(1999), Fingleton (2001, 2007), Egger and Pfaffermayr (2004), Moscone et al. ... in this line, referred to as spatial VAR models, are Carlino and deFina (1999), Di.
Testing for spatial effects in Seemingly Unrelated Regressions
Jesús Mur Departamento de Análisis Económico, Gran Vía 2-4, 50005 Zaragoza, Spain Universidad de Zaragoza,
Fernando A, López Facultad de C.C. de la Empresa Dpto. Métodos Cuantitativos e Informáticos Universidad Politécnica de Cartagena
Marcos Herrera
Working Paper Do not cite without permission from the authors
First Draft: January 2009
Revisions History:
Testing for spatial effects in Seemingly Unrelated Regressions.
Abstract The paper focuses on the case of a panel data set in which we observe, for the same group of individuals, a single equation in different periods of time. The information is treated by means of a spatial SUR model where, according to Anselin (1988a), each cross-section plays the role of an equation in the Zellner (1962) traditional approach. The problem that has been raised is the presence of spatial effects in the specification systems. Various useful tests are developed based on the principle of the Lagrange Multiplier in a maximum-likelihood framework. Also, we address the question of the time stability of the sequence of spatial dependence coefficients. The second part of the paper presents the results of a Monte Carlo experiment, solved with the aim of examining the behaviour of the methods developed in our work.
Keywords: Spatial dependence; Seemingly Unrelated Regressions; Monte Carlo. JEL Classification: C21; C50; R15
1
1. Introduction The seemingly unrelated regressions (SUR) equations are a traditional multivariate econometric formulation employed in very different fields, obviously including spatial analysis. The basis of the approach is very well-known due to the initial works of Zellner (1962), Theil (1971), Malinvaud (1970), Schmidt (1976) and Dwivedi and Srivastava (1978). Almost every econometric textbook includes a discussion of the SUR methodology, which is available in most popular econometric software packages. It hardly requires any further justification. SUR models are discussed in chapter 10 of the textbook by Anselin (1988a), which introduces the term of spatial SUR that ‘consists of an equation for each time period which is estimated for a cross-section of spatial units’ (p. 141). In each of these equations (cross-sections, in fact), some spatial elements may be introduced in either the form of mechanisms of intra-equation spatial error autocorrelation, or a spatially lagged dependent variable or both. Rey and Montouri (1999), Fingleton (2001, 2007), Egger and Pfaffermayr (2004), Moscone et al. (2007), Lauridsen et al. (2008) or LeGallo and Chasco (2008) have used this technique on different occasions. The main characteristics of the spatial SUR approach are the existence of a limited heterogeneity among the individuals (the regression coefficients are assumed to remain constant) and a certain inbalance between the cross-section dimension (where R is the number of individuals in the sample) and the number of cross-sections (T is the time dimension). Typically, in the Anselin’s approach the context of our paper, the former should be greater than the latter (or the ratio T/R should go to zero). If we need more flexibility, allowing for some individual heterogeneity, the next step may consist of specifying a spatial panel data model (Anselin et al, 2007, Baltagi, 2008, Elhorst, 2003, 2005, 2008). If the number of cross-sections is greater than the number of spatial units (the ratio R/T goes to zero) it can be preferable to change the approach, specifying, for example, one equation for each individual. Arora and Brown (1977), Hordijk and Nijkamp (1977), Hordijk (1979) or White and Hewings (1982) are examples in the same line of thought that allow for an absolute heterogeneity. 2
The covariance matrix among the errors terms of the different equations are used as a measure of spatial dependence, thus avoiding the necessity of specifying a weighting matrix. This is the method followed by Conley (1999), Chen and Conley (2001), Coakley et al. (2002), Pesaran (2005) or Conley and Molinari (2008) among others. Problems appear when R increases at a rate similar to T. As indicated by Driscoll and Kraay (1998), in this case it is necessary to introduce restrictions on the parameters to keep the dimensionality of the problem more manageable. Some recent applications in this line, referred to as spatial VAR models, are Carlino and deFina (1999), Di Giacinto (2003, 2006), Badinger et al. (2004), or Beenstock and Felsenstein (2007, 2008). Referring to the present paper, our preferred combination is a finite T and a large R. The case of a SUR model with spatial effects will be solely addressed, taking into account the question of specification and estimation. The problem is, first, to test for the presence of spatial effects in the SUR specification and, second, to confront the question of identifying the type of spatial process most adequate to the data. It should be acknowledged that, with regards to the latter question, the current situation is not very satisfactory, as LeGallo and Dall’erba (2006, p. 279) point out: ‘For our SUR specification with spatial autocorrelation and spatial regimes, no specification procedure has been formally suggested. Consequently, we apply here a sequential strategy similar to that applied for cross-sections’. The paper contains five sections. In the second section we develop a set of Lagrange Multipliers to test for the presence of spatial effects in a standard spatial SUR model. The third section contains some extension (robustness, instability and diagonality) to the basic case presented in section 2. In the fourth section we solve a Monte Carlo experiment designed to study the behaviour, in a finite sample context, of the tests previously obtained. Finally, the fifth section comments on the main conclusion reached in the paper. 