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Testing Spatial Dependence in Spatial Models with Endogenous Weights Matrices

  • Anil K. Bera , Osman Doğan EMAIL logo and Süleyman Taşpınar
Published/Copyright: October 9, 2018
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Journal of Econometric Methods
From the journal Journal of Econometric Methods Volume 8 Issue 1

Abstract

In this study, we propose simple test statistics for identifying the source of spatial dependence in spatial autoregressive models with endogenous weights matrices. Elements of the weights matrices are modelled in such a way that endogenity arises when the unobserved factors that affect elements of the weights matrices are correlated with the unobserved factors in the outcome equation. The proposed test statistics are robust to the presence of endogeneity in the weights and can be used to detect spatial dependence in the dependent variable and/or the disturbance terms. The robust test statistics are easy to calculate as computationally simple estimations are needed for their calculations. Our Monte Carlo results indicate that these tests have good size and power properties in finite samples. We also provide an empirical illustration to demonstrate the usefulness of the robust tests in identifying the source of spatial dependence.

Keywords: endogenous spatial weights matrix; inference; Lagrange multiplier test; LM test; parametric misspecification; Rao’s score test; robust LM test; SARAR model; specification testing
JEL Classification: C13; C21; C31

Acknowledgment

We are grateful to a co-editor and two anonymous referees for their constructive comments.

  1. Funding

    This research was supported, in part, by a grant of computer time from the City University of New York High Performance Computing Center under NSF Grants CNS-0855217 and CNS-0958379.

A First and Second Order Derivatives

From (9), the first order derivatives are

∂ ln ⁡ L ( θ ) ∂ λ = 1 σ ξ 2 ( R ( ρ ) W Y ) ′ ξ ( θ ) − t r ( W S − 1 ( λ ) ) , ∂ ln ⁡ L ( θ ) ∂ δ = 1 σ ξ 2 ε ( θ ) ′ ξ ( θ ) , ∂ ln ⁡ L ( θ ) ∂ ρ = 1 σ ξ 2 ( M ( S ( λ ) Y − X 1 β ) ) ′ ξ ( θ ) − t r ( M R − 1 ( ρ ) ) , ∂ ln ⁡ L ( θ ) ∂ σ ξ 2 = − n 2 σ ξ 2 + 1 2 σ ξ 4 ξ ′ ( θ ) ξ ( θ ) , ∂ ln ⁡ L ( θ ) ∂ β = 1 σ ξ 2 ( R ( ρ ) X 1 ) ′ ξ ( θ ) , ∂ ln ⁡ L ( θ ) ∂ α = − n 2 ∂ ln ⁡ | Σ ε | ∂ α − 1 2 ∂ t r ( Σ ε − 1 ε ′ ( Γ ) ε ( Γ ) ) ∂ α .

Note that ∂ln⁡|Σε|∂α has the jth element that is equal to tr(Σε−1∂Σε∂αj). Similarly, the jth element of ∂tr(Σε−1ε′(Γ)ε(Γ))∂α is equal to −tr(Σε−1∂Σε∂αjΣε−1ε′(Γ)ε(Γ)).

The derivation of the first order derivative with respect to Γ requires some well-known properties. Let A, B and C be matrices with appropriate dimensions, then

  1. vec(AB) = (I ⊗ A)vec(B) = (B′ ⊗ I)vec(A),

  2. vec′(B′)vec(A) = tr(BA) = tr(AB) = vec′(A′)vec(B),

where I is an identity matrix that has appropriate dimension. Using these properties, it can be shown that

∑ i = 1 n ( Z i ′ − X i ′ Γ ) Σ ε − 1 ( Z i − Γ ′ X i ) = vec ′ ( ( Z − X Γ ) ′ ) ( I n ⊗ Σ ε − 1 ) vec ( ( Z − X Γ ) ′ ) = vec ′ ( ( Z − X Γ ) ′ ) vec ( Σ ε − 1 ( Z − X Γ ) ′ ) = vec ′ ( ( Z − X Γ ) Σ ε − 1 ) vec ( ( Z − X Γ ) ) = vec ′ ( Z − X Γ ) ( Σ ε − 1 ⊗ I n ) vec ( Z − X Γ ) .

Also, vec(XΓ) = (Ip2 + p3 ⊗ X)vec(Γ) and vec′(XΓ) = vec′(Γ)(Ip2 + p3 ⊗ X′). Thus,

∂ ln ⁡ L ( θ ) ∂ vec ( Γ ) = ( I p 2 + p 3 ⊗ X ′ ) ( Σ ε − 1 ⊗ I n ) vec ( Z − X Γ ) − 1 σ ξ 2 δ ⊗ ( X ′ ξ ( θ ) ) = ( Σ ε − 1 ⊗ X ′ ) vec ( Z − X Γ ) − 1 σ ξ 2 δ ⊗ ( X ′ ξ ( θ ) ) .

The second order derivatives are listed as follows:

∂ 2 ln ⁡ L ( θ ) ∂ λ 2 = − 1 σ ξ 2 ( R ( ρ ) W Y ) ′ R ( ρ ) W Y − t r ( [ W S − 1 ( λ ) ] 2 ) , ∂ 2 ln ⁡ L ( θ ) ∂ λ ∂ ρ = − 1 σ ξ 2 ( M W Y ) ′ ξ ( θ ) − 1 σ ξ 2 ( R ( ρ ) W Y ) ′ M ( S ( λ ) Y − X 1 β ) , ∂ 2 ln ⁡ L ( θ ) ∂ λ ∂ σ ξ 2 = − 1 σ ξ 4 ( R ( ρ ) W Y ) ′ ξ ( θ ) , ∂ 2 ln ⁡ L ( θ ) ∂ λ ∂ β = − 1 σ ξ 2 X 1 ′ R ( ρ ) ′ R ( ρ ) W Y , ∂ 2 ln ⁡ L ( θ ) ∂ λ ∂ vec ( Γ ) = 1 σ ξ 2 δ ⊗ X ′ R ( ρ ) W Y , ∂ 2 ln ⁡ L ( θ ) ∂ λ ∂ δ = − 1 σ ξ 2 ε ′ ( Γ ) R ( ρ ) W Y , ∂ 2 ln ⁡ L ( θ ) ∂ λ ∂ α = 0 J × 1 ,

∂ 2 ln ⁡ L ( θ ) ∂ ρ 2 = − 1 σ ξ 2 ( S ( λ ) Y − X 1 β ) ′ M ′ M ( S ( λ ) Y − X 1 β ) − t r ( [ M R − 1 ( ρ ) ] 2 ) , ∂ 2 ln ⁡ L ( θ ) ∂ ρ ∂ σ ξ 2 = − 1 σ ξ 4 ( M ( S ( λ ) Y − X 1 β ) ) ′ ξ ( θ ) , ∂ 2 ln ⁡ L ( θ ) ∂ ρ ∂ β = − 1 σ ξ 2 X 1 ′ M ′ ξ ( θ ) − 1 σ ξ 2 X 1 ′ R ′ ( ρ ) M ( S ( λ ) Y − X 1 β ) , ∂ 2 ln ⁡ L ( θ ) ∂ ρ ∂ vec ( Γ ) = 1 σ ξ 2 δ ⊗ X ′ M ( S ( λ ) Y − X 1 β ) , ∂ 2 ln ⁡ L ( θ ) ∂ ρ ∂ δ = − 1 σ ξ 2 ε ′ ( Γ ) M ( S ( λ ) Y − X 1 β ) , ∂ 2 ln ⁡ L ( θ ) ∂ ρ ∂ α = 0 J × 1 ,

∂ 2 ln ⁡ L ( θ ) ∂ σ ξ 2 ∂ σ ξ 2 = n 2 σ ξ 4 − 1 σ ξ 6 ξ ′ ( θ ) ξ ( θ ) , ∂ 2 ln ⁡ L ( θ ) ∂ σ ξ 2 ∂ β = − 1 σ ξ 4 X 1 ′ R ′ ( ρ ) ξ ( θ ) , ∂ 2 ln ⁡ L ( θ ) ∂ σ ξ 2 ∂ vec ( Γ ) = 1 σ ξ 4 δ ⊗ X ′ ξ ( θ ) , ∂ 2 ln ⁡ L ( θ ) ∂ σ ξ 2 ∂ δ = − 1 σ ξ 4 ε ′ ( Γ ) ξ ( θ ) , ∂ 2 ln ⁡ L ( θ ) ∂ σ ξ 2 ∂ α = 0 J × 1 ,

∂ 2 ln ⁡ L ( θ ) ∂ β ∂ β ′ = − 1 σ ξ 2 X 1 ′ R ′ ( ρ ) R ( ρ ) X 1 , ∂ 2 ln ⁡ L ( θ ) ∂ β ∂ vec ′ ( Γ ) = 1 σ ξ 2 δ ′ ⊗ X 1 ′ R ′ ( ρ ) X , ∂ 2 ln ⁡ L ( θ ) ∂ β ∂ δ ′ = − 1 σ ξ 2 X 1 ′ R ′ ( ρ ) ε ( Γ ) , ∂ 2 ln ⁡ L ( θ ) ∂ β ∂ α ′ = 0 k 1 × J ,

∂ 2 ln ⁡ L ( θ ) ∂ vec ( Γ ) ∂ vec ′ ( Γ ) = − ( Σ ε − 1 ⊗ X ′ X ) − 1 σ ξ 2 ( δ δ ′ ⊗ X ′ X ) , ∂ 2 ln ⁡ L ( θ ) ∂ vec ( Γ ) ∂ δ ′ = − 1 σ ξ 2 ( I p 2 + p 3 ⊗ X ′ ξ ( θ ) ) + 1 σ ξ 2 δ ⊗ X ′ ε ( Γ ) , ∂ 2 ln ⁡ L ( θ ) ∂ vec ( Γ ) ∂ α ′ = ( I p 2 + p 3 ⊗ X ′ ε ( Γ ) ) ∂ vec ( Σ ε − 1 ) ∂ α ′ .

Note that above we used the fact that in the first order derivative with respect to vec(Γ), only the first term involves Σε−1 and that term can be written as (Σε−1⊗X′)vec(ε(Γ))=X′ε(Γ)Σε−1=(Ip2+p3⊗X′ε(Γ))vec(Σε−1).

∂ 2 ln ⁡ L ( θ ) ∂ δ ∂ δ ′ = − 1 σ ξ 2 ε ′ ( Γ ) ε ( Γ ) , ∂ 2 ln ⁡ L ( θ ) ∂ δ ∂ α ′ = 0 ( p 2 + p 3 ) × J ,

∂ 2 ln ⁡ L ( θ ) ∂ α ∂ α ′ = − n 2 ∂ 2 ln ⁡ | Σ ε | ∂ α ∂ α ′ − 1 2 ∂ 2 t r ( Σ ε − 1 ε ′ ( Γ ) ε ( Γ ) ) ∂ α ∂ α ′ .

Note that ∂2ln⁡|Σε|∂αj∂αk is equal to −tr(Σε−1∂Σε∂αjΣε−1∂Σε∂αk). Similarly, ∂2tr(Σε−1ε′(Γ)ε(Γ))∂αj∂αk is equal to tr(Σε−1[∂Σε∂αkΣε−1∂Σε∂αj+∂Σε∂αjΣε−1∂Σε∂αk]Σε−1ε′(Γ)ε(Γ)).

