity matrix as the weighting matrix to implement the GMM The asymptotic efficiency of the GMM estimator depends on the choice of the weighting matrix as discussed in Section Identification For θ 0 to be identified through the moment functions g(θ), lim n n 1 E[g(θ)] = 0 needs to have a unique solution at θ = θ 0 (Hansen, 1982) As S(λ)S 1 = I n + (λ 0 λ)g, it follows from (12) and (27) that u 1 (δ) = d 1 (δ) + [I n + (λ 0 λ)g]u 1 + [(φ 0 φ)i n + φ 0 (λ 0 λ)g]u 2 and u 2 (γ) = d 2 (γ) + u 2, where d 1 (δ) = [GX 1 β 0 + φ 0 GXγ 0, Xγ 0, X 1 ](δ 0 δ) and d 2 (γ) = X(γ 0 γ) For the linear moment functions, we have and lim n n 1 E[Q u 1 (δ)] = lim n n 1 Q d 1 (δ) = lim n n 1 Q [GX 1 β 0 + φ 0 GXγ 0, Xγ 0, X 1 ](δ 0 δ) lim n n 1 E[Q u 2 (γ)] = lim n n 1 Q d 2 (γ) = lim n n 1 Q X(γ 0 γ) Therefore, lim n n 1 E[g 1 (θ)] = 0 has a unique solution at θ = θ 0, if Q [GX 1 β 0 + φ 0 GXγ 0, Xγ 0, X 1 ] and Q X have full column rank for large enough n This sufficient rank condition implies the necessary rank condition that [GX 1 β 0 + φ 0 GXγ 0, Xγ 0, X 1 ] and X have full column rank and the rank of Q is at least max{dim(δ), K X }, for large enough n

16 7 Suppose [Xγ 0, X 1 ] has full column rank for large enough n 7 The necessary rank condition for identification does not hold if GX 1 β 0 + φ 0 GXγ 0 and [Xγ 0, X 1 ] are asymptotically linearly dependent 8 GX 1 β 0 + φ 0 GXγ 0 and [Xγ 0, X 1 ] are linearly dependent if there exist a constant scalar c 1 and a K 1 1 constant vector c 2 such that GX 1 β 0 + φ 0 GXγ 0 = c 1 Xγ 0 + X 1 c 2, which implies that d 1 (δ) = [(λ 0 λ)c 1 + (φ 0 φ)]xγ 0 + X 1 [(λ 0 λ)c 2 + (β 0 β)] Hence, the solutions of the linear moment equations lim n n 1 E[Q u 1 (δ)] = 0 are characterized by φ = φ 0 + (λ 0 λ)c 1 and β = β 0 + (λ 0 λ)c 2 (18) as long as Q [Xγ 0, X 1 ] has full column rank for large enough n In this case, φ 0 and β 0 can be identified if and only if λ 0 can be identified from the quadratic moment equations Given (18), we have E[u 1 (δ) Ξ j u 1 (δ)] = (λ 0 λ)(σ φ 0 σ 12 )tr(ξ (s) j G) and +(λ 0 λ) 2 [(σ φ 0 σ 12 + φ 2 0σ2)tr(G 2 Ξ j G) c 1 (σ 12 + φ 0 σ2)tr(ξ 2 (s) j G)] E[u 1 (δ) Ξ j u 2 (γ)] = (λ 0 λ)(σ 12 + φ 0 σ 2 2)tr(Ξ jg) E[u 2 (γ) Ξ j u 1 (δ)] = (λ 0 λ)(σ 12 + φ 0 σ 2 2)tr(Ξ j G) for j = 1,, m If (σ1 2 + φ 0σ 12 ) lim n n 1 tr(ξ (s) j G) 0 for some j {1,, m}, the quadratic moment equation has two roots λ = λ 0 and λ = λ 0 + lim n n 1 E[u 1 (δ) Ξ j u 1 (δ)] = 0 (σ φ 0σ 12 ) (σ φ 0σ 12 + φ 2 0 σ2 2 ) lim n [tr(g Ξ j G)/tr(Ξ (s) j G)] c 1 (σ 12 + φ 0 σ 2 2 ) 7 As Xγ 0 = X 1γ 10 + X 2γ 20, a necessary condition for (Xγ 0, X 1) to have full column rank is γ A necessary condition for GX 1β 0 + φ 0GXγ 0 and [Xγ 0, X 1] to be asymptotically linearly independent is (φ 0, β 0) 0

17 As (σ φ 0σ 12 + φ 2 0 σ2 2 ) > 0, if lim n [tr(g Ξ j G)/tr(Ξ (s) j for some j k, the moment equations 8 G)] lim n [tr(g Ξ k G)/tr(Ξ (s) G)] k lim n n 1 E[u 1 (δ) Ξ j u 1 (δ)] = 0 and lim n n 1 E[u 1 (δ) Ξ k u 1 (δ)] = 0 have a unique common root λ = λ 0 On the other hand, if (σ 12 + φ 0 σ 2 2 ) lim n n 1 tr(ξ j G) 0 for some j {1,, m}, the quadratic moment equation lim n n 1 E[u 1 (δ) Ξ j u 2 (γ)] = 0 has a unique root λ = λ 0 ; and if (σ 12 + φ 0 σ 2 2 ) lim n n 1 tr(ξ j G) 0 for some j {1,, m}, the quadratic moment equation lim n n 1 E[u 2 (γ) Ξ j u 1 (δ)] = 0 has a unique root λ = λ 0 To wrap up, the sufficient identification condition of θ 0 is summarized in the following assumption Assumption 16 lim n n 1 Q X and lim n n 1 Q [Xγ 0, X 1 ] both have full column rank, and at least one of the following conditions is satisfied (i) lim n n 1 Q [GX 1 β 0 + φ 0 GXγ 0, Xγ 0, X 1 ] has full column rank (ii) (σ1 2 + φ 0σ 12 ) lim n n 1 tr(ξ (s) j G) 0 for some j {1,, m}, and lim n n 1 [tr(ξ (s) 1 G),, tr(ξ(s) m G)] is linearly independent of lim n n 1 [tr(g Ξ 1 G),, tr(g Ξ m G)] (iii) (σ 12 + φ 0 σ2 2) lim n n 1 tr(ξ j G) 0 or (σ 12 + φ 0 σ2 2) lim n n 1 tr(ξ jg) 0 for some j {1,, m} 14 Asymptotic Properties 141 Consistency and Asymptotic Normality The GMM estimator defined in (17) falls into the class of Z-estimators (see Newey and McFadden, 1994) Therefore, to establish the consistency and asymptotic normality, it suffices to