2. Testing for Spatial Autocorrelation: a maximum-likelihood approach. Our specification follows the original design of Anselin (1988b): R individuals are observed in T different time periods; these are TR observations. Only one equation is specified with an 3
endogenous variable, called y, and set of explanatory variables, {x1, x2, ...,xk}. This equation is copied for each individual and time period. If there are unobserved effects, a set of (individual, temporal) dummy variables will be also introduced. The data of each individual shows some inertia which results in a structure of temporal dependence in the corresponding error term (Elhorst, 2001). The model described above corresponds to a SUR specification to which we add the existence of spatial effects. The general case of a model with an autoregressive structure both in the main equation and in the errors is presented first (known as a SARAR model). Next, the Spatial Lag Model (SLM) contains a spatial autoregressive term only in the main equation, whereas this autoregression appears only in the equation of the errors in the case of the Spatial Error Model (SEM). 2.1 A SARAR model. The characteristic of this model is that the spatial effects appear both in the main equations and in the equations of the errors:
y t = λ tW1 y t + x tβ + u t ⇒ A t y t = x tβ + u t ⇒ Bt u t = ε t u t = ρ tW2 u t + ε t A t = I R − λ tW1 B t = I R − ρ tW2
}
(1)
yt, ut and εt are vectors of order (Rx1), xt is a matrix of order (Rxk), β is a vector of parameters of order (kx1), IR is the identity matrix of order (RxR) and W1 and W2 are weighting matrices also of order (RxR). For the sake of simplicity, in the following we assume that the two weighting matrices are the same (W1=W2=W). The coefficients of the model, the same as the weighting matrix, W, are allowed to vary among the different cross-sections. However, in order to simplify the discussion, in the following we assume that vector β and matrix W remain the same, but that the parameters of dependence, ρt and λt may take different values in different cross-sections. Using a more compact notation:
4
Ay = Xβ+ u Bu = ε ε ~ N(0, Ω) ⎡ y1 ⎤ ⎢y ⎥ y = ⎢ 2 ⎥; L ⎢y ⎥ ⎣ T⎦
⎡x1 X=⎢0 ⎢L ⎢⎣ 0
0 x2 L 0
L L L L
TRXTk
TRx1
⎫⎪ ⎬ ⎪⎭
0⎤ 0 ⎥; L⎥ xT⎥⎦
(2) ⎡ u1 ⎤ u = ⎢ u 2 ⎥; ⎢ L⎥ ⎢⎣uT⎥⎦ TRx1
⎡ ε1 ⎤ ε = ⎢ ε2 ⎥ ⎢L⎥ ⎢⎣εT⎥⎦
TRx1
where A = ITR − Λ ⊗ W and B = ITR − ϒ ⊗ W ; Λ (respectively ϒ ) is a R by R diagonal matrix containing the parameters λt (respectively ρt) and ⊗ is the Kronecker product. The temporal dependence, introduced by the SUR mechanism, leads to the matrix Ω = Σ ⊗ I R where Σ is a TxT
{
}
matrix, Σ = σij;i, j = 1, 2,..., T . As usual, normality is assumed in the error terms. The logarithm of the likelihood function of the model of (2) is: −1
( Ay − Xβ) ' B ' ( Σ⊗IR ) B ( Ay − Xβ) (3) RT R l(y; θ) = − ln ( 2π) − ln Σ + ∑ Tt =1 ln Bt + ∑ Tt =1 ln At − 2 2 2 where θ ' = ⎣⎡β '; λ1;L; λ T; ρ1;L; ρT; vech(Σ) '⎤⎦ is the vector of parameters of the model, of order (k+2T+T(T+1)/2)x1. The ML estimation of these parameters results in a nonlinear optimization problem which can be solved applying standard techniques (Wang and Kockelman 2007). In Appendix, Section A.I more details are included in relation to the score vector and the information matrix. In order to test for the existence of spatial effects in the specification of (1), we can obtain the corresponding Lagrange. The null hypothesis of the absence of spatial effects implies that: H0 : λt = ρt = 0 (∀t) vs
HA : No H0
(4)
After a few calculi, the details of which appear in Section A.II of the Appendix, the final expression of the test is1: 1
Henceforth the following notation will be used:
LM = ⎡ g(θ) H 0 ⎤ ' ⎡⎢ I T,R (θ) H ⎤⎥ 0⎦ ⎣ ⎦ ⎣
−1
⎡ g(θ) ⎤ ~ χ 2(d.f .) H 0 ⎦ as ⎣
where g(θ) is the score (vector of first derivatives of the likelihood function), I T,R (θ) the information matrix and d.f. means degrees of freedom
5
LM SUR SARMA
= ⎡ g '( λ ) ⎢⎣ H0
⎡ I − I I −1 I I ⎤ g '(ρ) ⎤⎥ ⎢ λλ λβ ββ βλ λρ ⎥ H0 ⎦ Iρλ Iρρ ⎥⎦ ⎢⎣
(
−1 ⎡ g
(λ ) H ⎤ 0 ⎢ ⎥ χ 2(2T) (5) g ⎢ (ρ) H ⎥ as 0⎦ ⎣
)
ˆ 'U ˆ ⎤ ' τ where, ‘ o ’, is the Hadamard product, with g (λ) = ⎡ Σ −1 o ( Uˆ 'YL ) ⎤ ' τ and g (ρ) = ⎡ Σ −1 o U L ⎦ ⎣ ⎣ ⎦ H0 H0
ˆ is a (RxT) matrix whose columns contain the SUR residuals τ is a (Tx1) vector of ones, and U ˆ = [ uˆ uˆ L uˆ ] . U ˆ is the vector of spatially lagged corresponding to each cross-sections: U 1 2 T L ˆ = [ Wuˆ Wuˆ L Wuˆ ] ; YL is specified analogously. The terms of the residuals U 1 2 T L information matrix appear in expression (A.9) of the Appendix, Section A.II. 2.2 A SLM model.