For notational simplicity, denote H = MR−1, G = WS−1, and G¯=RGR−1. Define B(s) = B + B′ for any square matrix B. Then, we have

E [ ∂ 2 ln ⁡ L ( θ 0 ) ∂ θ ∂ θ ′ ] = 1 σ ξ 0 2 ( I ρ ρ I ρ λ − E [ ( ε δ 0 ) ′ H ′ R X 1 ] δ 0 ′ ⊗ E [ ( ε δ 0 ) ′ H ′ X ] − t r ( E [ H ′ ] ) 0 1 × J − E [ ( ε δ 0 ) ′ H ′ ε ] ∗ I λ λ I λ β ′ I λ vec ′ ( Γ ) − t r ( E [ G ¯ ′ ] ) 0 1 × J I λ δ ′ ∗ ∗ − E [ X 1 ′ R ′ R X 1 ] δ 0 ′ ⊗ E [ X 1 ′ R ′ X ] 0 k 1 × 1 0 k 1 × J 0 k 1 × ( p 2 + p 3 ) ∗ ∗ ∗ I vec ( Γ ) vec ′ ( Γ ) 0 k p × k p 0 k p × J 0 k p × ( p 2 + p 3 ) ∗ ∗ ∗ ∗ − n 2 σ ξ 0 2 0 1 × J 0 1 × ( p 2 + p 3 ) ∗ ∗ ∗ ∗ ∗ I α α ′ 0 J × ( p 2 + p 3 ) ∗ ∗ ∗ ∗ ∗ ∗ − n Σ ε 0 ) ,

where kp = (k2 + k3)(p2 + p3),

I ρ ρ = − E ( δ 0 ′ ε ′ H ′ H ε δ 0 ) − σ ξ 0 2 t r ( E [ H 2 + H H ′ ] ) , I ρ λ = − E [ ( ε δ 0 ) ′ H ′ G ¯ ( X 1 β 0 + ε δ 0 ) ] − σ ξ 0 2 t r ( E [ H ( s ) G ¯ ] ) , I λ λ = − E [ ( X 1 β 0 + ε δ 0 ) ′ G ¯ ′ G ¯ ( X 1 β 0 + ε δ 0 ) ] − σ ξ 0 2 t r ( E [ G ¯ 2 + G ¯ G ¯ ′ ] ) , I λ β ′ = − E [ ( X 1 β 0 + ε δ 0 ) ′ G ¯ ′ R X 1 ] , I λ vec ′ ( Γ ) = δ 0 ′ ⊗ E [ ( X 1 β 0 + ε δ 0 ) ′ G ¯ ′ X ] , I λ δ ′ = − E [ ( X 1 β 0 + ε δ 0 ) ′ G ¯ ′ ε ] , I vec ( Γ ) vec ′ ( Γ ) = − ( σ ξ 0 2 Σ ε 0 − 1 + δ 0 δ 0 ′ ) ⊗ X ′ X , I α α ′ [ k , j ] = − n σ ξ 0 2 2 t r ( Σ ε 0 − 1 ∂ Σ ε 0 ∂ α j Σ ε 0 − 1 ∂ Σ ε 0 ∂ α k ) .

B Detailed Expressions for Test Statistics

Let θ~ be the restricted MLE under the joint null H0 : ρ0 = λ0 = 0 and δ0 = 0. Then, evaluating the required first order conditions at θ~ yields

(48) L λ ( θ ~ ) = 1 n σ ~ ξ 2 ( W Y ) ′ ξ ~ , L ρ ( θ ~ ) = 1 n σ ~ ξ 2 ( M ( Y − X 1 β ~ ) ) ′ ξ ~ , L δ ( θ ~ ) = 1 n σ ~ ξ 2 ε ~ ′ ξ ~ .

Similarly, I(θ~) is given by

I ( θ ~ ) = 1 n σ ~ ξ 2 ( I ρ ρ I ρ λ 0 0 0 0 1 × J 0 ∗ I λ λ I λ β ′ 0 0 0 1 × J I λ δ ′ ∗ ∗ X 1 ′ X 1 0 0 k 1 × 1 0 k 1 × J 0 k 1 × ( p 2 + p 3 ) ∗ ∗ ∗ I vec ( Γ ) vec ′ ( Γ ) 0 k p × k p 0 k p × J 0 k p × ( p 2 + p 3 ) ∗ ∗ ∗ ∗ n 2 σ ~ ξ 2 0 1 × J 0 1 × ( p 2 + p 3 ) ∗ ∗ ∗ ∗ ∗ I α α ′ 0 J × ( p 2 + p 3 ) ∗ ∗ ∗ ∗ ∗ ∗ n Σ ~ ε ) ,

where kp = (k2 + k3)(p2 + p3),

I ρ ρ = σ ~ ξ 2 t r ( M 2 + M M ′ ) , I ρ λ = σ ~ ξ 2 t r ( M ( s ) W ) , I λ λ = ( X 1 β ~ ) ′ W ′ W X 1 β ~ + σ ~ ξ 2 t r ( W 2 + W W ′ ) , I λ β ′ = ( X 1 β ~ ) ′ W ′ X 1 , I λ δ ′ = ( X 1 β ~ ) ′ W ′ ε ~ , I vec ( Γ ) vec ′ ( Γ ) = ( σ ~ ξ 2 Σ ~ ε − 1 ) ⊗ X ′ X , I α α ′ [ k , j ] = n σ ~ ξ 2 2 t r ( Σ ~ ε 0 − 1 ∂ Σ ε 0 ∂ α j Σ ~ ε − 1 ∂ Σ ε 0 ∂ α k ) .

C Proofs of Propositions

Proof of Proposition 1

We first discuss the identification conditions stated in Assumption 5 and then give the asymptotic arguments for the consistency and asymptotic normality of the MLE.

Recall that

(49) 1 n [ Q ( θ ) − Q ( θ 0 ) ] = 1 2 E ( ln ⁡ | R ( ρ ) S ( λ ) S − 1 R − 1 R − 1 ′ S − 1 ′ S ′ ( λ ) R ′ ( ρ ) | 1 / n ) − 1 2 ln ⁡ ( σ ξ ∗ 2 / σ ξ 0 2 ) − 1 2 [ ∑ i = 1 p 2 + p 3 ( ω i − ln ⁡ ω i − 1 ) ] − 1 2 n ∑ i = 1 n X i ′ ( Γ 0 − Γ ) Σ ε − 1 ( Γ 0 − Γ ) ′ X i ,

where ωi is the ith eigenvalue of Σε01/2Σε−1Σε01/2. Since Σε01/2Σε−1Σε01/2 is positive definite, ωi for i = 1, 2, …, (p2 + p3) are positive. Therefore, (ωi − lnωi − 1) is non-negative, i.e. it is positive for ωi ≠ 1 and is zero for ωi = 1.

There are two cases that lead to 1n[Q(θ)−Q(θ0)]<0 for all θ ≠ θ0. Suppose E(H′H) is a positive definite matrix. Then, 1n[(λ0−λ),(β0−β)′,δ′(Γ−Γ0)′,(δ0−δ)′]E(H′H)[(λ0−λ),(β0−β)′,δ′(Γ−Γ0)′,(δ0−δ)′]′/σξ02>0 and λ0, β0, Γ0 and δ0 can be identified and (49) reduces to

(50) 1 n [ Q ( θ ) − Q ( θ 0 ) ] = 1 2 E [ ln ⁡ | R ( ρ ) R − 1 R − 1 ′ R ′ ( ρ ) | 1 / n ] − 1 2 ln ⁡ [ 1 n t r ( E [ R ( ρ ) R − 1 R − 1 ′ R ′ ( ρ ) ] ) + 1 n ( ρ 0 − ρ ) ′ δ 0 ′ Σ ε 0 σ ξ 0 2 δ 0 ( ρ 0 − ρ ) ] .

Since Σε0 is positive definite by assumption, 1n(ρ0−ρ)′δ0′Σε0σξ02δ0(ρ0−ρ)>0. Note that by the strict inequality of arithmetic and geometric means, we have 1ntr(E[R(ρ)R−1R−1′R′(ρ)])>E|R(ρ)R−1R−1′R′(ρ)|1/n. Then, by Jensen’s inequality ln⁡E(|R(ρ)R−1R−1′R′(ρ)|1/n)≥Eln⁡(|R(ρ)R−1R−1′R′(ρ)|1/n). The strict inequality of arithmetic and geometric means hold if R′(ρ)R(ρ) is linearly independent of R′R with probability one for any ρ ≠ ρ0.

In the second case, the rank condition for E(H′H) may be deficient. Hence λ0, β0, Γ0 and δ0 cannot be separately identified. In that case, 1n(ρ0−ρ)′δ0′Σε0σξ02δ0(ρ0−ρ)>0 can be used to identify ρ0. Then, using the strict inequality of arithmetic and geometric means hold by assuming that S′(λ)R′(ρ)S(λ)R(ρ) is linearly independent of S′R′SR with probability one for any ρ ≠ ρ0, λ0 can be identified. Then, 1n[(β0−β)′,δ′(Γ−Γ0)′,(δ0−δ)′]E([RX1,X,ε(Γ0)]′[RX1,X,ε(Γ0)])[(β0−β)′,δ′(Γ−Γ0)′,(δ0−δ)′]′/σξ02>0 to identify β0, Γ0 and δ0. These results are summarized in Assumption 5.

The asymptotic analysis will involve evaluation of the terms in the log-likelihood such as 1nX1′G′R′RGX1, 1nX1′G′M′MGX1, 1nX1′G′M′RGX1, 1nX1′G′R′G¯ε(Γ0), 1nX1′G′M′HG¯X1, 1nX1′G′M′G¯ε(Γ0), 1nX′G¯ε(Γ0), 1nX′HG¯ε(Γ0), 1nε′(Γ0)G¯′G¯ε(Γ0), 1nε′(Γ0)G¯′HG¯ε(Γ0), 1nε′(Γ0)G¯′H′HG¯ε(Γ0), 1nε′(Γ0)G¯′ξ(θ0), 1nε′(Γ0)G¯′G¯ξ(θ0), 1nε′(Γ0)G¯′HG¯ξ(θ0), 1nε′(Γ0)G¯′G¯ξ(θ0), 1nε′(Γ0)G¯′H′HG¯ξ(θ0), 1nξ′(θ0)G¯′G¯ξ(θ0), 1nξ′(θ0)H′Hξ(θ0), 1nξ′(θ0)G¯′H′HG¯ξ(θ0), 1nln⁡|S(λ)|=1n∑i=1n(∑l=1∞λllWiil) and 1nln⁡|R(ρ)|=1n∑i=1n(∑l=1∞ρllMiil). The last two equalities are derived in Qu and Lee (2013), and are useful to establish their convergence to their expectations under NED.

To prove consistency of the MLE θ^, we have to show uniform convergence of the elements of the log-likelihood to their expectations. Along with the identifiable uniqueness of the population parameters, this uniform convergence is sufficient for the consistency of the MLE. To show supθ∈Θ|1nln⁡L(θ)−1nQ(θ)|→p0, first pointwise convergence can be established using Proposition 1 in Qu and Lee (2015). In order to satisfy the conditions of their Proposition 1, counterparts of Claims C.1.5 C.1.6, C.2.5 and C.2.6 in Qu and Lee (2015) are needed for the matrices M and H. However, these counterparts are trivial to obtain. Hence, Proposition 1 can be used in our setup to show pointwise convergence of the log-likelihood function to its expectation. To show uniform convergence, stochastic equicontinuity of 1nln⁡L(θ) needs to be established. Note that except ln|S(λ) and ln|R(ρ)| all parameters enter the log-likelihood function as polynomials. Since the assumed parameter space is compact, uniform convergence of these terms follow. Equicontinuity of 1nln⁡|S(λ) and 1nln⁡|R(ρ)| can be established as in Qu and Lee (2015).