18 show that the GMM estimator satisfies the sufficient conditions for Z-estimators to be consistent and asymptotically normally distributed when properly normalized and centered A similar argument has been adopted by Lee (2007) to establish the asymptotic normality of the GMM estimator for the SAR model with exogenous regressors Let µ r,s = E(u r 1,i us 2,i ) for r + s = 3, 4 By Lemmas 11 and 12 in the Appendix, we have with Ω 11 = Var[g 1 (θ 0 )] = Σ (Q Q), and Ω = Var[g(θ 0 )] = Ω 12 = E[g 1 (θ 0 )g 2 (θ 0 ) ] = Ω 11 Ω 12 Ω 12 Ω 22 µ 3,0 µ 2,1 µ 2,1 µ 1,2 µ 2,1 µ 1,2 µ 1,2 µ 0,3 9 (19) (Q ω) = Ω 22 = Var[g 2 (θ 0 )] µ 4,0 3σ1 4 µ 3,1 3σ1 2σ 12 µ 3,1 3σ1 2σ 12 µ 2,2 σ1 2σ2 2 2σ2 12 µ 2,2 σ1 2 σ2 2 2σ2 12 µ 2,2 σ1 2σ2 2 2σ2 12 µ 1,3 3σ 12 σ 2 2 µ 2,2 σ1 2 σ2 2 (ω ω) 2σ2 12 µ 1,3 3σ 12 σ2 2 µ 0,4 3σ2 4 σ1 4 σ1 2σ 12 σ1 2σ 12 σ12 2 σ1 4 σ1 2σ 12 σ1 2σ 12 σ12 2 σ12 2 σ1 2 + σ2 2 σ 12 σ2 2 σ σ2 2 σ12 2 σ 12 σ 2 2 2, σ12 2 σ 12 σ2 2 σ1 2 σ2 2 σ 12 σ2 2 σ2 4 σ2 4 where ω = [vec D (Ξ 1 ),, vec D (Ξ m )] and 1 = tr(ξ 1 Ξ 1 ) tr(ξ 1 Ξ m ) tr(ξ m Ξ 1 ) tr(ξ m Ξ m ) and 2 = tr(ξ 1 Ξ 1) tr(ξ 1 Ξ m) tr(ξ mξ 1 ) tr(ξ mξ m ) Let D = E[ θ g(θ 0)] = [D 1, D 2], (110)

19 where and D 1 = E[ θ g 1(θ 0 )] = D 2 = E[ θ g 2(θ 0 )] = Q (GX 1 β 0 + φ 0 GXγ 0 ) Q Xγ 0 Q X Q X (σ φ 0σ 12 )tr(ξ (s) 1 G) 0 1 (K X +K 1 +1) (σ1 2 + φ 0σ 12 )tr(ξ (s) m G) 0 1 (KX +K 1 +1) (σ 12 + φ 0 σ2 2)tr(Ξ 1 G) 0 1 (K X +K 1 +1) (σ 12 + φ 0 σ2 2 )tr(ξ mg) 0 1 (KX +K 1 +1) (σ 12 + φ 0 σ2 2)tr(Ξ 1G) 0 1 (KX +K 1 +1) (σ 12 + φ 0 σ2 2 )tr(ξ mg) 0 1 (KX +K 1 +1) 0 m 1 0 m (KX +K 1 +1) The following proposition establishes the consistency and asymptotic normality of the GMM estimator 10 Proposition 11 Suppose Assumptions hold Then θ gmm defined in (17) is a consistent estimator of θ 0 and has the following asymptotic distribution n( θ gmm θ 0 ) d N(0, AsyVar( θ gmm )) where AsyVar( θ gmm ) = lim n [(n 1 D) F F(n 1 D)] 1 (n 1 D) F F(n 1 Ω)F F(n 1 D)[(n 1 D) F F(n 1 D)] 1 with Ω and D defined in (113) and (214) respectively Close inspection of AsyVar( θ gmm ) reveals that the optimal F F is (n 1 Ω) 1 by the generalized Schwarz inequality The following proposition establishes the consistency and asymptotic normality of the GMM estimator with the estimated optimal weighting matrix It also suggests a over-identifying restrictions (OIR) test based on the proposed GMM estimator

20 11 Proposition 12 Suppose Assumptions hold and n 1 ˆΩ is a consistent estimator of n 1 Ω defined in (113) Then, ˆθ gmm = arg min θ Θ g(θ) ˆΩ 1 g(θ) (111) is a consistent estimator of θ 0 and n(ˆθ gmm θ 0 ) d N(0, [ lim n n 1 D Ω 1 D] 1 ), where D is defined in (110) Furthermore g(ˆθ gmm ) ˆΩ 1 g(ˆθ gmm ) d χ 2 dim(g) dim(θ) 142 Asymptotic Efficiency When only the linear moment function g 1 (θ 0 ) is used for the GMM estimation, the GMM estimator defined in (111) reduces to the generalized spatial 3SLS in Kelejian and Prucha (2004) because ˆθ 3SLS = arg min g 1 (θ) ˆΩ 1 11 g 1(θ) = arg min u(θ) ( ˆΣ 1 P)u(θ) = [Z ( ˆΣ 1 P)Z] 1 Z ( ˆΣ 1 P)y, where ˆΣ is a consistent estimator of Σ, P = Q(Q Q) 1 Q, y = (y 1, y 2 ), and It follows from Proposition 12 that Z = Wy 1 y 2 X X n(ˆθ 3SLS θ 0 ) d N(0, [ lim n n 1 D 1Ω 1 11 D 1] 1 ) As D Ω 1 D D 1Ω 1 11 D 1 = (D 2 Ω 12Ω 1 11 D 1) (Ω 22 Ω 12Ω 1 11 Ω 12) 1 (D 2 Ω 12Ω 1 11 D 1), which is positive semi-definite, the proposed GMM estimator is asymptotically more efficient than the 3SLS estimator