The Spatial Lag Model is a particular case of the SARAR model: y t = λ tW y t + x tβ + ε t ⇒ A t y t = x tβ + ε t
(6)
A t = I R − λ tW
All the elements that intervene in this specification have already been defined. Using a more compact matrix notation:
}
Ay = Xβ+ ε ε ~ N(0, Ω)
(7)
where A = ITR − Λ ⊗ W and Ω is the same matrix as that indicated in (2). The logarithm of the likelihood function, introducing the SLM structure of (7), is: l(y; θ) = −
( Ay − Xβ) ' ( Σ ⊗ IR ) RT R ln ( 2π ) − ln Σ + ∑ Tt =1 ln A t − 2 2 2
θ ' = [β '; λ1;L; λ T; vech(Σ) ']
where
−1
( Ay − Xβ)
(8)
is the vector of parameters of the model, of order
(k+T+T(T+1)/2)x1. The details are shown in Appendix, Section A.III. The null hypothesis is that the spatial lag of the endogenous variable is, statistically, not relevant: H0 : λt = 0 (∀t)
vs
HA : No H0
The Lagrange Multiplier emerges as: 6
(9)
−1 ⎡ ⎤ LM SUR SLM = g '(λ ) H ⎣ I λλ − I λβIββIβλ ⎦ 0
−1
g (λ )
The terms of the score can be expressed as: g ( λ )
H0
H0
~ χ 2(T)
(10)
as
(
)
ˆ ' Y ⎤ ' τ whereas the = ⎡ Σ −1 o U L ⎦ ⎣
−1 elements that intervene in the matrix I λλ − I λβ Iββ Iβλ are defined in expression (A21) in the
Appendix, Section A.IV. 2.3 The SEM model
The SUR model with a structure of spatial dependence in the error term, SEM, is well known due to the seminal work of Anselin (1988b). However, to be thorough, we have include this case in our discussion. The structure of the model is simple: y t = x tβ + u t u t = ρ tW u t + ε t ⇒ B t u t = ε t
}
(11)
B t = I R − ρ tW More concisely: y = Xβ + u ; Bu = ε
with
ε ~ N(0, Ω = Σ ⊗ IR )
and
B = ITR − ϒ ⊗ W
(12)
The logarithm of the likelihood function: −1
( y − Xβ) ' B'( Σ⊗IR ) B ( y − Xβ) RT R l(y; θ) = − ln ( 2π ) − ln Σ + ∑Tt =1 ln B t − 2 2 2
(13)
where θ ' = ⎡⎣β '; ρ1;L; ρT; vech(Σ) '⎤⎦ is a vector of parameters, of order (k+T+T(T+1)/2)x1. More details appear in Section A.V of the Appendix. The null hypothesis that there is no SEM structure in the error terms of the SUR means that: H0 : ρt = 0 (∀t)
vs HA : No H0
(14)
The obtaining of the corresponding Multiplier is straightforward (refer to Section A.VI of the Appendix): LM SUR SEM = g '(ρ) H ⎣⎡ Iρρ ⎦⎤ 0
−1
g (ρ)
7
H0
~ χ 2(T)
as
(15)
As in previous cases, a more compact expression is proposed for the term of the score vector, g (ρ)
H0
(
)
ˆ 'U ˆ ⎤ ' τ and I = tr( W ' W ) I + tr( WW) Σ −1 o Σ . = ⎡ Σ −1 o U L ⎦ ρρ R ⎣
3. The robust Multipliers and some other extensions. The Lagrange Multipliers developed in the second section are flexible so they can be adapted to different situations. It is worth mentioning that they depend only on the estimation of the SUR model under the null of no spatial effects, which can be solved using standard techniques. Several extensions of the basic discussion of Section 2 follow. Firstly, we obtain the robust version of the raw Lagrange Multipliers; then we move to the problem of the lack of constancy of the parameters of spatial dependence, considering a temporal perspective. The final part of the section focuses on a simpler but more important question of SUR modelling, that of testing for the diagonality of the matrix Σ. 3.1 The robust Multipliers.
The tests of the second section will help us to improve the specification of spatial SUR models. The weakness of this battery of Lagrange Multipliers is that they are not robust to misspecification errors in the alternative (Davidson, 2000). As a consequence of this lack of robustness, it will be more difficult to identify the type of misspecification that affects the model (Bera and Yoon, 1993). In this section, we follow the reasoning of Anselin et al. (1996) in the SUR search of robustness for the raw LM SUR SLM and LM SEM tests. A brief introduction to the problem