For the asymptotic normality of the MLE θ^, first Proposition 2 in Qu and Lee (2015) can be used to establish asymptotic normality of 1n∂ln⁡L(θ0)∂θ, i.e. 1n∂ln⁡L(θ0)∂θ→dN(0, I). A mean value expansion of 1n∂ln⁡L(θ^)∂θ around θ0 along with uniform convergence of the Hessian matrix to its expectation (and applying Cramér’s theorem) yield the desired result. To show uniform convergence of the Hessian matrix to its expectation, Corollary 1 in Qu and Lee (2015) can be used. Thus, supθ∈Θ‖1n∂2ln⁡L(θ)∂θ∂θ′−E[1n∂2ln⁡L(θ)∂θ∂θ′]‖→p0. Furthermore, we assume that E[−1n∂2ln⁡L(θ)∂θ∂θ′] is a nonsingular matrix. Hence, by continuity of the inverse at a nonsingular matrix, the inverse Hessian converges uniformly to the inverse of the expectation of the Hessian. Then, the information matrix equality yields the desired result.    □

Proof of Proposition 2

The Taylor expansions of nLψ(θ~), nLϕ(θ~) and nLγ(θ~) around θ0 under HAψ and HAϕ can be written as

n L ψ ( θ ~ ) = n L ψ ( θ 0 ) − L ψ ψ ( θ 0 ) δ ψ − L ψ ϕ ( θ 0 ) δ ϕ + n L ψ γ ( θ 0 ) ( γ ~ − γ 0 ) + o p ( 1 ) , n L ϕ ( θ ~ ) = n L ϕ ( θ 0 ) − L ϕ ϕ ( θ 0 ) δ ϕ − L ϕ ψ ( θ 0 ) δ ψ + n L ϕ γ ( θ 0 ) ( γ ~ − γ 0 ) + o p ( 1 ) , n L γ ( θ ~ ) = n L γ ( θ 0 ) − L γ ψ ( θ 0 ) δ ψ − L γ ϕ ( θ 0 ) δ ϕ + n L γ γ ( θ 0 ) ( γ ~ − γ 0 ) + o p ( 1 ) .

Using the fact that nLγ(θ~)=0 in the Taylor expansions above, we can get the following results.

(51) n L ψ ( θ ~ ) = [ I k ψ ,   − I ψ γ I γ γ − 1 ] × [ 1 n ∂ ln ⁡ L ( θ 0 ) ∂ ψ 1 n ∂ ln ⁡ L ( θ 0 ) ∂ γ ] + [ I ψ ψ − I ψ γ I γ γ − 1 I γ ψ ] δ ψ + [ I ψ ϕ − I δ γ I γ γ − 1 I γ ϕ ] δ ϕ + o p ( 1 ) ,

(52) n L ϕ ( θ ~ ) = [ I k ϕ ,   − I ϕ γ I γ γ − 1 ] × [ 1 n ∂ ln ⁡ L ( θ 0 ) ∂ ϕ 1 n ∂ ln ⁡ L ( θ 0 ) ∂ γ ] + [ I ϕ ϕ − I ϕ γ I γ γ − 1 I γ ϕ ] δ ϕ + [ I ϕ ψ − I ψ γ I γ γ − 1 I γ ψ ] δ ψ + o p ( 1 ) .

Using the asymptotic normality of score functions from Proposition 1, we can show that nLψ(θ~)→dN(Iψ⋅γδψ+Iψϕ⋅γδϕ,Iψ⋅γ), where Iψ⋅γ=[Iψψ−IψγIγγ−1Iγψ] and Iψϕ⋅γ=[Iψϕ−IψγIγγ−1Iγϕ]. Then, under HAψ and HAϕ, this result implies that

(53) LM ψ ( θ ~ ) → d χ k ψ 2 ( φ 1 ) ,

where φ1=δψ′Iψ⋅γδψ+δψ′Iψϕ⋅γδϕ+δϕ′Iψϕ⋅γ′δψ+δϕ′Iψϕ⋅γ′Iψ⋅γ−1Iψϕ⋅γ(θ0)δϕ. Thus, the first result in Proposition 2 follows from (53).

Note that the adjusted score function in Proposition 2 can be written as

(54) n L ψ ⋆ ( θ ~ ) = [ n L ψ ( θ ~ ) − I ψ ϕ ⋅ γ ( θ ~ ) I ϕ ⋅ γ − 1 ( θ ~ ) n L ϕ ( θ ~ ) ] = [ I k ψ ,   I ψ ϕ ⋅ γ I ϕ ⋅ γ − 1 ] × [ n L δ ( θ ~ ) n L λ ( θ ~ ) ] + o p ( 1 ) .

Thus, we need to consider both (51) and (52) for the the last two results of Proposition 2. The combined system can be written as

(55) [ n L ψ ( θ ~ ) n L ϕ ( θ ~ ) ] = [ I k ψ 0 k ϕ × k ϕ − I ψ γ I γ γ − 1 0 k ϕ × k ψ I k ϕ − I ϕ γ I γ γ − 1 ] × [ n L ψ ( θ 0 ) n L ϕ ( θ 0 ) n L γ ( θ 0 ) ] + [ I ψ ⋅ γ δ ψ + I ψ ϕ ⋅ γ δ ϕ I ϕ ⋅ γ δ ϕ + I ϕ ψ ⋅ γ δ ψ ] + o p ( 1 ) .

Then, using the asymptotic normality of score functions from Proposition 1, we get

(56) [ n L ψ ( θ ~ ) n L ϕ ( θ ~ ) ] → d N ( [ I ψ ⋅ γ δ ψ + I ψ ϕ ⋅ γ δ ϕ I ϕ ⋅ γ δ ϕ + I ϕ ψ ⋅ γ δ ψ ] , [ I ψ ⋅ γ I ψ ϕ ⋅ γ I ϕ ψ ⋅ γ I ϕ ⋅ γ ] ) .

Using (56) in (54), we get the following result under H0ψ.

(57) n L ψ ⋆ ( θ ~ ) → d N ( 0 k ψ ,   I ψ ⋅ γ − I ψ ϕ ⋅ γ I ϕ ⋅ γ − 1 I ϕ ψ ⋅ γ ) .

Thus, LMψ⋆(θ~)→dχkψ2, which proves the second part of Proposition 2. Similarly, using (56) in (54), we get the following result under HAψ

(58) n L ψ ⋆ ( θ ~ ) → d N ( [ I ψ ⋅ γ − I ψ ϕ ⋅ γ I ϕ ⋅ γ − 1 I ϕ ψ ⋅ γ ] δ ψ ,   I ψ ⋅ γ − I ψ ϕ ⋅ γ I ϕ ⋅ γ − 1 I ϕ ψ ⋅ γ ) .

Then, the last result in Proposition 2 directly follows from the above result, namely

(59) LM ψ ⋆ ( θ ~ ) → d χ k ψ 2 ( φ 2 ) ,

where φ2=δψ′(Iψ⋅γ−Iψϕ⋅γIϕ⋅γ−1Iψϕ⋅γ′)δψ.

   □

D Simulation Results

The simulation results are based on the following weight matrices: (i) WS is the 49 × 49 contiguity weights matrix of 48 US states, (ii) WO is the 98 × 98 contiguity weights matrix corresponding to five nearest neighbors of each census tract in Toledo, Ohio, (iii) WC is the 361 × 361 weights matrix corresponding to whether the school districts are in the same county in Iowa in 2009, and (iv) WA is the 361 × 361 matrix corresponding to adjacency of 361 school districts in Iowa in 2009. LMμ⋆ and LMμ are robust and non-robust tests for H0 = ρ0 = λ0 = 0, (ii) LMκ is the joint test for H0 : ρ0 = λ0 = δ0 = 0. For all specifications, the nominal size of tests is set to 0.05, and the number of Monte Carlo repetitions to 1000. In each table, the first panel reveals the size properties for LMρ, LMρ⋆ and LMρa, while the first row in each panel corresponds to the size properties for LMλ, LMλ⋆ and LMλa. For LMμ, LMμ⋆ and LMκ, their size properties will be seen from the first row of the first panel in each table. The other rows show the power properties for all test statistics.

Table 3:

LM Tests (Bivariate Normal Distribution): δ0 = 0.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.047 0.062 0.061 0.035 0.053 0.057 0.050 0.053 0.049 0.050 0.057 0.053 0.047 0.049 0.058 0.061 0.066 0.048
0.1 0.0 0.068 0.058 0.055 0.116 0.097 0.090 0.086 0.089 0.103 0.100 0.049 0.050 0.156 0.108 0.102 0.118 0.111 0.108
0.3 0.0 0.390 0.061 0.060 0.661 0.412 0.377 0.551 0.530 0.502 0.659 0.053 0.050 0.850 0.531 0.504 0.778 0.758 0.752
0.5 0.0 0.846 0.029 0.038 0.982 0.779 0.742 0.961 0.957 0.943 0.985 0.040 0.035 1.000 0.917 0.889 0.996 0.995 0.997
0.0 0.1 0.058 0.065 0.070 0.064 0.055 0.062 0.064 0.069 0.072 0.090 0.073 0.072 0.103 0.061 0.055 0.092 0.090 0.085
0.1 0.1 0.175 0.063 0.060 0.198 0.094 0.087 0.179 0.173 0.161 0.302 0.079 0.079 0.337 0.101 0.096 0.285 0.277 0.251
0.3 0.1 0.570 0.055 0.050 0.751 0.390 0.384 0.698 0.675 0.621 0.875 0.095 0.094 0.937 0.497 0.464 0.910 0.903 0.880
0.5 0.1 0.932 0.037 0.043 0.986 0.748 0.720 0.983 0.974 0.965 0.998 0.072 0.069 1.000 0.873 0.839 0.999 0.998 0.999
0.0 0.3 0.386 0.188 0.195 0.274 0.070 0.069 0.337 0.335 0.308 0.628 0.314 0.307 0.455 0.074 0.078 0.537 0.535 0.520
0.1 0.3 0.595 0.178 0.185 0.541 0.107 0.115 0.539 0.532 0.506 0.879 0.326 0.317 0.799 0.136 0.135 0.786 0.790 0.796
0.3 0.3 0.907 0.125 0.124 0.943 0.351 0.337 0.930 0.927 0.902 0.995 0.257 0.250 0.997 0.461 0.429 0.994 0.992 0.988
0.5 0.3 0.988 0.064 0.070 0.998 0.687 0.649 0.997 0.995 0.996 1.000 0.202 0.188 1.000 0.797 0.765 1.000 1.000 1.000
0.0 0.5 0.845 0.429 0.398 0.695 0.097 0.100 0.787 0.788 0.753 0.986 0.723 0.681 0.925 0.107 0.125 0.966 0.965 0.958
0.1 0.5 0.911 0.384 0.378 0.866 0.127 0.133 0.897 0.893 0.866 0.996 0.666 0.633 0.987 0.151 0.154 0.993 0.992 0.985
0.3 0.5 0.981 0.291 0.289 0.985 0.310 0.312 0.985 0.983 0.986 1.000 0.563 0.525 1.000 0.390 0.393 1.000 1.000 1.000
0.5 0.5 0.999 0.185 0.170 0.999 0.514 0.482 1.000 1.000 0.999 1.000 0.402 0.362 1.000 0.643 0.628 1.000 1.000 1.000
WC
WA
0.0 0.0 0.048 0.056 0.048 0.046 0.051 0.047 0.039 0.039 0.053 0.053 0.049 0.048 0.062 0.056 0.053 0.052 0.046 0.046
0.1 0.0 0.468 0.054 0.055 0.609 0.226 0.231 0.491 0.494 0.456 0.323 0.060 0.063 0.464 0.213 0.191 0.369 0.367 0.327
0.3 0.0 1.000 0.046 0.051 1.000 0.968 0.959 1.000 1.000 1.000 1.000 0.060 0.063 1.000 0.922 0.898 1.000 1.000 0.999
0.5 0.0 1.000 0.041 0.057 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.047 0.049 1.000 1.000 1.000 1.000 1.000 1.000
0.0 0.1 0.488 0.184 0.176 0.391 0.073 0.078 0.377 0.373 0.349 0.337 0.164 0.159 0.236 0.062 0.060 0.271 0.261 0.241
0.1 0.1 0.955 0.209 0.207 0.963 0.223 0.217 0.958 0.955 0.932 0.865 0.130 0.121 0.886 0.193 0.191 0.846 0.847 0.808
0.3 0.1 1.000 0.152 0.150 1.000 0.945 0.949 1.000 1.000 1.000 1.000 0.132 0.130 1.000 0.909 0.891 1.000 1.000 1.000
0.5 0.1 1.000 0.048 0.052 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.108 0.104 1.000 0.997 0.993 1.000 1.000 1.000
0.0 0.3 1.000 0.882 0.874 0.998 0.084 0.086 1.000 1.000 0.998 0.997 0.788 0.730 0.977 0.061 0.063 0.991 0.991 0.989
0.1 0.3 1.000 0.878 0.873 1.000 0.243 0.252 1.000 1.000 1.000 1.000 0.757 0.749 1.000 0.193 0.174 1.000 1.000 1.000
0.3 0.3 1.000 0.750 0.726 1.000 0.858 0.851 1.000 1.000 1.000 1.000 0.689 0.631 1.000 0.825 0.802 1.000 1.000 1.000
0.5 0.3 1.000 0.185 0.172 1.000 0.997 0.997 1.000 1.000 1.000 1.000 0.524 0.466 1.000 0.988 0.984 1.000 1.000 1.000
0.0 0.5 1.000 1.000 1.000 1.000 0.131 0.133 1.000 1.000 1.000 1.000 0.994 0.989 1.000 0.107 0.115 1.000 1.000 1.000
0.1 0.5 1.000 0.993 0.990 1.000 0.227 0.235 1.000 1.000 1.000 1.000 0.992 0.979 1.000 0.208 0.218 1.000 1.000 1.000
0.3 0.5 1.000 0.914 0.894 1.000 0.697 0.680 1.000 1.000 1.000 1.000 0.968 0.946 1.000 0.676 0.646 1.000 1.000 1.000
0.5 0.5 1.000 0.343 0.311 1.000 0.942 0.936 1.000 1.000 1.000 1.000 0.794 0.727 1.000 0.941 0.922 1.000 1.000 1.000
Table 4:

LM Tests (Bivariate Normal Distribution): δ0 = 0.25.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.054 0.066 0.043 0.056 0.078 0.044 0.062 0.043 0.269 0.047 0.102 0.060 0.073 0.126 0.053 0.086 0.041 0.539
0.1 0.0 0.079 0.064 0.059 0.192 0.200 0.092 0.177 0.091 0.372 0.114 0.081 0.057 0.294 0.278 0.093 0.267 0.127 0.701
0.3 0.0 0.370 0.059 0.042 0.730 0.544 0.346 0.671 0.512 0.745 0.659 0.085 0.050 0.917 0.738 0.428 0.888 0.754 0.952
0.5 0.0 0.843 0.056 0.029 0.983 0.861 0.709 0.971 0.937 0.978 0.984 0.072 0.049 1.000 0.956 0.837 0.999 0.995 1.000
0.0 0.1 0.068 0.063 0.084 0.095 0.074 0.044 0.097 0.067 0.303 0.129 0.066 0.095 0.196 0.134 0.048 0.147 0.101 0.599
0.1 0.1 0.173 0.059 0.071 0.287 0.177 0.075 0.230 0.162 0.445 0.340 0.068 0.094 0.504 0.256 0.073 0.460 0.300 0.736
0.3 0.1 0.594 0.040 0.055 0.842 0.564 0.348 0.782 0.664 0.851 0.846 0.061 0.113 0.968 0.705 0.428 0.954 0.887 0.977
0.5 0.1 0.924 0.042 0.038 0.992 0.823 0.667 0.985 0.970 0.994 0.995 0.036 0.074 1.000 0.945 0.790 1.000 0.999 1.000
0.0 0.3 0.407 0.127 0.199 0.382 0.102 0.074 0.351 0.304 0.529 0.654 0.171 0.319 0.625 0.140 0.061 0.607 0.563 0.836
0.1 0.3 0.593 0.127 0.209 0.632 0.189 0.102 0.591 0.528 0.729 0.875 0.180 0.338 0.888 0.288 0.106 0.863 0.799 0.942
0.3 0.3 0.883 0.094 0.154 0.946 0.500 0.312 0.933 0.900 0.955 0.987 0.156 0.321 0.996 0.616 0.336 0.997 0.993 0.998
0.5 0.3 0.989 0.049 0.084 0.998 0.740 0.568 0.998 0.995 0.998 1.000 0.106 0.230 1.000 0.879 0.667 1.000 1.000 1.000
0.0 0.5 0.837 0.349 0.450 0.748 0.127 0.097 0.821 0.810 0.863 0.981 0.525 0.695 0.958 0.149 0.116 0.966 0.964 0.981
0.1 0.5 0.917 0.291 0.404 0.876 0.178 0.123 0.898 0.879 0.914 0.997 0.519 0.685 0.994 0.263 0.130 0.995 0.995 0.999
0.3 0.5 0.992 0.202 0.302 0.995 0.403 0.255 0.992 0.989 0.996 1.000 0.409 0.609 1.000 0.543 0.321 1.000 1.000 1.000
0.5 0.5 0.996 0.098 0.162 0.999 0.627 0.463 0.998 0.998 0.999 1.000 0.267 0.435 1.000 0.755 0.555 1.000 1.000 1.000
WC
WA
0.0 0.0 0.057 0.096 0.045 0.088 0.146 0.040 0.124 0.041 0.995 0.055 0.191 0.060 0.154 0.292 0.054 0.217 0.052 0.994
0.1 0.0 0.527 0.108 0.058 0.792 0.511 0.200 0.742 0.537 1.000 0.437 0.164 0.051 0.805 0.686 0.166 0.773 0.430 0.998
0.3 0.0 1.000 0.100 0.046 1.000 0.992 0.953 1.000 1.000 1.000 0.996 0.172 0.068 1.000 0.990 0.877 1.000 1.000 1.000
0.5 0.0 1.000 0.138 0.045 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.120 0.059 1.000 1.000 0.997 1.000 1.000 1.000
0.0 0.1 0.549 0.084 0.218 0.603 0.138 0.054 0.511 0.429 0.999 0.439 0.058 0.202 0.603 0.257 0.049 0.509 0.345 0.997
0.1 0.1 0.979 0.074 0.217 0.995 0.550 0.228 0.986 0.956 1.000 0.929 0.055 0.203 0.990 0.676 0.168 0.969 0.890 1.000
0.3 0.1 1.000 0.058 0.188 1.000 0.986 0.933 1.000 1.000 1.000 1.000 0.067 0.219 1.000 0.991 0.813 1.000 1.000 1.000
0.5 0.1 1.000 0.054 0.071 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.042 0.151 1.000 1.000 0.998 1.000 1.000 1.000
0.0 0.3 1.000 0.722 0.901 1.000 0.154 0.069 1.000 1.000 1.000 0.999 0.463 0.838 0.998 0.255 0.063 0.998 0.993 1.000
0.1 0.3 1.000 0.703 0.894 1.000 0.472 0.183 1.000 1.000 1.000 1.000 0.427 0.827 1.000 0.614 0.127 1.000 1.000 1.000
0.3 0.3 1.000 0.524 0.778 1.000 0.953 0.788 1.000 1.000 1.000 1.000 0.367 0.763 1.000 0.969 0.705 1.000 1.000 1.000
0.5 0.3 1.000 0.097 0.262 1.000 1.000 0.994 1.000 1.000 1.000 1.000 0.219 0.586 1.000 0.999 0.978 1.000 1.000 1.000
0.0 0.5 1.000 0.997 0.999 1.000 0.176 0.130 1.000 1.000 1.000 1.000 0.949 0.992 1.000 0.275 0.128 1.000 1.000 1.000
0.1 0.5 1.000 0.980 0.997 1.000 0.349 0.158 1.000 1.000 1.000 1.000 0.941 0.996 1.000 0.509 0.119 1.000 1.000 1.000
0.3 0.5 1.000 0.821 0.944 1.000 0.816 0.607 1.000 1.000 1.000 1.000 0.847 0.975 1.000 0.908 0.550 1.000 1.000 1.000
0.5 0.5 1.000 0.234 0.460 1.000 0.969 0.887 1.000 1.000 1.000 1.000 0.569 0.887 1.000 0.981 0.883 1.000 1.000 1.000
Table 5:

LM Tests (Bivariate Normal Distribution): δ0 = 0.5.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.037 0.097 0.034 0.101 0.161 0.018 0.113 0.029 0.883 0.070 0.138 0.052 0.234 0.334 0.028 0.259 0.047 0.997
0.1 0.0 0.093 0.138 0.043 0.331 0.371 0.053 0.286 0.085 0.935 0.200 0.165 0.056 0.586 0.537 0.056 0.563 0.160 0.998
0.3 0.0 0.391 0.117 0.042 0.839 0.726 0.297 0.785 0.518 0.989 0.738 0.154 0.049 0.982 0.905 0.370 0.977 0.815 1.000
0.5 0.0 0.835 0.113 0.027 0.993 0.917 0.657 0.992 0.948 0.998 0.981 0.126 0.038 1.000 0.982 0.801 1.000 0.999 1.000
0.0 0.1 0.090 0.055 0.061 0.190 0.199 0.026 0.176 0.052 0.914 0.201 0.080 0.110 0.426 0.318 0.029 0.373 0.162 0.999
0.1 0.1 0.215 0.069 0.072 0.449 0.319 0.055 0.412 0.169 0.944 0.460 0.069 0.129 0.774 0.489 0.051 0.732 0.368 1.000
0.3 0.1 0.626 0.061 0.047 0.911 0.695 0.290 0.858 0.649 0.990 0.897 0.071 0.119 0.993 0.862 0.324 0.989 0.923 1.000
0.5 0.1 0.913 0.060 0.035 0.997 0.896 0.658 0.993 0.972 1.000 0.997 0.065 0.084 1.000 0.968 0.741 1.000 0.999 1.000
0.0 0.3 0.471 0.094 0.250 0.509 0.166 0.050 0.445 0.363 0.957 0.777 0.109 0.445 0.853 0.279 0.046 0.788 0.684 1.000
0.1 0.3 0.630 0.103 0.220 0.758 0.315 0.060 0.702 0.559 0.975 0.906 0.109 0.413 0.974 0.491 0.060 0.952 0.885 1.000
0.3 0.3 0.899 0.064 0.164 0.980 0.611 0.233 0.941 0.896 0.997 0.995 0.091 0.399 1.000 0.794 0.243 0.998 0.992 1.000
0.5 0.3 0.979 0.044 0.082 1.000 0.811 0.512 0.995 0.989 1.000 1.000 0.065 0.277 1.000 0.935 0.623 1.000 1.000 1.000
0.0 0.5 0.859 0.253 0.493 0.848 0.201 0.082 0.850 0.816 0.983 0.992 0.436 0.800 0.989 0.292 0.082 0.984 0.976 1.000
0.1 0.5 0.935 0.223 0.457 0.945 0.274 0.086 0.923 0.889 0.994 0.999 0.370 0.786 0.999 0.437 0.084 1.000 0.999 1.000
0.3 0.5 0.986 0.142 0.334 0.996 0.502 0.202 0.999 0.991 1.000 1.000 0.281 0.690 1.000 0.693 0.240 1.000 1.000 1.000
0.5 0.5 1.000 0.062 0.181 0.999 0.680 0.389 0.998 0.998 1.000 1.000 0.182 0.528 1.000 0.823 0.451 1.000 1.000 1.000
WC
WA
0.0 0.0 0.095 0.210 0.049 0.356 0.456 0.020 0.416 0.074 1.000 0.237 0.361 0.091 0.754 0.796 0.025 0.792 0.142 1.000
0.1 0.0 0.693 0.223 0.052 0.964 0.867 0.184 0.940 0.698 1.000 0.753 0.377 0.099 0.993 0.966 0.120 0.992 0.683 1.000
0.3 0.0 1.000 0.240 0.053 1.000 1.000 0.948 1.000 1.000 1.000 0.999 0.359 0.088 1.000 1.000 0.841 1.000 1.000 1.000
0.5 0.0 1.000 0.302 0.031 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.301 0.090 1.000 1.000 0.998 1.000 1.000 1.000
0.0 0.1 0.717 0.051 0.292 0.889 0.429 0.021 0.823 0.619 1.000 0.788 0.104 0.363 0.966 0.746 0.033 0.945 0.689 1.000
0.1 0.1 0.988 0.058 0.296 1.000 0.797 0.133 1.000 0.989 1.000 0.987 0.096 0.390 1.000 0.936 0.069 0.999 0.984 1.000
0.3 0.1 1.000 0.044 0.243 1.000 0.996 0.917 1.000 1.000 1.000 1.000 0.088 0.349 1.000 1.000 0.764 1.000 1.000 1.000
0.5 0.1 1.000 0.105 0.065 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.106 0.267 1.000 1.000 0.997 1.000 1.000 1.000
0.0 0.3 1.000 0.556 0.950 1.000 0.363 0.062 1.000 1.000 1.000 1.000 0.272 0.954 1.000 0.661 0.060 1.000 1.000 1.000
0.1 0.3 1.000 0.534 0.949 1.000 0.681 0.101 1.000 1.000 1.000 1.000 0.238 0.941 1.000 0.896 0.074 1.000 1.000 1.000
0.3 0.3 1.000 0.330 0.866 1.000 0.979 0.771 1.000 1.000 1.000 1.000 0.166 0.889 1.000 0.996 0.595 1.000 1.000 1.000
0.5 0.3 1.000 0.057 0.391 1.000 1.000 0.988 1.000 1.000 1.000 1.000 0.075 0.773 1.000 1.000 0.946 1.000 1.000 1.000
0.0 0.5 1.000 0.980 1.000 1.000 0.281 0.105 1.000 1.000 1.000 1.000 0.845 0.999 1.000 0.635 0.094 1.000 1.000 1.000
0.1 0.5 1.000 0.951 1.000 1.000 0.544 0.082 1.000 1.000 1.000 1.000 0.818 0.999 1.000 0.798 0.065 1.000 1.000 1.000
0.3 0.5 1.000 0.741 0.984 1.000 0.868 0.494 1.000 1.000 1.000 1.000 0.652 0.996 1.000 0.964 0.366 1.000 1.000 1.000
0.5 0.5 1.000 0.156 0.604 1.000 0.974 0.869 1.000 1.000 1.000 1.000 0.343 0.954 1.000 0.998 0.798 1.000 1.000 1.000
Table 6:

Robust LM Tests for Endogeneity in Weights: δ0 = 0.

WS
WO
WC
WA
λ 0 ρ 0 LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆
0.0 0.0 0.050 0.058 0.052 0.048 0.054 0.048 0.051 0.052
0.1 0.0 0.069 0.071 0.074 0.075 0.067 0.070 0.053 0.049
0.3 0.0 0.074 0.065 0.050 0.054 0.058 0.061 0.073 0.072
0.5 0.0 0.054 0.057 0.050 0.059 0.071 0.084 0.082 0.090
0.0 0.1 0.063 0.075 0.062 0.067 0.060 0.054 0.071 0.057
0.1 0.1 0.056 0.055 0.065 0.062 0.060 0.061 0.069 0.073
0.3 0.1 0.071 0.070 0.075 0.070 0.065 0.065 0.092 0.086
0.5 0.1 0.048 0.060 0.078 0.081 0.082 0.088 0.099 0.091
0.0 0.3 0.067 0.067 0.092 0.089 0.087 0.062 0.138 0.075
0.1 0.3 0.082 0.075 0.082 0.061 0.111 0.060 0.123 0.070
0.3 0.3 0.066 0.061 0.101 0.092 0.104 0.064 0.145 0.091
0.5 0.3 0.067 0.079 0.101 0.093 0.088 0.073 0.123 0.101
0.0 0.5 0.105 0.085 0.137 0.089 0.178 0.078 0.245 0.092
0.1 0.5 0.087 0.080 0.170 0.102 0.154 0.072 0.247 0.103
0.3 0.5 0.092 0.089 0.158 0.114 0.135 0.093 0.218 0.114
0.5 0.5 0.102 0.089 0.133 0.114 0.102 0.077 0.177 0.085
Table 7:

Robust LM Tests for Endogeneity in Weights: δ0 = 0.25.

WS
WO
WC
WA
λ 0 ρ 0 LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆
0.0 0.0 0.427 0.392 0.713 0.678 0.998 0.997 1.000 0.999
0.1 0.0 0.401 0.383 0.702 0.657 0.997 0.994 0.998 0.997
0.3 0.0 0.401 0.370 0.649 0.621 0.996 0.995 0.996 0.996
0.5 0.0 0.322 0.293 0.599 0.560 0.960 0.952 0.988 0.980
0.0 0.1 0.391 0.393 0.699 0.705 0.997 0.995 0.996 0.998
0.1 0.1 0.383 0.394 0.659 0.659 0.990 0.994 0.995 0.996
0.3 0.1 0.357 0.360 0.630 0.642 0.991 0.993 0.996 0.997
0.5 0.1 0.308 0.284 0.530 0.552 0.962 0.956 0.979 0.977
0.0 0.3 0.319 0.391 0.595 0.695 0.992 0.997 0.994 0.999
0.1 0.3 0.331 0.401 0.572 0.677 0.971 0.993 0.984 0.996
0.3 0.3 0.308 0.358 0.553 0.644 0.969 0.990 0.980 0.997
0.5 0.3 0.247 0.284 0.488 0.570 0.919 0.929 0.957 0.982
0.0 0.5 0.262 0.407 0.464 0.672 0.954 0.996 0.946 0.997
0.1 0.5 0.249 0.391 0.438 0.663 0.949 0.988 0.942 0.995
0.3 0.5 0.255 0.362 0.434 0.635 0.921 0.976 0.952 0.991
0.5 0.5 0.211 0.293 0.363 0.503 0.850 0.895 0.904 0.968
Table 8:

Robust LM Tests for Endogeneity in Weights: δ0 = 0.5.

WS
WO
WC
WA
λ 0 ρ 0 LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆ LM δ ⋆ SAR LM δ ⋆
0.0 0.0 0.956 0.938 1.000 0.997 1.000 1.000 1.000 1.000
0.1 0.0 0.965 0.946 1.000 1.000 1.000 1.000 1.000 1.000
0.3 0.0 0.929 0.893 0.999 0.999 1.000 1.000 1.000 1.000
0.5 0.0 0.891 0.835 0.994 0.991 1.000 1.000 1.000 1.000
0.0 0.1 0.961 0.959 0.999 0.998 1.000 1.000 1.000 1.000
0.1 0.1 0.947 0.932 1.000 1.000 1.000 1.000 1.000 1.000
0.3 0.1 0.939 0.926 0.996 0.992 1.000 1.000 1.000 1.000
0.5 0.1 0.858 0.809 0.997 0.995 1.000 1.000 1.000 1.000
0.0 0.3 0.935 0.954 0.998 0.998 1.000 1.000 1.000 1.000
0.1 0.3 0.909 0.925 0.997 0.998 1.000 1.000 1.000 1.000
0.3 0.3 0.884 0.903 0.996 0.999 1.000 1.000 1.000 1.000
0.5 0.3 0.795 0.798 0.986 0.990 1.000 1.000 1.000 1.000
0.0 0.5 0.858 0.931 0.993 0.998 1.000 1.000 1.000 1.000
0.1 0.5 0.848 0.919 0.984 0.998 1.000 1.000 1.000 1.000
0.3 0.5 0.822 0.872 0.982 0.992 1.000 1.000 1.000 1.000
0.5 0.5 0.731 0.756 0.956 0.976 1.000 1.000 1.000 1.000
Table 9:

LM Tests (Conditional Moment Assumptions Hold Without Normality): δ0 = 0.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.051 0.056 0.062 0.053 0.063 0.061 0.063 0.067 0.059 0.052 0.053 0.047 0.048 0.046 0.043 0.047 0.040 0.047
0.1 0.0 0.064 0.053 0.051 0.102 0.090 0.093 0.084 0.083 0.083 0.113 0.058 0.050 0.157 0.104 0.095 0.131 0.123 0.127
0.3 0.0 0.413 0.059 0.061 0.675 0.403 0.381 0.594 0.578 0.517 0.654 0.045 0.050 0.876 0.535 0.517 0.800 0.783 0.754
0.5 0.0 0.825 0.047 0.046 0.979 0.780 0.747 0.960 0.950 0.937 0.990 0.027 0.038 0.999 0.907 0.881 0.999 0.999 0.999
0.0 0.1 0.077 0.060 0.057 0.067 0.063 0.063 0.073 0.075 0.063 0.092 0.053 0.063 0.086 0.051 0.060 0.073 0.077 0.077
0.1 0.1 0.155 0.072 0.078 0.213 0.118 0.114 0.183 0.174 0.157 0.323 0.068 0.069 0.350 0.125 0.134 0.301 0.299 0.260
0.3 0.1 0.600 0.056 0.058 0.778 0.392 0.370 0.707 0.686 0.645 0.857 0.068 0.073 0.937 0.492 0.447 0.896 0.881 0.867
0.5 0.1 0.931 0.042 0.041 0.992 0.762 0.714 0.982 0.976 0.970 0.997 0.065 0.063 0.999 0.876 0.846 1.000 1.000 1.000
0.0 0.3 0.361 0.185 0.179 0.269 0.086 0.091 0.316 0.321 0.290 0.633 0.288 0.280 0.459 0.070 0.075 0.538 0.535 0.500
0.1 0.3 0.607 0.176 0.179 0.533 0.118 0.117 0.562 0.551 0.490 0.831 0.288 0.276 0.772 0.127 0.126 0.781 0.780 0.743
0.3 0.3 0.905 0.143 0.137 0.938 0.331 0.319 0.920 0.914 0.884 0.992 0.281 0.265 0.994 0.466 0.463 0.994 0.994 0.992
0.5 0.3 0.986 0.075 0.074 0.996 0.629 0.601 0.996 0.995 0.992 1.000 0.171 0.166 1.000 0.797 0.770 1.000 1.000 1.000
0.0 0.5 0.844 0.447 0.420 0.684 0.119 0.127 0.797 0.802 0.767 0.978 0.679 0.648 0.923 0.110 0.118 0.970 0.967 0.951
0.1 0.5 0.934 0.402 0.382 0.876 0.125 0.133 0.905 0.900 0.880 0.992 0.645 0.617 0.980 0.153 0.169 0.987 0.989 0.986
0.3 0.5 0.984 0.263 0.255 0.980 0.308 0.284 0.982 0.980 0.978 0.999 0.568 0.520 1.000 0.417 0.408 1.000 0.999 0.998
0.5 0.5 0.998 0.138 0.152 0.999 0.533 0.516 0.999 0.999 0.998 1.000 0.393 0.379 1.000 0.611 0.580 1.000 1.000 1.000
WC
WA
0.0 0.0 0.054 0.055 0.053 0.051 0.054 0.049 0.053 0.050 0.047 0.042 0.037 0.035 0.052 0.050 0.050 0.049 0.048 0.052
0.1 0.0 0.499 0.050 0.048 0.615 0.209 0.192 0.506 0.505 0.460 0.345 0.052 0.053 0.476 0.200 0.196 0.382 0.388 0.326
0.3 0.0 1.000 0.043 0.043 1.000 0.968 0.969 1.000 1.000 1.000 0.996 0.046 0.047 1.000 0.911 0.885 0.999 0.999 0.997
0.5 0.0 1.000 0.045 0.059 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.028 0.033 1.000 1.000 1.000 1.000 1.000 1.000
0.0 0.1 0.482 0.190 0.190 0.377 0.061 0.058 0.408 0.411 0.364 0.324 0.150 0.143 0.231 0.058 0.057 0.269 0.268 0.247
0.1 0.1 0.965 0.165 0.167 0.973 0.246 0.235 0.952 0.952 0.927 0.870 0.141 0.143 0.868 0.186 0.181 0.840 0.836 0.793
0.3 0.1 1.000 0.172 0.166 1.000 0.951 0.939 1.000 1.000 1.000 1.000 0.144 0.140 1.000 0.898 0.867 1.000 1.000 1.000
0.5 0.1 1.000 0.042 0.047 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.128 0.122 1.000 0.998 0.997 1.000 1.000 1.000
0.0 0.3 1.000 0.865 0.845 1.000 0.082 0.082 1.000 1.000 0.999 0.998 0.744 0.726 0.984 0.061 0.062 0.995 0.995 0.989
0.1 0.3 1.000 0.866 0.861 1.000 0.241 0.243 1.000 1.000 1.000 1.000 0.761 0.715 1.000 0.201 0.204 1.000 1.000 1.000
0.3 0.3 1.000 0.722 0.703 1.000 0.862 0.854 1.000 1.000 1.000 1.000 0.707 0.655 1.000 0.830 0.807 1.000 1.000 1.000
0.5 0.3 1.000 0.199 0.187 1.000 0.991 0.992 1.000 1.000 1.000 1.000 0.517 0.480 1.000 0.989 0.980 1.000 1.000 1.000
0.0 0.5 1.000 0.996 0.994 1.000 0.134 0.148 1.000 1.000 1.000 1.000 0.985 0.978 1.000 0.107 0.117 1.000 1.000 1.000
0.1 0.5 1.000 0.999 0.998 1.000 0.206 0.213 1.000 1.000 1.000 1.000 0.990 0.983 1.000 0.220 0.206 1.000 1.000 1.000
0.3 0.5 1.000 0.911 0.894 1.000 0.687 0.677 1.000 1.000 1.000 1.000 0.962 0.937 1.000 0.657 0.629 1.000 1.000 1.000
0.5 0.5 1.000 0.360 0.345 1.000 0.927 0.923 1.000 1.000 1.000 1.000 0.794 0.729 1.000 0.940 0.931 1.000 1.000 1.000
Table 10:

LM Tests (Conditional Moment Assumptions Hold Without Normality): δ0 = 0.25.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.050 0.076 0.044 0.050 0.076 0.046 0.063 0.038 0.254 0.044 0.092 0.048 0.068 0.099 0.046 0.087 0.046 0.486
0.1 0.0 0.064 0.075 0.055 0.146 0.162 0.080 0.128 0.069 0.298 0.119 0.084 0.044 0.269 0.215 0.072 0.229 0.109 0.633
0.3 0.0 0.375 0.066 0.045 0.710 0.542 0.324 0.641 0.521 0.756 0.648 0.082 0.053 0.897 0.690 0.398 0.859 0.739 0.941
0.5 0.0 0.852 0.063 0.050 0.989 0.862 0.704 0.977 0.945 0.984 0.981 0.068 0.064 1.000 0.934 0.829 1.000 0.997 1.000
0.0 0.1 0.066 0.064 0.073 0.094 0.079 0.055 0.084 0.060 0.259 0.125 0.073 0.097 0.152 0.109 0.051 0.151 0.101 0.567
0.1 0.1 0.165 0.055 0.062 0.292 0.172 0.096 0.231 0.168 0.418 0.337 0.059 0.091 0.511 0.287 0.091 0.434 0.278 0.742
0.3 0.1 0.558 0.059 0.064 0.820 0.522 0.311 0.743 0.643 0.809 0.842 0.055 0.089 0.962 0.671 0.384 0.944 0.862 0.983
0.5 0.1 0.918 0.041 0.029 0.993 0.812 0.641 0.980 0.962 0.984 0.994 0.045 0.075 0.999 0.927 0.778 0.998 0.997 1.000
0.0 0.3 0.373 0.102 0.181 0.335 0.101 0.074 0.298 0.288 0.476 0.668 0.196 0.341 0.637 0.133 0.085 0.621 0.601 0.841
0.1 0.3 0.600 0.125 0.186 0.640 0.179 0.097 0.586 0.521 0.715 0.857 0.131 0.280 0.890 0.276 0.115 0.857 0.822 0.945
0.3 0.3 0.881 0.082 0.133 0.942 0.452 0.277 0.916 0.875 0.930 0.995 0.147 0.297 0.999 0.622 0.365 0.998 0.995 0.999
0.5 0.3 0.990 0.041 0.083 0.997 0.704 0.540 0.995 0.993 0.995 1.000 0.128 0.230 1.000 0.848 0.692 1.000 1.000 1.000
0.0 0.5 0.842 0.324 0.448 0.753 0.126 0.113 0.806 0.799 0.851 0.983 0.504 0.682 0.966 0.153 0.114 0.971 0.965 0.988
0.1 0.5 0.933 0.303 0.415 0.899 0.180 0.123 0.906 0.896 0.933 0.997 0.517 0.677 0.993 0.238 0.125 0.996 0.993 0.998
0.3 0.5 0.981 0.182 0.267 0.989 0.413 0.276 0.988 0.983 0.990 1.000 0.400 0.583 1.000 0.514 0.317 1.000 1.000 1.000
0.5 0.5 1.000 0.087 0.143 1.000 0.601 0.448 1.000 0.999 1.000 1.000 0.261 0.410 1.000 0.730 0.533 1.000 1.000 1.000
WC
WA
0.0 0.0 0.052 0.109 0.058 0.078 0.122 0.046 0.101 0.049 0.981 0.051 0.160 0.063 0.128 0.237 0.048 0.176 0.060 0.990
0.1 0.0 0.512 0.110 0.061 0.758 0.517 0.222 0.700 0.525 0.998 0.408 0.148 0.050 0.765 0.608 0.144 0.732 0.384 0.997
0.3 0.0 1.000 0.107 0.048 1.000 0.990 0.943 1.000 1.000 1.000 0.998 0.133 0.062 1.000 0.992 0.847 1.000 1.000 1.000
0.5 0.0 1.000 0.114 0.038 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.120 0.047 1.000 1.000 1.000 1.000 1.000 1.000
0.0 0.1 0.497 0.075 0.178 0.572 0.139 0.061 0.471 0.425 0.996 0.382 0.055 0.168 0.540 0.262 0.054 0.461 0.302 0.991
0.1 0.1 0.963 0.078 0.208 0.992 0.475 0.206 0.979 0.960 1.000 0.902 0.051 0.181 0.976 0.603 0.148 0.950 0.877 1.000
0.3 0.1 1.000 0.066 0.168 1.000 0.987 0.916 1.000 1.000 1.000 1.000 0.051 0.167 1.000 0.979 0.825 1.000 1.000 1.000
0.5 0.1 1.000 0.048 0.059 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.034 0.123 1.000 1.000 1.000 1.000 1.000 1.000
0.0 0.3 1.000 0.686 0.885 1.000 0.155 0.062 1.000 1.000 1.000 0.999 0.434 0.780 0.998 0.252 0.087 0.998 0.997 1.000
0.1 0.3 1.000 0.659 0.873 1.000 0.453 0.195 1.000 1.000 1.000 1.000 0.413 0.784 1.000 0.542 0.134 1.000 1.000 1.000
0.3 0.3 1.000 0.517 0.723 1.000 0.916 0.777 1.000 1.000 1.000 1.000 0.349 0.716 1.000 0.957 0.685 1.000 1.000 1.000
0.5 0.3 1.000 0.099 0.252 1.000 0.999 0.996 1.000 1.000 1.000 1.000 0.214 0.538 1.000 0.998 0.979 1.000 1.000 1.000
0.0 0.5 1.000 0.984 0.999 1.000 0.152 0.108 1.000 1.000 1.000 1.000 0.933 0.989 1.000 0.260 0.118 1.000 1.000 1.000
0.1 0.5 1.000 0.974 0.997 1.000 0.359 0.184 1.000 1.000 1.000 1.000 0.917 0.993 1.000 0.456 0.140 1.000 1.000 1.000
0.3 0.5 1.000 0.823 0.925 1.000 0.764 0.563 1.000 1.000 1.000 1.000 0.818 0.967 1.000 0.880 0.565 1.000 1.000 1.000
0.5 0.5 1.000 0.244 0.435 1.000 0.954 0.895 1.000 1.000 1.000 1.000 0.557 0.829 1.000 0.979 0.888 1.000 1.000 1.000
Table 11:

LM Tests (Conditional Moment Assumptions Hold Without Normality): δ0 = 0.5.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.036 0.100 0.034 0.061 0.120 0.022 0.097 0.025 0.797 0.044 0.151 0.046 0.126 0.236 0.030 0.179 0.029 0.993
0.1 0.0 0.099 0.119 0.060 0.244 0.272 0.070 0.236 0.076 0.875 0.145 0.150 0.035 0.405 0.440 0.048 0.394 0.111 0.995
0.3 0.0 0.380 0.138 0.049 0.765 0.612 0.263 0.714 0.470 0.966 0.686 0.152 0.036 0.967 0.815 0.321 0.944 0.739 1.000
0.5 0.0 0.824 0.114 0.037 0.984 0.862 0.604 0.975 0.926 0.999 0.980 0.120 0.043 1.000 0.972 0.746 1.000 0.996 1.000
0.0 0.1 0.074 0.072 0.064 0.154 0.145 0.039 0.134 0.065 0.808 0.145 0.093 0.081 0.325 0.267 0.040 0.280 0.100 0.991
0.1 0.1 0.190 0.062 0.063 0.366 0.259 0.063 0.314 0.156 0.878 0.365 0.083 0.083 0.657 0.427 0.059 0.567 0.297 0.998
0.3 0.1 0.572 0.078 0.043 0.855 0.584 0.238 0.794 0.607 0.972 0.864 0.076 0.077 0.982 0.793 0.302 0.971 0.883 1.000
0.5 0.1 0.911 0.061 0.041 0.994 0.833 0.587 0.983 0.952 1.000 0.996 0.071 0.078 1.000 0.952 0.727 1.000 0.999 1.000
0.0 0.3 0.394 0.085 0.204 0.454 0.157 0.056 0.391 0.340 0.886 0.716 0.110 0.346 0.774 0.206 0.055 0.713 0.613 0.998
0.1 0.3 0.630 0.086 0.194 0.712 0.265 0.080 0.658 0.541 0.961 0.881 0.099 0.377 0.937 0.391 0.062 0.901 0.829 1.000
0.3 0.3 0.893 0.059 0.121 0.950 0.529 0.226 0.934 0.875 0.995 0.989 0.083 0.313 0.999 0.706 0.269 0.998 0.989 1.000
0.5 0.3 0.985 0.053 0.094 0.995 0.723 0.450 0.993 0.986 0.999 1.000 0.048 0.235 1.000 0.891 0.561 1.000 1.000 1.000
0.0 0.5 0.874 0.225 0.442 0.849 0.176 0.091 0.848 0.824 0.984 0.988 0.346 0.702 0.990 0.252 0.090 0.986 0.981 1.000
0.1 0.5 0.927 0.208 0.407 0.932 0.230 0.084 0.926 0.898 0.987 0.996 0.326 0.692 0.996 0.350 0.090 0.998 0.995 1.000
0.3 0.5 0.990 0.126 0.270 0.988 0.441 0.185 0.992 0.987 0.998 1.000 0.279 0.586 1.000 0.600 0.237 1.000 1.000 1.000
0.5 0.5 0.999 0.067 0.157 1.000 0.599 0.360 1.000 0.999 1.000 1.000 0.185 0.442 1.000 0.759 0.443 1.000 1.000 1.000
WC
WA
0.0 0.0 0.057 0.204 0.038 0.167 0.325 0.030 0.263 0.040 1.000 0.106 0.367 0.050 0.444 0.658 0.026 0.599 0.062 1.000
0.1 0.0 0.595 0.200 0.044 0.888 0.746 0.158 0.877 0.582 1.000 0.564 0.374 0.047 0.956 0.900 0.114 0.951 0.527 1.000
0.3 0.0 1.000 0.178 0.054 1.000 0.996 0.887 1.000 1.000 1.000 0.999 0.343 0.061 1.000 0.997 0.764 1.000 1.000 1.000
0.5 0.0 1.000 0.293 0.041 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.328 0.040 1.000 1.000 1.000 1.000 1.000 1.000
0.0 0.1 0.611 0.065 0.217 0.757 0.312 0.036 0.672 0.497 1.000 0.605 0.134 0.213 0.874 0.629 0.037 0.835 0.495 1.000
0.1 0.1 0.976 0.062 0.199 0.996 0.679 0.125 0.989 0.960 1.000 0.954 0.124 0.240 1.000 0.870 0.090 0.999 0.944 1.000
0.3 0.1 1.000 0.052 0.195 1.000 0.987 0.837 1.000 1.000 1.000 1.000 0.126 0.221 1.000 0.999 0.710 1.000 1.000 1.000
0.5 0.1 1.000 0.107 0.053 1.000 1.000 0.997 1.000 1.000 1.000 1.000 0.110 0.185 1.000 1.000 0.988 1.000 1.000 1.000
0.0 0.3 1.000 0.450 0.882 1.000 0.283 0.062 1.000 1.000 1.000 1.000 0.158 0.814 1.000 0.576 0.053 1.000 1.000 1.000
0.1 0.3 1.000 0.468 0.886 1.000 0.565 0.112 1.000 1.000 1.000 1.000 0.157 0.821 1.000 0.809 0.086 1.000 1.000 1.000
0.3 0.3 1.000 0.311 0.764 1.000 0.944 0.675 1.000 1.000 1.000 1.000 0.113 0.753 1.000 0.982 0.571 1.000 1.000 1.000
0.5 0.3 1.000 0.045 0.285 1.000 0.997 0.972 1.000 1.000 1.000 1.000 0.095 0.644 1.000 1.000 0.930 1.000 1.000 1.000
0.0 0.5 1.000 0.922 0.998 1.000 0.261 0.097 1.000 1.000 1.000 1.000 0.753 0.997 1.000 0.482 0.094 1.000 1.000 1.000
0.1 0.5 1.000 0.917 0.996 1.000 0.449 0.102 1.000 1.000 1.000 1.000 0.701 0.989 1.000 0.710 0.108 1.000 1.000 1.000
0.3 0.5 1.000 0.633 0.934 1.000 0.802 0.473 1.000 1.000 1.000 1.000 0.581 0.981 1.000 0.921 0.418 1.000 1.000 1.000
0.5 0.5 1.000 0.142 0.465 1.000 0.953 0.834 1.000 1.000 1.000 1.000 0.289 0.884 1.000 0.983 0.810 1.000 1.000 1.000
Table 12:

LM Tests (Bivariate t5 Distribution): δ0 = 0.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.031 0.041 0.040 0.037 0.061 0.049 0.042 0.036 0.093 0.038 0.048 0.051 0.044 0.056 0.060 0.044 0.039 0.109
0.1 0.0 0.061 0.052 0.050 0.084 0.085 0.081 0.072 0.064 0.138 0.088 0.041 0.045 0.120 0.078 0.085 0.109 0.106 0.168
0.3 0.0 0.367 0.051 0.039 0.572 0.328 0.304 0.477 0.479 0.494 0.588 0.056 0.065 0.769 0.400 0.376 0.689 0.672 0.684
0.5 0.0 0.809 0.061 0.066 0.938 0.648 0.618 0.902 0.888 0.896 0.974 0.051 0.059 0.998 0.776 0.759 0.987 0.985 0.986
0.0 0.1 0.052 0.063 0.063 0.065 0.074 0.073 0.078 0.073 0.131 0.082 0.064 0.064 0.080 0.060 0.065 0.077 0.071 0.141
0.1 0.1 0.152 0.062 0.060 0.176 0.092 0.087 0.151 0.135 0.202 0.249 0.060 0.060 0.288 0.092 0.091 0.231 0.221 0.292
0.3 0.1 0.570 0.064 0.069 0.715 0.284 0.277 0.636 0.628 0.630 0.801 0.068 0.075 0.885 0.373 0.361 0.833 0.831 0.845
0.5 0.1 0.919 0.053 0.050 0.971 0.607 0.568 0.956 0.952 0.949 0.993 0.048 0.054 0.999 0.759 0.735 0.995 0.994 0.996
0.0 0.3 0.348 0.154 0.156 0.289 0.090 0.087 0.312 0.316 0.370 0.580 0.229 0.220 0.463 0.072 0.082 0.491 0.501 0.526
0.1 0.3 0.541 0.141 0.126 0.531 0.122 0.115 0.514 0.507 0.507 0.823 0.228 0.233 0.783 0.115 0.120 0.766 0.767 0.759
0.3 0.3 0.864 0.113 0.113 0.901 0.277 0.273 0.864 0.856 0.852 0.983 0.194 0.188 0.983 0.339 0.356 0.978 0.980 0.977
0.5 0.3 0.979 0.072 0.071 0.993 0.540 0.517 0.986 0.982 0.986 1.000 0.171 0.164 1.000 0.637 0.604 1.000 1.000 1.000
0.0 0.5 0.820 0.333 0.311 0.694 0.100 0.127 0.743 0.750 0.751 0.957 0.505 0.470 0.906 0.109 0.113 0.937 0.935 0.929
0.1 0.5 0.893 0.283 0.276 0.843 0.126 0.138 0.849 0.861 0.843 0.983 0.447 0.433 0.972 0.161 0.158 0.972 0.973 0.973
0.3 0.5 0.980 0.217 0.216 0.979 0.255 0.253 0.978 0.979 0.976 1.000 0.425 0.411 1.000 0.311 0.320 1.000 1.000 0.999
0.5 0.5 0.994 0.132 0.133 0.995 0.443 0.409 0.992 0.992 0.994 1.000 0.272 0.269 1.000 0.506 0.489 1.000 1.000 1.000
WC
WA
0.0 0.0 0.035 0.037 0.044 0.039 0.044 0.043 0.033 0.032 0.138 0.047 0.033 0.037 0.034 0.033 0.040 0.033 0.038 0.131
0.1 0.0 0.459 0.056 0.059 0.561 0.182 0.179 0.456 0.453 0.486 0.286 0.058 0.068 0.367 0.150 0.150 0.293 0.285 0.358
0.3 0.0 1.000 0.043 0.042 1.000 0.875 0.865 1.000 1.000 1.000 0.994 0.046 0.061 0.998 0.771 0.748 0.994 0.995 0.997
0.5 0.0 1.000 0.070 0.082 1.000 0.997 0.997 1.000 1.000 1.000 1.000 0.055 0.069 1.000 0.992 0.989 1.000 1.000 1.000
0.0 0.1 0.413 0.145 0.140 0.338 0.064 0.064 0.332 0.334 0.388 0.278 0.087 0.092 0.221 0.054 0.057 0.212 0.218 0.294
0.1 0.1 0.960 0.149 0.150 0.967 0.157 0.158 0.944 0.942 0.921 0.819 0.100 0.098 0.849 0.146 0.164 0.771 0.769 0.766
0.3 0.1 1.000 0.106 0.111 1.000 0.829 0.814 1.000 1.000 1.000 1.000 0.115 0.119 1.000 0.696 0.671 1.000 1.000 1.000
0.5 0.1 1.000 0.046 0.053 1.000 0.996 0.994 1.000 1.000 1.000 1.000 0.068 0.077 1.000 0.986 0.977 1.000 1.000 1.000
0.0 0.3 0.999 0.723 0.702 0.998 0.078 0.090 0.998 0.998 0.998 0.993 0.558 0.545 0.981 0.077 0.071 0.986 0.990 0.985
0.1 0.3 1.000 0.653 0.642 1.000 0.194 0.197 1.000 1.000 1.000 1.000 0.521 0.502 1.000 0.140 0.139 1.000 1.000 1.000
0.3 0.3 1.000 0.478 0.475 1.000 0.719 0.711 1.000 1.000 1.000 1.000 0.496 0.455 1.000 0.652 0.629 1.000 1.000 1.000
0.5 0.3 1.000 0.145 0.135 1.000 0.959 0.962 1.000 1.000 1.000 1.000 0.310 0.300 1.000 0.948 0.932 1.000 1.000 1.000
0.0 0.5 1.000 0.954 0.948 1.000 0.122 0.126 1.000 1.000 1.000 1.000 0.931 0.907 1.000 0.090 0.100 1.000 1.000 1.000
0.1 0.5 1.000 0.936 0.927 1.000 0.191 0.186 1.000 1.000 1.000 1.000 0.911 0.890 1.000 0.159 0.163 1.000 1.000 1.000
0.3 0.5 1.000 0.754 0.731 1.000 0.541 0.528 1.000 1.000 1.000 1.000 0.801 0.782 1.000 0.553 0.538 1.000 1.000 1.000
0.5 0.5 1.000 0.240 0.234 1.000 0.829 0.826 1.000 1.000 1.000 1.000 0.593 0.560 1.000 0.834 0.805 1.000 1.000 1.000
Table 13:

LM Tests (Bivariate t5 Distribution): δ0 = 0.25.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.037 0.068 0.037 0.040 0.084 0.040 0.056 0.031 0.331 0.043 0.086 0.053 0.059 0.101 0.052 0.081 0.051 0.536
0.1 0.0 0.058 0.082 0.051 0.129 0.157 0.079 0.134 0.077 0.384 0.110 0.088 0.064 0.237 0.214 0.083 0.211 0.128 0.635
0.3 0.0 0.347 0.080 0.059 0.620 0.426 0.248 0.531 0.418 0.698 0.597 0.109 0.064 0.861 0.605 0.322 0.809 0.671 0.916
0.5 0.0 0.799 0.078 0.048 0.961 0.699 0.518 0.932 0.891 0.949 0.974 0.090 0.058 0.997 0.862 0.685 0.993 0.987 0.997
0.0 0.1 0.063 0.065 0.060 0.094 0.088 0.056 0.084 0.064 0.354 0.110 0.066 0.064 0.149 0.115 0.067 0.133 0.080 0.571
0.1 0.1 0.138 0.054 0.062 0.232 0.167 0.080 0.201 0.146 0.430 0.271 0.047 0.073 0.414 0.216 0.084 0.338 0.247 0.685
0.3 0.1 0.562 0.061 0.057 0.760 0.437 0.233 0.711 0.606 0.785 0.800 0.072 0.073 0.935 0.571 0.298 0.892 0.819 0.952
0.5 0.1 0.883 0.074 0.055 0.972 0.679 0.519 0.956 0.926 0.974 0.989 0.067 0.080 0.999 0.817 0.646 0.999 0.997 1.000
0.0 0.3 0.351 0.107 0.168 0.346 0.091 0.066 0.326 0.302 0.575 0.607 0.138 0.258 0.607 0.110 0.062 0.534 0.507 0.789
0.1 0.3 0.530 0.096 0.141 0.587 0.166 0.087 0.535 0.485 0.693 0.811 0.139 0.249 0.863 0.234 0.097 0.806 0.752 0.921
0.3 0.3 0.861 0.067 0.127 0.919 0.386 0.246 0.887 0.853 0.930 0.985 0.106 0.237 0.992 0.521 0.284 0.988 0.982 0.998
0.5 0.3 0.983 0.067 0.092 0.994 0.602 0.446 0.993 0.985 0.996 1.000 0.088 0.168 1.000 0.743 0.558 1.000 0.999 1.000
0.0 0.5 0.802 0.255 0.350 0.740 0.103 0.089 0.749 0.740 0.833 0.963 0.353 0.520 0.950 0.151 0.100 0.945 0.933 0.979
0.1 0.5 0.889 0.220 0.311 0.881 0.173 0.112 0.878 0.858 0.923 0.991 0.354 0.525 0.986 0.240 0.140 0.986 0.981 0.996
0.3 0.5 0.979 0.136 0.208 0.984 0.346 0.237 0.975 0.969 0.988 0.999 0.269 0.426 0.999 0.465 0.291 0.998 0.997 0.999
0.5 0.5 0.997 0.083 0.121 0.998 0.526 0.381 0.997 0.997 0.999 1.000 0.176 0.318 1.000 0.638 0.467 1.000 1.000 1.000
WC
WA
0.0 0.0 0.038 0.090 0.039 0.064 0.125 0.029 0.085 0.037 0.926 0.043 0.162 0.067 0.119 0.237 0.057 0.170 0.051 0.939
0.1 0.0 0.464 0.119 0.058 0.703 0.447 0.190 0.628 0.473 0.984 0.354 0.140 0.047 0.664 0.517 0.114 0.632 0.330 0.975
0.3 0.0 1.000 0.133 0.062 1.000 0.942 0.821 1.000 1.000 1.000 0.995 0.162 0.054 1.000 0.959 0.733 0.999 0.999 1.000
0.5 0.0 1.000 0.169 0.062 1.000 1.000 0.997 1.000 1.000 1.000 1.000 0.174 0.050 1.000 1.000 0.985 1.000 1.000 1.000
0.0 0.1 0.498 0.061 0.150 0.561 0.157 0.070 0.463 0.415 0.971 0.362 0.045 0.145 0.517 0.191 0.047 0.407 0.285 0.967
0.1 0.1 0.948 0.072 0.153 0.974 0.419 0.146 0.953 0.922 0.998 0.879 0.066 0.154 0.956 0.463 0.104 0.926 0.830 0.998
0.3 0.1 1.000 0.058 0.125 1.000 0.932 0.788 1.000 1.000 1.000 1.000 0.077 0.120 1.000 0.944 0.661 1.000 1.000 1.000
0.5 0.1 1.000 0.046 0.039 1.000 0.998 0.989 1.000 1.000 1.000 1.000 0.063 0.107 1.000 1.000 0.967 1.000 1.000 1.000
0.0 0.3 1.000 0.504 0.747 0.999 0.127 0.075 0.998 1.000 1.000 0.998 0.250 0.621 0.996 0.216 0.064 0.995 0.995 0.999
0.1 0.3 1.000 0.451 0.731 1.000 0.357 0.140 1.000 1.000 1.000 1.000 0.249 0.610 1.000 0.477 0.149 1.000 1.000 1.000
0.3 0.3 1.000 0.338 0.571 1.000 0.852 0.642 1.000 1.000 1.000 1.000 0.211 0.562 1.000 0.893 0.532 1.000 1.000 1.000
0.5 0.3 1.000 0.092 0.198 1.000 0.981 0.938 1.000 1.000 1.000 1.000 0.124 0.395 1.000 0.987 0.916 1.000 1.000 1.000
0.0 0.5 1.000 0.925 0.979 1.000 0.146 0.114 1.000 1.000 1.000 1.000 0.763 0.949 1.000 0.233 0.107 1.000 1.000 1.000
0.1 0.5 1.000 0.872 0.955 1.000 0.292 0.159 1.000 1.000 1.000 1.000 0.736 0.932 1.000 0.396 0.137 1.000 1.000 1.000
0.3 0.5 1.000 0.632 0.813 1.000 0.662 0.455 1.000 1.000 1.000 1.000 0.596 0.861 1.000 0.789 0.442 1.000 1.000 1.000
0.5 0.5 1.000 0.148 0.306 1.000 0.903 0.774 1.000 1.000 1.000 1.000 0.360 0.671 1.000 0.948 0.779 1.000 1.000 1.000
Table 14:

LM Tests (Bivariate t5 Distribution): δ0 = 0.5.

WS
WO
λ 0 ρ 0 LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ LMρ LM ρ a LM ρ ⋆ LMλ LM λ a LM λ ⋆ LMμ LM μ ⋆ LMκ
0.0 0.0 0.027 0.099 0.039 0.092 0.161 0.033 0.106 0.030 0.835 0.061 0.126 0.035 0.162 0.247 0.024 0.189 0.042 0.972
0.1 0.0 0.079 0.106 0.039 0.225 0.281 0.058 0.239 0.078 0.870 0.182 0.138 0.036 0.457 0.424 0.058 0.446 0.141 0.984
0.3 0.0 0.349 0.117 0.035 0.724 0.598 0.219 0.687 0.404 0.969 0.701 0.154 0.025 0.944 0.777 0.273 0.925 0.709 0.996
0.5 0.0 0.779 0.130 0.034 0.964 0.820 0.509 0.946 0.880 0.995 0.975 0.148 0.047 0.997 0.938 0.642 0.997 0.986 1.000
0.0 0.1 0.086 0.070 0.057 0.165 0.146 0.038 0.136 0.055 0.848 0.186 0.071 0.095 0.366 0.237 0.038 0.300 0.146 0.978
0.1 0.1 0.182 0.082 0.045 0.365 0.291 0.059 0.309 0.144 0.902 0.406 0.083 0.084 0.669 0.395 0.058 0.577 0.320 0.988
0.3 0.1 0.554 0.065 0.046 0.818 0.552 0.183 0.758 0.560 0.966 0.847 0.100 0.092 0.968 0.729 0.215 0.948 0.826 0.999
0.5 0.1 0.891 0.095 0.033 0.982 0.798 0.491 0.969 0.939 0.998 0.993 0.074 0.073 0.998 0.925 0.600 0.998 0.996 1.000
0.0 0.3 0.378 0.079 0.169 0.466 0.170 0.046 0.396 0.282 0.880 0.716 0.069 0.293 0.804 0.244 0.046 0.711 0.618 0.997
0.1 0.3 0.591 0.071 0.157 0.698 0.258 0.060 0.614 0.491 0.961 0.856 0.086 0.285 0.927 0.381 0.055 0.889 0.796 0.995
0.3 0.3 0.859 0.050 0.110 0.943 0.491 0.176 0.903 0.835 0.988 0.987 0.078 0.263 0.997 0.640 0.213 0.993 0.976 0.999
0.5 0.3 0.978 0.050 0.068 0.998 0.709 0.397 0.994 0.985 0.999 0.999 0.060 0.199 1.000 0.835 0.509 1.000 1.000 1.000
0.0 0.5 0.839 0.172 0.377 0.848 0.186 0.081 0.804 0.777 0.973 0.979 0.254 0.613 0.982 0.231 0.077 0.969 0.955 1.000
0.1 0.5 0.904 0.158 0.356 0.918 0.262 0.083 0.891 0.855 0.986 0.994 0.260 0.612 0.995 0.329 0.090 0.994 0.988 1.000
0.3 0.5 0.971 0.091 0.216 0.986 0.455 0.176 0.974 0.962 0.996 0.998 0.183 0.506 1.000 0.592 0.214 0.999 0.999 1.000
0.5 0.5 0.999 0.072 0.156 0.999 0.576 0.323 0.999 0.999 1.000 1.000 0.105 0.358 1.000 0.729 0.404 1.000 1.000 1.000
WC
WA
0.0 0.0 0.071 0.212 0.049 0.225 0.370 0.036 0.296 0.059 1.000 0.160 0.360 0.050 0.549 0.662 0.029 0.639 0.113 0.999
0.1 0.0 0.602 0.217 0.040 0.876 0.686 0.121 0.854 0.545 1.000 0.673 0.360 0.063 0.964 0.880 0.086 0.965 0.574 1.000
0.3 0.0 0.998 0.235 0.039 1.000 0.989 0.802 1.000 1.000 1.000 1.000 0.386 0.049 1.000 0.996 0.639 1.000 1.000 1.000
0.5 0.0 1.000 0.318 0.036 1.000 1.000 0.996 1.000 1.000 1.000 1.000 0.409 0.062 1.000 0.999 0.986 1.000 1.000 1.000
0.0 0.1 0.657 0.061 0.169 0.802 0.327 0.027 0.708 0.530 0.999 0.703 0.145 0.198 0.915 0.656 0.029 0.883 0.554 1.000
0.1 0.1 0.981 0.064 0.182 0.998 0.651 0.125 0.991 0.965 1.000 0.967 0.131 0.210 0.999 0.845 0.074 0.999 0.932 1.000
0.3 0.1 1.000 0.097 0.118 1.000 0.972 0.773 1.000 1.000 1.000 1.000 0.168 0.184 1.000 0.988 0.574 1.000 1.000 1.000
0.5 0.1 1.000 0.163 0.047 1.000 1.000 0.989 1.000 1.000 1.000 1.000 0.173 0.137 1.000 1.000 0.969 1.000 1.000 1.000
0.0 0.3 1.000 0.312 0.828 1.000 0.296 0.050 1.000 1.000 1.000 0.997 0.104 0.799 1.000 0.571 0.050 0.999 0.996 1.000
0.1 0.3 1.000 0.301 0.799 1.000 0.571 0.099 1.000 1.000 1.000 1.000 0.094 0.743 1.000 0.821 0.060 1.000 1.000 1.000
0.3 0.3 1.000 0.195 0.646 1.000 0.911 0.586 1.000 1.000 1.000 1.000 0.085 0.699 1.000 0.975 0.428 1.000 1.000 1.000
0.5 0.3 1.000 0.054 0.226 1.000 0.990 0.934 1.000 1.000 1.000 1.000 0.063 0.509 1.000 0.998 0.882 1.000 1.000 1.000
0.0 0.5 1.000 0.816 0.986 1.000 0.273 0.084 1.000 1.000 1.000 1.000 0.540 0.983 1.000 0.512 0.085 1.000 1.000 1.000
0.1 0.5 1.000 0.751 0.980 1.000 0.461 0.091 1.000 1.000 1.000 1.000 0.486 0.973 1.000 0.696 0.078 1.000 0.999 1.000
0.3 0.5 1.000 0.456 0.889 1.000 0.776 0.392 1.000 1.000 1.000 1.000 0.375 0.944 1.000 0.919 0.341 1.000 1.000 1.000
0.5 0.5 1.000 0.110 0.415 1.000 0.923 0.731 1.000 1.000 1.000 1.000 0.155 0.795 1.000 0.987 0.745 1.000 1.000 1.000

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Published Online: 2018-10-09

©2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Research Articles
  2. Testing Spatial Dependence in Spatial Models with Endogenous Weights Matrices
  3. Uniformity and the Delta Method
  4. On the Size Distortion of a Test for Equality between the ATE and FE Estimands
  5. Nonparametric estimation of natural direct and indirect effects based on inverse probability weighting
  6. Regression Discontinuity and Heteroskedasticity Robust Standard Errors: Evidence from a Fixed-Bandwidth Approximation
  7. Testing for a Functional Form of Mean Regression in a Fully Parametric Environment
  8. Dif-in-Dif Estimators of Multiplicative Treatment Effects
  9. Broken or Fixed Effects?
  10. Misspecified Discrete Choice Models and Huber-White Standard Errors
  11. Review Article
  12. Local Average and Quantile Treatment Effects Under Endogeneity: A Review
https://doi.org/10.1515/jem-2017-0015
Keywords for this article
endogenous spatial weights matrix; inference; Lagrange multiplier test; LM test; parametric misspecification; Rao’s score test; robust LM test; SARAR model; specification testing

Articles in the same Issue

  1. Research Articles
  2. Testing Spatial Dependence in Spatial Models with Endogenous Weights Matrices
  3. Uniformity and the Delta Method
  4. On the Size Distortion of a Test for Equality between the ATE and FE Estimands
  5. Nonparametric estimation of natural direct and indirect effects based on inverse probability weighting
  6. Regression Discontinuity and Heteroskedasticity Robust Standard Errors: Evidence from a Fixed-Bandwidth Approximation
  7. Testing for a Functional Form of Mean Regression in a Fully Parametric Environment
  8. Dif-in-Dif Estimators of Multiplicative Treatment Effects
  9. Broken or Fixed Effects?
  10. Misspecified Discrete Choice Models and Huber-White Standard Errors
  11. Review Article
  12. Local Average and Quantile Treatment Effects Under Endogeneity: A Review
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