21 12 The asymptotic efficiency of the proposed GMM estimator depends on the choices of Q and Ξ 1,, Ξ m Following Lee (2007), our discussion on the asymptotic efficiency focuses on two cases: (i) u = (u 1, u 2 ) N(0, Σ I n ), and (ii) Ξ j has a zero diagonal for all j = 1,, m Let P be a subset of all Ξ s satisfying Assumption 15 such that diag(ξ) = 0 for all Ξ P The sub-class of quadratic moment functions using Ξ P is of a particular interest because these quadratic moment functions could be robust against unknown form of heteroskedasticity as shown in Lin and Lee (2010) Let g (θ) = [g1(δ), g2(θ) ], (112) where g1 (δ) = (I 2 Q ) u(θ) and g 2(θ) = [u 1 (δ) Ξ u 1 (δ), u 1 (δ) Ξ u 2 (γ), u 2 (γ) Ξ u 1 (δ), u 2 (γ) Ξ u 2 (γ)] In cases (i) and (ii), Ω = Var[g (θ 0 )] = Σ (Q Q ) 0 0 Ω 22 (113) where Ω 22 = σ1 4 σ1 2σ 12 σ1 2σ 12 σ12 2 σ12 2 σ1 2σ2 2 σ 12 σ2 2 σ12 2 σ 12 σ2 2 tr(ξ Ξ ) + σ1 4 σ1 2σ 12 σ1 2σ 12 σ12 2 σ1 2σ2 2 σ12 2 σ 12 σ2 2 σ1 2σ2 2 σ 12 σ2 2 tr(ξ Ξ ) σ 4 2 σ 4 2 The following proposition gives the infeasible best GMM (BGMM) estimator θ bgmm = arg min θ Θ g (θ) Ω 1 g (θ) (114) with the optimal Q and Ξ in cases (i) and (ii) respectively Proposition 13 Suppose Assumptions hold Let G = WS 1 (i) Suppose u N(0, Σ I n ) The BGMM estimator defined in (114) with Q = [GX, X] and Ξ = G n 1 tr(g)i n is the most efficient one in the class of GMM estimators defined in (17)

22 13 (ii) Without the normality assumption on u, the BGMM estimator defined in (114) with Q = [GX, X] and Ξ = G diag(g) is the most efficient one in the sub-class of GMM estimators defined in (17) with Ξ j P for all j = 1,, m Under normality, the model can be efficiently estimated by the ML estimator To get some intuition of the optimal Q and Ξ in case (i), we compare the linear and quadratic moment functions utilized by the GMM estimator with the first order partial derivatives of the log likelihood function Let G(λ) = WS(λ) 1, where S(λ) = I n λw The log likelihood function based on the joint normal distribution of y = (y 1, y 2 ) is 9 L(θ, Σ) = n ln(2π) 1 2 ln Σ I n + ln S(λ) 1 2 u(θ) (Σ I n ) 1 u(θ) with the first order partial derivatives and λ L(θ, ˆΣ) = [(φxγ + X 1 β) G(λ), 0 1 n ] ( ˆΣ I n ) 1 u(θ) + ˆσ2 2 ˆΣ u 1(δ) [G(λ) n 1 tr(g(λ))i n ]u 1 (δ) ˆσ2 12 ˆΣ u 2(γ) [G(λ) n 1 tr(g(λ))i n ]u 1 (δ) +φ ˆσ2 2 ˆΣ u 1(δ) [G(λ) n 1 tr(g(λ))i n ]u 2 (γ) φ ˆσ 12 ˆΣ u 2(γ) [G(λ) n 1 tr(g(λ))i n ]u 2 (γ) φ L(θ, Σ) = [γ X, 0 1 n ]( ˆΣ I n ) 1 u(θ) β L(θ, Σ) = [X 1, 0 K1 n]( ˆΣ I n ) 1 u(θ) γ L(θ, Σ) = [0 K X n, X ]( ˆΣ I n ) 1 u(θ) where ˆΣ is the ML estimator for Σ given by ˆΣ = ˆσ2 1 ˆσ 12 ˆσ 12 ˆσ 2 2 = n 1 u 1(δ) u 1 (δ) u 1 (δ) u 2 (γ) u 1 (δ) u 2 (γ) u 2 (γ) u 2 (γ) 9 The detailed derivation of the log likelihood function and its partial derivatives can be found in Appendix 11

23 14 Close inspection reveals the similarity between the ML and BGMM estimators under normality, as the first order partial derivatives of the log likelihood function can be treated as linear combinations of the moment functions Q (λ) u 1 (δ), Q (λ) u 2 (γ), u 1 (δ) Ξ (λ)u 1 (δ), u 1 (δ) Ξ (λ)u 2 (γ), u 2 (γ) Ξ (λ)u 1 (δ), and u 2 (γ) Ξ (λ)u 2 (γ) with Q (λ) = [G(λ)X, X] and Ξ (λ) = G(λ) n 1 tr(g(λ))i n The optimal Q and Ξ are not feasible as G involves the unknown parameter λ 0 Suppose there exists a n-consistent preliminary estimator ˆλ for λ 0 (say, the 2SLS estimator with IV matrix Q = [WX, X]) Then, the feasible optimal ˆQ and ˆΞ can obtained by replacing λ 0 in Q and Ξ by ˆλ Furthermore, suppose ˆσ 2 1, ˆσ 12, ˆσ 2 2 are consistent preliminary estimators for σ1 2, σ 12, σ2 2 Then, n 1 ˆΩ is a consistent estimator of n 1 Ω defined in (113) with the unknown parameters λ, σ1 2, σ 12, σ2 2 in Ω replaced by ˆλ, ˆσ 2 1, ˆσ 12, ˆσ 2 2 Then, the feasible BGMM estimator is given by ˆθ bgmm = arg min θ Θ ĝ (θ) ˆΩ 1 ĝ (θ), (115) where ĝ (θ) is obtained by replacing Q and Ξ in g (θ) with ˆQ and ˆΞ Following a similar argument in the proof of Proposition 3 in Lee (2007), the feasible BGMM estimator ˆθ bgmm can be shown to have the same limiting distribution as its infeasible counterpart θ bgmm Proposition 14 Suppose Assumptions hold, ˆλ is a n-consistent estimator of λ 0, and ˆΣ is a consistent estimator of Σ The feasible BGMM estimator ˆθ bgmm defined in (115) is asymptotically equivalent to the corresponding infeasible BGMM estimator θ bgmm Under Assumption 13, G = WS 1 = W + λ 0 W 2 + λ 2 0 W3 + Thus, G can be approximated by the leading order terms of the series expansion, ie W, W 2, W 3, Therefore, a convenient alternative to the BGMM estimator under normality for empirical researchers would be the GMM estimator with Q = [WX,, W m X, X] and Ξ 1 = W, Ξ 2 = W 2 n 1 tr(w 2 )I n,, Ξ m = W m n 1 tr(w m )I n, for some fixed m