follows. The likelihood function of the SARAR model discussed in Section 2.1, L [ ϕ; λ; ρ] , depends on three groups of parameters: those associated with the basic SUR structure, ϕ ' = [β '; vech(Σ)] , those linked to the spatial lag of the endogenous variable, λ ' = [ λ1;L; λ T] , and those that introduce spatial dependence into the error terms, ρ ' = ⎣⎡ρ1;L; ρ T ⎦⎤ . If, for example, vector ρ is zero, the likelihood function collapses in L1 [ ϕ; λ ] and the unidirectional test, whose null hypothesis is that H0: λ=0,
8
2 leads to the LM SUR SLM statistic of (10), distributed as a centered chi-squared χ (T, 0) . Under
alternative hypotheses of the type H A : λ = ξ
R , with ξ ≠ 0 , this distribution will have a non-
−1 centrality parameter, χ 2(T, π1) , equal to π1 = ξ ' I λ⋅ϕξ with I λ⋅ϕ = I λλ − I λϕI ϕϕ I ϕλ . The presence of
this parameter is what gives power to the test. If the true likelihood function is assumed to be L2 [ ϕ; ρ] , where λ is zero and ρ is different from zero, the LM SUR SLM statistic will have a non-centrality bias even if the null hypothesis, H0: λ=0, holds. In particular, for alternatives of the type H A : ρ = ζ
R ; ζ ≠ 0 , the asymptotic distribution
−1 −1 of the test statistic is a χ 2(T, π 2) , where π 2 = ζ ' I λρ⋅ϕI λ⋅ϕ I ϕρ . This I λρ⋅ϕζ being I λρ⋅ϕ = I ρλ − I λϕI ϕϕ
factor of non-centrality will be positive if the terms of the score associated with λ and ρ ( g ( λ ) g ( ρ)
and H0
, respectively) are correlated. Under the composite hypothesis that the two sets of parameters H0
are zero, expressions (A9) and (A.10) show that Iλρ is a non-null diagonal matrix, but not null. Given that matrix Iϕλ is also different from zero but Iϕρ=0, then I λρ⋅ϕ = I ρλ ≥ 0 . The consequence is that the parameter of non centrality π2 will be positive and the LM SUR SLM statistic will, unduly, reject the hypothesis that the vector of parameters λ is zero when the data has been generated by a SUR with a SEM structure. The discussion for the LM SUR SEM statistic of (15), under the assumption that the data has been generated by HA : λ = ζ
L1 [ ϕ; λ ] ,
is analogous. To be precise, under alternatives of the type
R ; ζ ≠ 0 , the asymptotic distribution of this statistic is a χ 2(T, π 3) , where
−1 −1 π 3 = ζ ' I λρ⋅ϕI ρ⋅ϕ I λρ⋅ϕζ with I ρ⋅ϕ = I ρρ − I ρϕI ϕϕ I ϕρ . For the same reasons highlighted before, it will
be true that I λρ⋅ϕ = I ρλ ≥ 0 and π3>0. This result induces the LM SUR SEM statistic to wrongly reject the hypothesis that the vector of parameters ρ is zero when the data has been generated by a SUR with a SLM structure. 9
The proposal of Bera and Yoon (1993, p. 652) ‘is to construct a size-resistant test ... where we adapt the statistic for the nuisance parameters’. The correction consists in adjusting the bias that
affects both the score and the covariance matrix that intervene in the raw Lagrange Multiplier. In our case:
}
H0 : λ t = 0 H A : No H0
(16)
−1 ⎡ SUR ⎡ ⎤ ⎤ 2 −1 −1 −1 LM* SLM = ⎢g (λ) H0 − Iλρ⋅ϕIρ⋅ϕg (ρ) H0 ⎥ ' ⎡⎣ Iλ⋅ϕ−Iλρ⋅ϕIρ⋅ϕIλρ⋅ϕ⎤⎦ ⎢g (λ ) H0 − Iλρ⋅ϕIρ⋅ϕg (ρ) H0 ⎥ ~ χ (T) ⎣ ⎦ ⎣ ⎦ as
where: = ⎡ Σ −1 o ⎣ g (λ) = ⎡ Σ −1 o ⎣ H0 g ( ρ)
H0
( Uˆ ' Uˆ L )⎤⎦ ' τ ( Uˆ 'YL )⎤⎦ ' τ
(17)
I λρ⋅ϕ = I ρλ I ρ⋅ϕ = I ρρ −1 I λρ⋅ϕ = I ρλ I λ⋅ϕ = I λλ − I λϕI ϕϕ I λϕ −1 0 ⎤ −1 = ⎡ I ββ I ϕϕ ⎢ I σσ⎥⎦ ⎣
The information needed to obtain the value of (16) appears in expressions (A9) and (A10) of the Appendix. Likewise:
}
H0 : ρt = 0 H A : No H 0
(18) −1 ⎡
SUR ⎡ ⎤ ⎤ 2 −1 −1 −1 LM* SEM = ⎢g (ρ) H0 − Iλρ⋅ϕIλ⋅ϕg (λ ) H0 ⎥ ' ⎣⎡ Iρ⋅ϕ− I λρ⋅ϕIλ⋅ϕIλρ⋅ϕ⎦⎤ ⎢g (ρ) H 0 − Iλρ⋅ϕIλ⋅ϕg (λ ) H 0 ⎥ ~ χ (T) ⎣ ⎦ ⎣ ⎦ as
where: = ⎡ Σ −1 o ⎣ g (λ) = ⎡ Σ −1 o ⎣ H0 g ( ρ)
H0
I λρ⋅ϕ = I ρλ I λϕ = ⎡⎣I λβ 0⎤⎦
( Uˆ ' Uˆ L )⎤⎦ ' τ ( Uˆ 'YL )⎤⎦ ' τ −1 I λ⋅ϕ = I λλ − I λϕI ϕϕ I λϕ −1 = I ϕϕ
⎡ Iββ 0 ⎤ ⎢ I σσ ⎥⎦ ⎣
(19)
−1
Anselin et al (1996) show that the corrections suggested by Bera and Yoon (1993) work well in relation to the spatial Lagrange Multipliers obtained from a single cross-section estimation. Our expectancy is that the same happens in a SUR context.
10
3.2 Testing for the time-constancy of the parameters of spatial dependence.
The coefficients of spatial dependence of the SUR specifications of Section 2 are allowed to vary in each cross-section. If the cross-sections refer to consecutive and close time periods, it is reasonable to question if these coefficients remain the same over time. In order to appreciate further the testing procedure for this case, let us introduce in the first place the model of the null. Under the restriction of time constancy of the spatial coefficients, the SARAR model of (2) becomes: Ay = Xβ + u ⎪⎫ Bu = ε ⎬ ε ~ N(0, Ω) ⇒ Ω = Σ ⊗ IR ⎪⎭ ˆ A = ITR −λ ⊗ W = IT ⊗ ( IR −λW) = IT ⊗ A ˆ B = ITR −ρ⊗ W = IT ⊗ ( IR −ρW) = IT ⊗ B
(20)
ˆ and Bˆ are two matrices of order (RxR). The number of parameters to estimate reduces to where A
(k+2+T(T+1)/2). If we adopt a SLM model, the restricted version is: Ay = Xβ+ ε ε ~ N(0, Ω) ⇒ Ω = Σ⊗ IR
}
ˆ A = ITR −λ ⊗ W = IT ⊗ ( IR −λW) = IT ⊗ A
(21)
The number of parameters to estimate is (k+1+T(T+1)/2), the same as in the SEM case, under the assumption of constancy: y = Xβ+ u ⎫⎪ Bu = ε ⎬ ε ~ N(0, Ω) ⇒ Ω = Σ⊗ IR ⎪⎭ ˆ B = ITR −ρ⊗ W = IT ⊗ ( IR −ρW) = IT ⊗ B
(22)
It is evident that the hypothesis of temporal homogeneity in the spatial dependence parameters has important consequences, so it should be tested adequately. One simple solution is the likelihood ratio which compares the likelihood of the ample model against that obtained with its restricted version (that is, the model of 2 against the model of 20 in the SARAR case; the model of 7 against that of 21 in the SLM case and the model of 11 against that of 22 in the SEM case). This 11
procedure requires the estimation of the model of the null hypothesis and of the model of the alternative hypothesis. The Lagrange Multiplier is a more attractive procedure because only the estimation of the restricted model is needed. Below, the main results are shown. As mentioned, in the SARAR case, we compare the unrestricted specification of expression (2) with the restricted version of (20). These two models are related by a set of 2(T-1) restrictions that lead to the composite null hypothesis: H0 : λ1 = λ2 =... = λT and ρ1 = ρ2 =... = ρT
vs
HA : No H0
(23)
The expression of the Lagrange Multiplier is the usual quadratic form of the score vector (which appears in expression A3) on the inverse of the information matrix (as indicated in A5, A6 in the Appendix), in both cases having been evaluated under the null hypothesis of (23). That is: ⎡ ⎤ ⎡ T,R (θ) ⎤ LM SUR SARMA (λ , ρ) = ⎣ g(θ) H 0 ⎦ ' ⎢⎣ I H 0 ⎥⎦
−1
⎡ g(θ) ⎤ χ 2(2(T − 1)) H 0 ⎦ as ⎣
(24)
with2:
g(θ) H0
⎡ ⎤ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂β ⎥ ⎢ ∂l ⎥ ⎢ ∂λ ⎥ t ⎢ t =1,...T ⎥ = ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂ρ t ⎥ ⎢ t =1,...T ⎥ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂σ ij ⎥ ⎣⎢i, j=1,...T ⎦⎥
⎡ ⎡g (β) ⎤ ⎢ H0 ⎢ ⎥ ⎢ − tr ⎢ g (λ ) H ⎥ ⎢ 0 =⎢ = g (ρ) ⎥ ⎢ − tr ⎢ H0 ⎥ ⎢ ⎢g ⎥ ⎢ ⎢⎣ (Σ) H 0 ⎥⎦ ⎢ ⎣
0
⎤ ⎤ y⎥ ˆ ⊗ BW ⎦⎥ ⎥ ⎥ ⎥ ⎤ ⊗ W ⎥ uˆ ⎥ ⎦ ⎥ ⎥ ⎦
( Aˆ W ) + εˆ ' ⎡⎣⎢( Σ ) ( ) t =1,...T −1 ⎡ Σ −1Ett ˆ ˆ W ' + ε ( B ) ⎢⎣( ) −1
−1E tt
t =1,...T
0
(25)
H0
In the case of the SLM of (6), there are (T-1) restrictions in the null hypothesis: H0 : λ1 = λ2 = ... = λT
vs HA : No H0
(26)
The expression of the Multiplier does not vary: ⎡ ⎤ ⎡ T,R (θ) ⎤ LM SUR SLM (λ ) = ⎣ g(θ) H 0 ⎦ ' ⎢⎣ I H 0 ⎥⎦
−1
⎡ g(θ) ⎤ χ 2(T − 1) H 0 ⎦ as ⎣
(27)
with: 2
For the sake of brevity, in what follows we use the following notation:
∂l
∂λ
=
∂l ∂l ∂l ∂l ∂l , being λ = [ λ1; λ 2;K ; λ T ] and ρ = ⎡⎣ρ1; ρ 2;K; ρ T ⎤⎦ two = = ; ; ∂ λ t ∂ρ ∂ρ t ∂vech(Σ) ∂ σ ij
t =1,...T
t =1,...T
i, j=1,...T
vectors of order (Tx1).