24 15 15 Monte Carlo Experiments We conduct a small Monte Carlo simulation experiment to study the finite sample performance of the proposed GMM estimator The DGP considered in the experiment follows equations (11) and (12) with K 1 = K 2 = 1 In the DGP, we set λ 0 = 06 and γ 0 = (0, 1), and generate X = [X 1, X 2 ] and u = (u 1, u 2 ) as X 1 N(0, I n ), X 2 N(0, I n ), and u N(0, Σ I n ), where Σ = 1 σ 12 σ 12 1 We conduct 1000 replications in the simulation experiment for different specifications with n {245, 490}, σ 12 {01, 05, 09}, and (φ 0, β 0 ) {(05, 05), (02, 02)} From the reduce form equation (28), E(Wy 1 ) = GX 1 β 0 + φ 0 GXγ 0 Therefore, φ 0 = β 0 = 05 corresponds to the case that the IVs based on E(Wy 1 ) are relatively informative and φ 0 = β 0 = 02 corresponds to the case that the IVs based on E(Wy 1 ) are less informative Let W 0 denote the spatial weights matrix for the study of crimes across 49 districts in Columbus, Ohio, in Anselin (1988) For n = 245, we set W = I 5 W 0, and for n = 490, we set W = I 10 W 0 Let Ĝ = W(I n ˆλW) 1, where ˆλ is the 2SLS estimator of λ 0 using the IV matrix Q = [WX, W 2 X, X] Let ˆQ = [ĜX, X] In the experiment, we consider the following estimators (a) The 2SLS estimator of equation (11) with the linear moment function ˆQ u 1 (δ) (b) The 3SLS estimator of equations (11) and (12) with the linear moment function (I 2 ˆQ) u(θ) (c) The single-equation GMM (GMM-1) estimator of equation (11) with the linear moment function ˆQ u 1 (δ) and the quadratic moment function u 1 (δ) [Ĝ n 1 tr(ĝ)i n ]u 1 (δ) (d) The system GMM (GMM-2) estimator of equations (11) and (12) with the linear moment function (I 2 ˆQ) u(θ) and the quadratic moment functions u 1 (δ) [Ĝ n 1 tr(ĝ)i n ]u 1 (δ), u 1 (δ) [Ĝ n 1 tr(ĝ)i n ]u 2 (γ), u 2 (γ) [Ĝ n 1 tr(ĝ)i n ]u 1 (δ), and u 2 (γ) [Ĝ n 1 tr(ĝ)i n ]u 2 (γ)

25 16 Although the 2SLS estimator and the single-equation GMM estimator only use limited information in equation (11) and thus may not be as efficient as their counterparts (ie the 3SLS estimator and the system GMM estimator respectively) that use full information in the whole system, these estimators require weaker assumptions on the reduced form equation (12) and thus may be desirable under certain circumstances The estimation results of equation (11) are reported in Tables We report the mean and standard deviation (SD) of the empirical distributions of the estimates To facilitate the comparison of different estimators, we also report their root mean square errors (RMSE) The main observations from the experiment are summarized as follows (i) The 2SLS and 3SLS estimators of λ 0 are upwards biased with large SDs when the IVs for Wy 1 are less informative For example, when n = 245 and σ 12 = 01, the 2SLS and 3SLS estimates of λ 0 reported in Table 13 are upwards biased by about 10% The biases and SDs reduce as sample size increases The 3SLS estimators of λ 0 and β 0 perform better as σ 12 increases (ii) The single-equation GMM (GMM-1) estimator of λ 0 is upwards biased when the IVs for Wy 1 are less informative When n = 245 and σ 12 = 01, the GMM-1 estimates of λ 0 reported in Table 13 are upwards biased by about 6% The bias reduces as sample size increases The GMM-1 estimator of λ 0 reduces the SD of the 2SLS estimator The SD reduction is more significant when the IVs for Wy 1 are less informative In Table 11, when σ 12 = 01, the GMM-1 estimator reduces the SD of the 2SLS estimator by about 60% In Table 3, when σ 12 = 01, the GMM-1 estimator reduces the SD of the 2SLS estimator by about 65% (iii) The system GMM (GMM-2) estimator of λ 0 is upwards biased when the sample size is moderate (n = 245) and the IVs for Wy 1 are less informative The bias reduces as σ 12 increases When n = 490, the GMM-2 estimator is essentially unbiased even if the IVs are weak The GMM-2 estimators of λ 0 and β 0 have smaller SDs than the corresponding GMM-1 estimators The reduction in the SD is more significant when the endogeneity problem is more severe (ie σ 12 is larger) and/or the IVs for Wy 1 are less informative For example,

26 17 in Table 12, when σ 12 = 09, the GMM-2 estimator of λ 0 reduces the SD of the GMM-1 estimator by about 42% In Table 14, when σ 12 = 09, the GMM-2 estimator of λ 0 reduces the SD of the GMM-1 estimator by about 75% In both cases, the GMM-2 estimator of β 0 reduces the SD of the corresponding GMM-1 estimator by about 56% 16 Conclusion In this paper, we propose a general GMM framework for the estimation of SAR models with endogenous regressors We introduce a new set of quadratic moment conditions to construct the GMM estimator, based on the correlation structure of the spatially lagged dependent variable with the model disturbance term and with the endogenous regressor We establish the consistency and asymptotic normality of the proposed GMM estimator and discuss the optimal choice of moment conditions We also conduct a Monte Carlo experiment to show the GMM estimator works well in finite samples The proposed GMM estimator utilizes correlation across equations (11) and (12) to construct moment equations and thus can be considered as a full information estimator If we only use the moment equations based on u 1 (δ), ie, the residual function of equation (11), the proposed GMM estimator becomes a single-equation GMM estimator Although the single-equation GMM estimator may not be as efficient as the full information GMM estimator, the single-equation GMM estimator requires weaker assumptions on the reduced form equation (12) and thus may be desirable under certain circumstances The Monte Carlo experiment shows that the full information GMM estimator improves the efficiency of the single-equation GMM estimator when the endogeneity problem is severe and/or the IVs for the spatially lagged dependent variable are weak

27 18 Appendix 11 Likelihood Function of the SAR Model with Endogenous Regressors Let µ y (θ) = S 1 (λ)(φxγ + X 1 β) Xγ and R(φ, λ) = S 1 (λ) φs 1 (λ) 0 I n, where S(λ) = I n λw From the reduced form equations (12) and (27), y = (y 1, y 2 ) = µ y (θ 0 ) + R(φ 0, λ 0 )u where u = (u 1, u 2 ) Under normality, u N(0, Σ I n ), and thus y N(µ y, R(Σ I n )R ), where µ y = µ y (θ 0 ) and R = R(φ 0, λ 0 ) Hence, the log likelihood function of (11) and (12) is given by L(θ, Σ) = n ln(2π) 1 2 ln R(φ, λ)(σ I n)r (φ, λ) 1 2 [y µ y(θ)] [R(φ 0, λ 0 )(Σ I n )R(φ 0, λ 0 ) ] 1 [y µ y (θ)] As u(θ) = R 1 (φ, λ)[y µ y (θ)] and R 1 (φ, λ) = S(λ) Then, the log likelihood function can be written as L(θ, Σ) = n ln(2π) 1 2 ln (Σ I n) + ln S(λ) 1 2 u(θ) (Σ I n ) 1 u(θ) The first order partial derivatives of the log likelihood function are λ L(θ, Σ) = tr(g(λ)) + [y 1W, 0](Σ I n ) 1 u(θ) φ L(θ, Σ) = [y 2, 0](Σ I n ) 1 u(θ) β L(θ, Σ) = [X 1, 0](Σ I n ) 1 u(θ) γ L(θ, Σ) = [0, X ](Σ I n ) 1 u(θ) and (Σ I n ) 1 L(θ, Σ) = 1 2 (Σ I n) 1 2 u(θ)u(θ), (116) where G(λ) = WS(λ) 1 Since Wy 1 = G(λ)(u 1 (δ) + φu 2 (γ)) + G(λ)(φXγ + X 1 β) and y 2 =

28 19 Xγ + u 2 (γ), then λ L(θ, Σ) = tr(g(λ)) + [(φxγ + X 1β) G(λ), 0] (Σ I n ) 1 u(θ) (117) +[(u 1 (δ) + φu 2 (γ)) G(λ), 0] (Σ I n ) 1 u(θ) and φ L(θ, Σ) = [γ X, 0](Σ I n ) 1 u(θ) + [u 2 (γ), 0](Σ I n ) 1 u(θ) (118) From (116), the ML estimator for Σ is given by ˆΣ = ˆσ2 1 ˆσ 12 ˆσ 12 ˆσ 2 2 = n 1 u 1(δ) u 1 (δ) u 1 (δ) u 2 (γ) u 1 (δ) u 2 (γ) u 2 (γ) u 2 (γ) Substitution of ˆΣ into (117) and (118) gives λ L(θ, ˆΣ) = [(φxγ + X 1 β) G(λ), 0] ( ˆΣ I n ) 1 u(θ) + ˆσ2 2 ˆΣ u 1(δ) [G(λ) n 1 tr(g(λ))i n ]u 1 (δ) ˆσ2 12 ˆΣ u 2(γ) [G(λ) n 1 tr(g(λ))i n ]u 1 (δ) +φ ˆσ2 2 ˆΣ u 1(δ) [G(λ) n 1 tr(g(λ))i n ]u 2 (γ) φ ˆσ 12 ˆΣ u 2(γ) [G(λ) n 1 tr(g(λ))i n ]u 2 (γ) and Appendix 12 φ L(θ, Σ) = [γ X, 0]( ˆΣ I n ) 1 u(θ) Lemmas For ease of reference, we list some useful results without proofs Lemmas can be found (or are straightforward extensions of the lemmas) in Lee (2007) Lemma 17 is a special case of Lemma 3 in Yang and Lee (2014) Lemmas 18 and 19 are from Breusch et al (1999)

29 Lemma 11 Let A and B be n n nonstochastic matrices such that tr(a) = tr(b) = 0 Then, 20 (i) E(u 1Au 1 u 1Bu 1 ) = (µ 4,0 3σ1)vec 4 D (A) vec D (B) + σ1tr(ab 4 (s) ) (ii) E(u 1Au 1 u 1Bu 2 ) = (µ 3,1 3σ1σ 2 12 )vec D (A)vec D (B) + σ1σ 2 12 tr(ab (s) ) (iii) E(u 1Au 1 u 2Bu 2 ) = (µ 2,2 σ1σ σ12)vec 2 D (A) vec D (B) + σ12tr(ab 2 (s) ) (iv) E(u 1Au 2 u 1Bu 2 ) = (µ 2,2 σ1σ σ12)vec 2 D (A) vec D (B) + σ1σ 2 2tr(AB 2 ) + σ12tr(ab) 2 (v) E(u 1Au 2 u 2Bu 2 ) = (µ 1,3 3σ 12 σ2)vec 2 D (A) vec D (B) + σ 12 σ2tr(ab 2 (s) ) (vi) E(u 2Au 2 u 2Bu 2 ) = (µ 0,4 3σ2)vec 4 D (A) vec D (B) + σ2tr(ab 4 (s) ) Lemma 12 Let A be an n n nonstochastic matrix and c be an n 1 nonstochastic vector Then, (i) E(u 1Au 1 u 1c) = µ 3,0 vec D (A) c (ii) E(u 1Au 1 u 2c) = E(u 1Au 2 u 1c) = µ 2,1 vec D (A) c (iii) E(u 1Au 2 u 2c) = E(u 2Au 2 u 1c) = µ 1,2 vec D (A) c (iv) E(u 2Au 2 u 2c) = µ 0,3 vec D (A) c Lemma 13 Let A be an n n nonstochastic matrix with row and columns sums uniformly bounded in absolute value Then, (i) n 1 u 1 Au 1 = O p (1), n 1 u 1 Au 2 = O p (1); and (ii) n 1 [u 1 Au 1 E(u 1 Au 1)] = o p (1), n 1 [u 1 Au 2 E(u 1 Au 2)] = o p (1) Lemma 14 Let A be an n n nonstochastic matrix with row and columns sums uniformly bounded in absolute value Let c be an n 1 nonstochastic vector with uniformly bounded elements Then, n 1/2 c Au r = O p (1) and n 1 c Au r = o p (1) Furthermore, if lim n n 1 c AA c exists and is positive definite, then n 1/2 c d Au r N(0, σ 2 r lim n n 1 c AA c), for r = 1, 2 Lemma 15 Suppose n 1 [Γ(θ) Γ 0 (θ)] = o p (1) uniformly in θ Θ, where Γ 0 (θ) is uniquely identified at θ 0 Define ˆθ = arg min θ Θ Γ(θ) and ˆθ = arg min θ Θ Γ (θ) If n 1 [Γ(θ) Γ (θ)] = o p (1) uniformly in θ Θ then both ˆθ and ˆθ are consistent estimators of θ 0 Furthermore, assume