12
g(θ) H0
⎡ ⎤ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂β ⎥ ⎡ ⎤ ⎢ ⎥ ⎡ g (β ) ⎤ ⎢ 0 ⎥ H0 ⎥ ⎢ ∂l ⎥ ⎢ ⎢ st ' −1 R ⎢ ⎥ ˆ W + ∑ s=1 σ εˆ sW y t ⎥⎥ =⎢ = ⎢ − tr A ⎥ = g (λ ) ⎢ ⎥ H 0 ⎢ ∂λ t ⎥ ⎢ ⎥ t =1,...T ⎢ t =1,...T ⎥ ⎢⎣ g (Σ ) H 0 ⎥⎦ ⎢ ⎥⎦ 0 ⎣ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂ σ ij ⎥ ⎢⎣i, j=1,...T ⎥⎦
(
)
(28)
Lastly, in the SEM of (11) we obtain: H0 : ρ1 = ρ2 = . . . = ρT
vs
HA : No H0
(29)
with: ⎡ ⎤ ⎡ T,R (θ) ⎤ LM SUR SEM (ρ) = ⎣ g(θ) H 0 ⎦ ' ⎢⎣ I H 0 ⎥⎦
−1
⎡ g(θ) ⎤ χ 2(T − 1) H 0 ⎦ as ⎣
(30)
where:
g(θ) H0
⎡ ⎤ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂β ⎥ ⎤ ⎡g ⎤ ⎡ ⎢ ⎥ 0 ⎥ (β) H ⎥ ⎢ ⎢ 0 ⎢ ∂l ⎥ ⎢ ⎥ st − 1 R ' =⎢ ⎥ = ⎢ g (ρ) ⎥ = ⎢ − tr Bˆ W + ∑ s=1 σ εˆ sW uˆ t ⎥ ⎢ H0 ⎥ ⎢ ∂ρ t ⎥ ⎥ t =1,...T ⎢ ⎥ ⎢ g t 1,...T = ⎢ ⎥ (Σ) H ⎢⎣ ⎥⎦ 0⎦ 0 ⎣ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂ σ ij ⎥ ⎣⎢i, j=1,...T ⎦⎥ H
(
)
(31)
0
More details can be found in Section A.VII of the Appendix. 3.3 Testing for the diagonality of the matrix Σ.
Up to now, our main concern has been analyzing the spatial structure of the SUR model. In this section we focus on the assumption of non-diagonality of matrix Σ. If the hypothesis of diagonality cannot be rejected, the SUR model simplifies in a set of (unrelated) cross-sections. On the contrary, the non-diagonality of matrix Σ reinforces the need to also consider the temporal evolution of the spatial system. This problem is well known in SUR literature where different proposals can be found. Among them, those that best fit the approach developed here are the Likelihood Ratio and the Lagrange Multiplier tests (Breusch and Pagan, 1980). The only aspect to bear in mind is that, in this case, both 13
tests must be used in models where a certain spatial structure has been introduced. For example, in the case of the SARAR of (2): Ay = Xβ+ u ⎫⎪ Bu = ε ⎬ ε ~ N(0, Ω) ⇒ Ω = Σ⊗ IR ⎪⎭
(32)
matrix Σ describes the temporal correlation that exists between the errors of the T cross-sections once the spatial structure that intervenes in the specification has been filtered. Then, the test statistic is obtained in the usual way: ⎡σ12 0 K 0 ⎤ ⎫ ⎢ ⎥⎪ 0 σ12 K 0 ⎥ ⎪ ⎢ Σ = : ⎪ H0 2 ⎢K K K K ⎥ ⎬ → LM Σ = R ∑ Tt =1 ∑ st −=11 r 2ts χ (T(T −1) 2) as ⎢0 0 K ⎥⎪ σT2 ⎦ ⎪ ⎣ No H 0 ⎪⎭ HA :
(33)
with: r ts =
∑R r =1 ( εˆ tr − εˆ t )( εˆ sr − εˆ s ) ∑R r =1 ( εˆ tr − εˆ t )
2
∑R r =1 ( εˆ sr − εˆ s )
2
(34)
rts is the coefficient of correlation between the residuals of cross-sections t and s. The Likelihood Ratio is defined as usual: LR Σ = 2 ⎡⎣ l(y; θ)|H A − l(y; θ)|H0 ⎤⎦ = R ⎡ ln Σ% H0 − ln Σ% H A ⎤ χ 2(T(T −1) 2) ⎣ ⎦ as
(35)
being l(y; θ)|H0 y l(y; θ)|HA the value of the log-likelihood function under the null and alternative hypothesis, respectively, and Σ% H 0 and Σ% H A the corresponding covariance matrices. Under mild regularity conditions, the two statistics converge asymptotically to the χ 2 with T(T-1)/2 degrees of freedom (as shown, for example, in Breusch and Pagan, 1980). Breusch and Pagan (1980) use the least squares residuals because they discuss the case of a static model, but this isn´t so in our case. In the framework of (33), and under the null hypothesis of diagonality, there is a panel of ML residuals obtained from the estimation of a spatial model (SARAR, SLM or SEM) for each of the T cross-sections. In other words, if we introduce spatial dependencies in the cross-sections, obtaining 14
the asymptotic distribution under the null of diagonality
is not immediate. As a useful
approximation, Lee (2004) shows the (quasi) maximum likelihood residuals of an (spatial) econometric model with correct specified behaviour, asymptotically (with R), as a Gaussian random variable. Under this setting, the test statistic of (33) will converge to the chi-squared distribution, as assumed.
4. A Monte Carlo study: design and empirical results. In this section we present the results of an extensive Monte Carlo exercise which focus on the tests developed in Sections 2 and 3. The first sub-section examines the robustness of the diagonal test in the presence of spatial dependence in SURE models. The second sub-section deals with the behaviour of the raw and robust Lagrange Multipliers. The third part of the Monte Carlo considers the test of instability of the spatial dependence parameters. The design of the experiment considers the factors that, in our opinion, may have greater impact on tests: the structure of temporal dependence among the errors of the cross-sections, the intensity of the spatial dependence, its stability (or lack of stability) and the size of the sample. To evaluate the power of the test, two types of alternatives are considered: SLM : y t = (I R − ρt W) −1 (α + β x t + e t ) ; t = 1,..., T ⎫⎪ ⎬ SEM : y t = α + β x t + (I R − λ t W) −1 e t ; t = 1,..., T ⎪⎭
(36)
The structure of the equations is very simple: they contain an intercept, with a value of 2 ( α t = 2 ), and only one regressor (xt) drawn from a N(0,1) distribution. The value of the coefficient β is 3, which assures that, in absence of spatial effects ( ρ t = 0 / λ t = 0 ), the R2 coefficient of the
Least Square (LS) regression of the equation will be close to 0.9. The error terms e = (e1 ,..., eT ) have been obtained from a multivariate normal distribution, centred on zero and covariance matrix defined as: Σ k = (σ k,ij ) with σ k,ij = 0.95s k |i − j|
15
(37)
The parameter sk determines the intensity of the dependency between the residuals of the equations i and j. Three values for sk have been used, (106, 20, 0.2), which introduce different levels of time dependence: null (s1=106; Σ1=IT); moderate (s2=20; in matrix Σ2) and extreme (s3=0.2; in matrix Σ3). Moreover, we have simulated different values for T, namely T= 2, 4 and 10, combined with two different sample sizes corresponding to regular lattices of (7x7), R=49, and (9x9), R=81, respectively. For each cross-section a row-standardized weighting matrix has been defined using a rook scheme. Each experiment has been repeated 2000 times. 4.1. The diagonality test.
There are several studies where the hypothesis of non-diagonality of the matrix Σ, have been revised (Shiba and Tsurumi 1988, Kurata 2001, Dufour and Kahlaf, 2002, Tsay 2004). This correlation hypothesis, as well as the different equations of a SUR model are crucial as they clarify the specification. In this part of the paper we highlight the size problems of the two most popular tests, LR and LM, when a spatial dependency structure is present in each of the transversal cuts. The design of our experiment considers only one regressor and a balanced relation between R and T. In spite of this favourable design, Table 1 and Figure 1 show that both tests have problems with size in the presence of spatial dependence. If there is independence among the SUR equations, the LR Σ test appears to slightly overestimate the 5% nominal size, especially when the number of cross-section (T=10) increases. These results are in accordance with well documented literature on LR-based multivariate tests. Under spatial dependence, the two tests, LM Σ and LR Σ , have serious problems with empirical size, even for low to moderate values of the spatial dependence parameter. There is a strong tendency to greatly over-reject high values of the spatial dependence parameter, whatever the type of process (SLM or SEM) used. The number of cross-sections (T) also has clear harmful effects.
16
Table 1: Empirical size of the LM and LR test in spatial dependent processes. R=81*.
T=2 LM Σ LR Σ
SLM Model T=4 T=10 LM Σ LR Σ LM Σ LR Σ
T=2 LM Σ LR Σ
SEM Model T=4 T=10 LM Σ LR Σ LM Σ LR Σ
0.00
0.061
0.065
0.051
0.055
0.058
0.088
0.061
0.065
0.051
0.055
0.058
0.088
0.10
0.051
0.054
0.047
0.060
0.066
0.103
0.054
0.055
0.048
0.059
0.069
0.107
0.20
0.065
0.07
0.073
0.085
0.108
0.161
0.059
0.061
0.073
0.082
0.074
0.128
0.30
0.073
0.079
0.097
0.107
0.207
0.274
0.066
0.069
0.084
0.093
0.142
0.192
0.40
0.076
0.078
0.261
0.286
0.516
0.600
0.088
0.093
0.105
0.120
0.251
0.322
0.50
0.173
0.186
0.315
0.348
0.959
0.975
0.102
0.107
0.164
0.185
0.500
0.582
0.60
0.213
0.235
0.377
0.415
0.978
0.988
0.133
0.142
0.277
0.302
0.799
0.848
0.70
0.289
0.296
0.538
0.578
1.000
1.000
0.191
0.198
0.445
0.473
0.980
0.986
0.80
0.719
0.729
0.811
0.821
1.000
1.000
0.279
0.286
0.728
0.740
1.000
1.000
0.90
0.798
0.788
0.992
0.992
1.000
1.000
0.477
0.485
0.935
0.939
1.000
1.000
0.95
0.842
0.848
0.994
0.995
1.000
1.000
0.592
0.596
0.990
0.991
1.000
1.000
ρ
(*) Equivalent results are obtained for another values of R.
If the successive cross-sections repeat a similar pattern of spatial dependence, a false impression of persistence in the values corresponding to each individual through time emerge. The tests interpret this situation as a symptom of temporal dependence. In order to correct this weakness, as indicated in section 3.4, it is necessary to filter the spatial structure in the different cross-section beforehand. By doing this, assumption of diagonality in the Σ matrix can be properly analysed. Figure 1: Empirical size of the LM Σ and LR Σ tests in spatially dependent processes. SLM case