30 2 that 1 n Γ(θ) converges uniformly to a matrix which is nonsingular at θ θ θ 0 and 1 Γ(θ) = θ O p (1) If 1 n [ 2 Γ (θ) θ θ 2 Γ(θ)] = o θ θ p (1) and 1 n [ Γ (θ) Γ(θ)] = o θ θ p (1) uniformly in Θ, then n(ˆθ θ 0 ) n(ˆθ θ 0 ) = o p (1) n 21 Lemma 16 Let A and B be n n nonstochastic matrices with row and columns sums uniformly bounded in absolute value, c 1 and c 2 be n 1 nonstochastic vectors with uniformly bounded elements G is either G, G n 1 tr(g)i n or G diag(g), and Ĝ is obtained by replacing λ 0 in G by its n-consistent estimator ˆλ Suppose Assumption 13 holds Then, n 1 c 1 (Ĝ G)c 2 = o p (1), n 1/2 c 1 (Ĝ G)Au r = o p (1), n 1 u ra (Ĝ G)Bu s = o p (1), and n 1/2 u r(ĝ G)u s = o p (1), for r, s = 1, 2 Lemma 17 Let A r,s be an n n nonstochastic matrix with row and column sums uniformly bounded in absolute value for r, s = 1, 2 Let c 1 and c 2 be n 1 nonstochastic vectors with uniformly bounded elements Let σ 2 = Var(ɛ), where ɛ = 2 r=1 c ru r + 2 s=1 2 r=1 (u sa r,s u r E[u sa r,s u r ]) Suppose σ 2 = O(n) and n 1 σ 2 is bounded away from zero Then, σ 1 ɛ d N(0, 1) Lemma 18 Consider the set of moment conditions E[g(θ 0 )] = 0 with g(θ) = [g 1 (θ), g 2 (θ) ] Define D i = E[ θ g i (θ)] and Ω ij = E[g i (θ)g j (θ) ] for i, j = 1, 2 The following statements are equivalent (i) g 2 is redundant given g 1 ; (ii) D 2 = Ω 21 Ω 1 11 D 1 and (iii) there exists a matrix A such that D 2 = Ω 21 A and D 1 = Ω 11 A Lemma 19 Consider the set of moment conditions E[g(θ 0 )] = 0 with g(θ) = [g 1 (θ), g 2 (θ), g 3 (θ) ] Then (g 2, g 3 ) is redundant given g 1 if and only if g 2 is redundant given g 1 and g 3 is redundant given g 1

31 22 Appendix 13 Proofs Proof of Proposition 11: To prove consistency, first we need to show the uniform convergence of n 2 g(θ) F Fg(θ) in probability For some typical row F i of F F i g(θ) = f 1,i Q u 1 (δ) + f 2,i Q u 2 (γ) + u 1 (δ) f 1,ij Ξ j u 1 (δ) + u 1 (δ) f 2,ij Ξ j u 2 (γ) j=1 +u 2 (γ) f 3,ij Ξ j u 1 (δ) + u 2 (γ) f 4,ij Ξ j u 2 (γ) j=1 where F i = (f 1,i, f 2,i, f 1,i1,, f 1,im,, f 4,i1,, f 4,im ) and f 1,i and f 2,i are row sub-vectors As u 1 (δ) = d 1 (δ) + r 1 (δ), where d 1 (δ) = (λ 0 λ)g(φ 0 Xγ 0 + X 1 β 0 ) + (φ 0 φ)xγ 0 + X 1 (β 0 β) and r 1 (δ) = u 1 + (λ 0 λ)(gu 1 + φ 0 Gu 2 ) + (φ 0 φ)u 2, we have u 1 (δ) f 1,ij Ξ j u 1 (δ) = d 1 (δ) f 1,ij Ξ j d 1 (δ) + l 1 (δ) + q 1 (δ) j=1 where l 1 (δ) = d 1 (δ) ( m j=1 f 1,ijΞ (s) j j=1 j=1 ) r 1 (δ) and q 1 (δ) = r 1 (δ) ( m j=1 f 1,ijΞ j ) r 1 (δ) It follows by Lemmas 13 and 14 that n 1 l 1 (δ) = o p (1) and n 1 q 1 (δ) n 1 E[q 1 (δ)] = o p (1) uniformly in Θ, where E[q 1 (δ)] = (λ 0 λ)[σ σ 12 (2φ 0 φ) + σ 2 2φ 0 (φ 0 φ)] +(λ 0 λ) 2 (σ 2 2φ σ 12 φ 0 + σ 2 1) j=1 f 1,ij tr(g Ξ j G) j=1 j=1 f 1,ij tr(gξ (s) j ) Hence, n 1 u 1 (δ) ( m j=1 f 1,ijΞ j ) u 1 (δ) n 1 E[u 1 (δ) ( m j=1 f 1,ijΞ j ) u 1 (δ)] = o p (1) uniformly in Θ, where E[u 1 (δ) ( m j=1 f 1,ijΞ j ) u 1 (δ)] = d 1 (δ) ( m j=1 f 1,ijΞ j ) d 1 (δ) + E[q 1 (δ)] As u 2 (γ) = d 2 (γ) + u 2, where d 2 (γ) = X(γ 0 γ), we have u 1 (γ) f 2,ij Ξ j u 2 (δ) = d 1 (γ) f 2,ij Ξ j d 2 (δ) + l 2 (θ) + q 2 (θ) j=1 where l 2 (θ) = r 1 (δ) ( m j=1 f 2,ijΞ j ) d 2 (γ) + d 1 (δ) ( m j=1 f 2,ijΞ j ) u 2 and j=1 q 2 (δ) = r 1 (δ) f 2,ij Ξ j u 2 j=1

32 It follows by Lemmas 13 and 14 that n 1 l 2 (θ) = o p (1) and n 1 q 2 (θ) n 1 E[q 2 (θ)] = o p (1) uniformly in Θ, where E[q 2 (θ)] = (λ 0 λ)(σ 12 + σ 2 2φ 0 ) f 2,ij tr(gξ j ) Hence, n 1 u 1 (γ) ( m j=1 f 2,ijΞ j ) u 2 (δ) n 1 E[u 1 (γ) ( m j=1 f 2,ijΞ j ) u 2 (δ)] = o p (1) uniformly in Θ, where E[u 1 (γ) ( m j=1 f 2,ijΞ j ) u 2 (δ)] = d 1 (γ) ( m j=1 f 2,ijΞ j ) d 2 (δ) + E[q 2 (θ)] Similarly, n 1 u 2 (γ) f 3,ij Ξ j u 1 (δ) n 1 E[u 2 (γ) f 3,ij Ξ j u 1 (δ)] = o p (1), j=1 n 1 u 2 (γ) f 4,ij Ξ j u 2 (γ) n 1 E[u 2 (γ) f 4,ij Ξ j u 2 (γ)] = o p (1), j=1 n 1 f 1,i Q u 1 (δ) n 1 E[f 1,i Q u 1 (δ)] = o p (1), n 1 f 2,i Q u 2 (γ) n 1 E[f 2,i Q u 2 (γ)] = o p (1) j=1 j=1 j=1 and 23 uniformly in Θ Therefore, n 1 Fg(θ) n 1 FE[g(θ)] = o p (1) uniformly in Θ, and hence, 1 n 2 g(θ) F Fg(θ) converges in probability to a well defined limit uniformly in Θ As g(θ) is a quadratic function of θ, n 1 FE[g(θ)] is uniformly equicontinuous on Θ by Assumption 14 The identification condition and the uniform equicontinuity of n 1 FE[g(θ)] imply that the identification uniqueness condition for n 2 E[g(θ) ]F FE[g(θ)] must be satisfied The consistency of ˆθ follows by Theorem 151 of Peracchi (2001) For the asymptotic normality of θ gmm, by the mean value theorem, n( θgmm θ 0 ) = [ n 1 θ g( θ gmm ) F n 1 F θ g( θ) ] 1 n 1 θ g( θ gmm ) F n 1/2 Fg(θ 0 )

33 24 where θ = t θ gmm + (1 t)θ 0 for some t [0, 1] and Q Wy 1 Q y 2 Q X Q X u 1 (δ) Ξ (s) 1 Wy 1 u 1 (δ) Ξ (s) 1 y 2 u 1 (δ) Ξ (s) 1 X 1 0 u 1 (δ) Ξ (s) m Wy 1 u 1 (δ) Ξ (s) m y 2 u 1 (δ) Ξ (s) m X 1 0 u 2 (γ) Ξ 1 Wy 1 u 2 (γ) Ξ 1 y 2 u 2 (γ) Ξ 1 X 1 u 1 (δ) Ξ 1 X θ g(θ) = u 2 (γ) Ξ mwy 1 u 2 (γ) Ξ my 2 u 2 (γ) Ξ mx 1 u 1 (δ) Ξ m X u 2 (γ) Ξ 1 Wy 1 u 2 (γ) Ξ 1 y 2 u 2 (γ) Ξ 1 X 1 u 1 (δ) Ξ 1 X u 2 (γ) Ξ m Wy 1 u 2 (γ) Ξ m y 2 u 2 (γ) Ξ m X 1 u 1 (δ) Ξ mx u 2 (γ) Ξ (s) 1 X u 2 (γ) Ξ (s) m X Using Lemmas 13 and 14, it follows by a similar argument in the proof of Proposition 1 in Lee (2007) that n 1 θ g(ˆθ) n 1 D = o p (1) and n 1 θ g( θ) n 1 D = o p (1) with D given by (110) By Lemma 17 and the Cramer-Wald device, we have n 1/2 Fg(θ 0 ) d N(0, lim n n 1 FΩF ) with Ω given by (113) The desired result follows Proof of Proposition 12: Note that n 1 g(θ) ˆΩ 1 g(θ) = n 1 g(θ) Ω 1 g(θ) + n 1 g(θ) ( ˆΩ 1 Ω 1 )g(θ) With F = (n 1 Ω) 1/2, uniform convergence of n 1 g(θ) Ω 1 g(θ) in probability follows by a similar argument in the proof of Proposition 11 On the other hand, n 1 g(θ) ( ˆΩ 1 Ω 1 )g(θ) ( n 1 g(θ) ) 2 (n 1 ˆΩ) 1 (n 1 Ω) 1 where is the Euclidean norm for vectors and matrices By a similar argument in the proof of Proposition 11, we have n 1 g(θ) n 1 E[g(θ)] = o p (1) and n 1 E[g(θ)] = O(1) uniformly in Θ,

34 which in turn implies that n 1 g(θ) = O p (1) uniformly in Θ Therefore, n 1 g(θ) ( ˆΩ 1 Ω 1 )g(θ) = o p (1) uniformly in Θ The consistency of ˆθ gmm follows For the asymptotic normality of n(ˆθ gmm θ 0 ), note that from the proof of Proposition 11 we have n 1 θ g(ˆθ gmm ) n 1 D = o p (1), since ˆθ gmm is consistent Let θ = tˆθ gmm + (1 t)θ 0 for some t [0, 1], then by the mean value theorem, 25 n(ˆθ gmm θ 0 ) [ = n 1 θ g(ˆθ gmm ) (n 1 ˆΩ) 1 n 1 ] 1 θ g(ˆθ) n 1 θ g(ˆθ gmm ) (n 1 ˆΩ) 1 n 1/2 g(θ 0 ) [ = n 1 D ( n 1 Ω ) 1 1 n D] 1 n 1 D ( n 1 Ω ) 1 n 1/2 g(θ 0 ) + o p (1) which concludes the first part of the proof, since in the proof of Proposition 11 it is established that n 1/2 g(θ 0 ) converges in distribution For the overidentification test, by the mean value theorem, for some t [0, 1] and θ = tˆθ gmm + (1 t)θ 0 n 1/2 g(ˆθ gmm ) = n 1/2 g(θ 0 ) + n 1/2 θ g( θ)(ˆθ gmm θ 0 ) = n 1/2 g(θ 0 ) n 1 D n(ˆθ gmm θ 0 ) + o p (1) = An 1/2 g(θ 0 ) + o p (1) [ where A = I dim(g) n 1 D n 1 D ( n 1 Ω ) 1 1 n D] 1 n 1 D ( n 1 Ω ) 1 Therefore g(ˆθ gmm ) ˆΩ 1 g(ˆθ gmm ) = n 1/2 g(ˆθ gmm ) (n 1 Ω) 1 n 1/2 g(ˆθ gmm ) + o p (1) = n 1/2 g(θ 0 ) A (n 1 Ω) 1 An 1/2 g(θ 0 ) + o p (1) = [ ( n 1 Ω ) 1/2 n 1/2 g(θ 0 )] B[ ( n 1 Ω ) 1/2 n 1/2 g(θ 0 )] + o p (1) where B = I dim(g) ( n 1 Ω ) [ 1/2 n 1 D n 1 D ( n 1 Ω ) 1 1 n D] 1 n 1 D ( n 1 Ω ) 1/2 Therefore g(ˆθ gmm ) ˆΩ 1 g(ˆθ gmm ) d χ 2 tr(b), where tr(b) = dim(g) dim(θ)

35 26 Proof of Proposition 13: To establish the asymptotic efficiency, we use an argument by Breusch et al (1999) to show that any additional moment conditions g defined in (16) given g defined in (24) will be redundant Following Breusch et al (1999), g is redundant given g if the asymptotic variance of an estimator based on moment equations E[g(θ)] = 0 and E[g (θ)] = 0 is the same as an estimator based on E[g (θ)] = 0 In cases (i) and (ii), Ω # = E[g(θ 0 )g (θ 0 ) ] = (Σ Q Q ) 0 0 Ω # 22 where Ω # 22 = σ1 4 σ1 2σ 12 σ1 2σ 12 σ 2 12 σ1 2 σ 12 σ12 2 σ1 2σ2 2 σ2 2σ tr(ξ 1 Ξ ) 12 σ1 2 σ 12 σ1 2σ2 2 σ12 2 σ2 2σ 12 tr(ξ m Ξ ) σ12 2 σ2 2σ 12 σ2 2σ 12 σ2 4 σ1 4 σ1 2σ 12 σ1 2σ 12 σ 2 12 σ1 2 + σ 12 σ1 2σ2 2 σ12 2 σ2 2σ tr(ξ 1 12 Ξ ) σ1 2 σ 12 σ12 2 σ1 2σ2 2 σ2 2σ 12 tr(ξ mξ ) σ12 2 σ2 2σ 12 σ2 2σ 12 σ2 4 Let A = 1 σ1 2σ2 2 σ2 12 σ2 2(Cβ 0 + φ 0 γ 0 ) σ2 2γ 0 σ2 2C σ 12I KX σ 12 (Cβ 0 + φ 0 γ 0 ) σ 12 γ 0 σ 12 C σ1 2I K X σ φ 0 σ σ φ 0 σ , where C = [I K1, 0] and X 1 = XC Then D = Ω # A, where D is defined in (110) Based on Lemma 18 g is redundant given g Furthermore, Lemma 19 tells us that any subset of g is

36 27 redundant given g Proof of Proposition 14: To show the desired result, we only need to show ˆΓ(θ) = ĝ (θ) ˆΩ 1 ĝ (θ) and Γ(θ) = g (θ) Ω 1 g (θ) satisfy the conditions of Lemma 15 First, n 1 [ĝ 1 (θ) g 1 (θ)] = n 1 [I 2 ( ˆQ Q )] u(θ), n 1 [ĝ2,rs (θ) g 2,rs (θ)] = n 1 u r (θ) (ˆΞ Ξ )u s (θ), θ g (θ) = Q Wy 1 Q y 2 Q X Q X u 1 (δ) Ξ (s) Wy 1 u 1 (δ) Ξ (s) y 2 u 1 (δ) Ξ (s) X 1 0 u 2 (γ) Ξ Wy 1 u 2 (γ) Ξ y 2 u 2 (γ) Ξ X 1 u 1 (δ) Ξ X u 2 (γ) Ξ Wy 1 u 2 (γ) Ξ y 2 u 2 (γ) Ξ X 1 u 1 (δ) Ξ X u 2 (γ) Ξ (s) X, and 2 θ θ g (θ) = 0 θ u 1(δ) Ξ u θ 1 (δ) θ u 1(δ) Ξ θ u 2 (γ) θ u 2(γ) Ξ θ u 1 (δ) θ u 2(γ) Ξ θ u 2 (γ) where Q = [GX, X], Ξ is either G n 1 tr(g)i n or G diag(g), and θ u 2 (γ) = [0, 0, 0, X], θ u 1 (δ) = [Wy 1, y 2, X 1, 0], By inspection of each term of the above matrices, we conclude n 1 [ĝ (θ) g (θ)] = o p (1), n 1 [ ĝ (θ) g (θ)] = o θ θ p (1) and n 1 [ 2 ĝ (θ) θ θ 2 g (θ)] = θ θ o p (1) uniformly in Θ by Lemma 16 Second, as Ĝ G = (ˆλ λ 0 )G 2 + (ˆλ λ 0 ) 2 ĜG 2, we have n 1 tr(ˆξ ˆΞ ) n 1 tr(ξ Ξ ) = o p (1) and n 1 tr(ˆξ ˆΞ ) n 1 tr(ξ Ξ ) = o p (1) Therefore, as ˆΣ is a consistent estimator of Σ, we have n 1 ( ˆΩ Ω ) = o p (1) Hence, we can conclude that n 1 [ˆΓ(θ) Γ(θ)] = o p (1) and n 1 [ 2 ˆΓ(θ) θ θ 2 Γ(θ)] = o θ θ p (1) uniformly in Θ Finally, since n 1/2 g (θ 0 ) = O p (1) by a similar argument in the proof of Proposition 11 and n 1/2 [ĝ (θ 0 )

37 28 g (θ 0 )] = o p (1) by Lemma 16, n 1/2 [ θ ˆΓ(θ 0 ) θ Γ(θ 0)] = 2 θ ĝ (θ 0 ) ˆΩ 1 n 1/2 [ĝ (θ 0 ) g (θ 0 )] + 2[ θ ĝ (θ 0 ) ˆΩ 1 θ g (θ 0 ) Ω 1 ]n 1/2 g (θ 0 ) = o p (1) The desired result follows Appendix 14 Tables Table 11: 2SLS, 3SLS and GMM Estimation (n = 245) λ 0 = 03 φ 0 = 10 β 0 = 05 σ 12 = 01 2SLS 0601(0128)[0128] 0497(0068)[0068] 0496(0066)[0066] 3SLS 0601(0126)[0126] 0497(0068)[0068] 0496(0066)[0066] GMM (0052)[0052] 0499(0066)[0066] 0498(0065)[0065] GMM (0051)[0051] 0497(0067)[0068] 0498(0065)[0065] σ 12 = 05 2SLS 0601(0138)[0138] 0496(0068)[0068] 0495(0066)[0066] 3SLS 0602(0111)[0111] 0496(0068)[0068] 0498(0057)[0057] GMM (0046)[0046] 0501(0066)[0066] 0498(0065)[0065] GMM (0045)[0045] 0495(0068)[0068] 0499(0057)[0057] σ 12 = 09 2SLS 0601(0170)[0170] 0495(0070)[0070] 0494(0067)[0067] 3SLS 0603(0059)[0059] 0497(0068)[0068] 0500(0029)[0029] GMM (0050)[0050] 0503(0066)[0066] 0498(0065)[0065] GMM (0023)[0023] 0495(0070)[0070] 0500(0028)[0028] Mean(SD)[RMSE]