SEM case
1,0
1,0
0,8
0,8
0,6
0,6
0,4
0,4
0,2
0,2
0,0
0,0 0
0,1
0,2
0,3
0,4 T=2
0,5 T=4
0,6
0,7
0,8
0,9
0
1
0,1
0,2
0,3
0,4 T=2
T=10
17
0,5 T=4
0,6 T=10
0,7
0,8
0,9
1
4.2. The tests of spatial dependence.
Below we present the size and power of the spatial dependence tests developed in Sections 2 and 3. The empirical size of the test, at a nominal level of 5% appears in Table 2. The estimates are very close to the nominal value; any tendency to over or underestimate the size can be noticed. Table 2: Empirical size of the spatial dependence tests, 5% nominal level. * SUR SUR * SUR SUR LM SEM LM SLM LM SLM LM SARAR R LM SUR SEM 49 0.045 0.048 0.056 0.055 0.043 Σ1 81 0.053 0.049 0.045 0.044 0.046 49 0.048 0.051 0.056 0.058 0.053 T=2 Σ2 81 0.050 0.047 0.053 0.052 0.051 49 0.050 0.051 0.056 0.056 0.052 Σ3 81 0.048 0.049 0.052 0.057 0.057 49 0.045 0.041 0.056 0.050 0.048 Σ1 81 0.042 0.042 0.051 0.052 0.047 49 0.043 0.046 0.053 0.056 0.048 T=4 Σ2 81 0.050 0.053 0.054 0.056 0.051 49 0.041 0.044 0.055 0.053 0.051 Σ3 81 0.043 0.048 0.053 0.056 0.049 49 0.036 0.040 0.055 0.060 0.047 Σ1 81 0.044 0.040 0.049 0.051 0.047 49 0.041 0.040 0.062 0.063 0.049 T=10 Σ2 81 0.044 0.045 0.055 0.052 0.047 49 0.038 0.040 0.070 0.075 0.056 Σ3 81 0.045 0.046 0.067 0.069 0.059 See expression (37) for the meaning of Σj (j=1,2,3)
Tables 3, 4 and 5 present the results corresponding to the power of the tests, also at a nominal value of 5%. Case 1 combines time stability in the spatial dependence coefficients and temporal independence among the SUR equations. Case 2 maintains the situation of time stability of the spatial parameters but introduces an extreme temporal correlation in the SUR structure. Cases 3 and 4 are characterized by the time heterogeneity of the spatial dependence parameters. In Case 3, the T cross-sections are independent, as in Case 1, and spatial dependence only exists in the first cross-section (the remaining cross-sections are spatially independent). Case 4 combines a very strong correlation in the SUR equations with the instability in the spatial coefficients of Case 3. To be brief, we present only the results for the case where R=81 (9x9 lattice), for a representative range of values of the spatial correlation coefficient.
18
Table 3: Empirical power T=2. R=81. Percentage of rejections of the null hypothesis, 5% sig. level. SEM Model SLM Model Case 1: Temporal independence (Σ1 = IT ) and stability in spatial dependence ρi = ρ j (∀i, j) SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.044 0.058 0.079 0.123 0.181 0.306 0.501 0.775 1.000
0.046 0.053 0.049 0.057 0.048 0.040 0.051 0.048 0.059
SUR
* SUR
LM SLM LM SLM
0.065 0.239 0.545 0.732 0.814 0.973 1.000 1.000 1.000
0.075 0.228 0.523 0.681 0.735 0.943 1.000 1.000 1.000
SUR
LM SARMA
λ
0.054 0.175 0.424 0.617 0.708 0.939 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
LM SEM LM SEM
0.075 0.289 0.609 0.897 0.986 0.999 1.000 1.000 1.000
0.063 0.236 0.482 0.786 0.961 0.995 1.000 1.000 1.000
LM SLM
SUR
LM SLM
* SUR
0.071 0.097 0.190 0.380 0.532 0.725 0.911 0.996 0.997
0.056 0.065 0.064 0.063 0.102 0.102 0.109 0.129 0.378
SUR
LM SARMA
0.071 0.215 0.500 0.829 0.975 0.998 1.000 1.000 1.000
Case 2: Extreme temporal correlation (Σ3 ) and estability in spatial dependence ρi = ρ j (∀i, j) SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.046 0.188 0.077 0.195 0.238 0.383 0.551 0.792 1.000
0.053 0.086 0.079 0.058 0.066 0.050 0.051 0.049 0.057
SUR
* SUR
LM SLM LM SLM
0.643 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.652 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SUR
LM SARMA
λ
0.530 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
LM SEM LM SEM
0.073 0.263 0.626 0.893 0.983 1.000 1.000 1.000 1.000
0.078 0.271 0.612 0.889 0.983 0.999 1.000 1.000 1.000
LM SLM
SUR
LM SLM
* SUR
0.047 0.068 0.068 0.079 0.127 0.160 0.195 0.284 0.459
0.053 0.065 0.061 0.080 0.117 0.106 0.185 0.194 0.265
Case 3: Temporal independence (Σ1 = IT ) and instability in spatial dependence ρ1 ≠ 0; ρi = 0 SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.042 0.041 0.062 0.076 0.108 0.211 0.631 0.883 1.000
0.047 0.050 0.047 0.039 0.043 0.040 0.048 0.055 0.057
SUR
* SUR
LM SLM LM SLM
0.059 0.079 0.223 0.620 0.502 0.871 1.000 1.000 1.000
0.061 0.075 0.196 0.590 0.438 0.822 1.000 1.000 1.000
SUR
LM SARMA
λ
0.055 0.060 0.161 0.502 0.384 0.793 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
LM SEM LM SEM
0.068 0.174 0.355 0.617 0.843 0.942 0.993 1.000 1.000
0.067 0.153 0.279 0.506 0.669 0.891 0.981 0.995 1.000
LM SLM
SUR
LM SLM
* SUR
0.060 0.069 0.131 0.192 0.385 0.440 0.561 0.830 0.985
0.052 0.056 0.060 0.075 0.070 0.091 0.104 0.257 0.101
Case 4: Extreme temporal correlation (Σ3 ) and instability in spatial dependence ρ1 ≠ 0; ρi = 0 ρ 0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
SUR
* SUR
LM SEM LM SEM
0.074 0.266 0.486 0.463 0.683 0.697 0.775 0.821 0.991
0.047 0.064 0.062 0.071 0.052 0.062 0.058 0.059 0.061
SUR
* SUR
LM SLM LM SLM
0.460 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.429 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SUR
LM SARMA
λ
0.343 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
19
SUR
* SUR
LM SEM LM SEM
0.755 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.637 0.986 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SUR
* SUR
LM SLM
LM SLM
0.253 0.632 0.786 0.791 0.945 0.948 0.946 0.921 0.995
0.082 0.112 0.161 0.202 0.204 0.217 0.268 0.342 0.260
SUR
LM SARMA
0.067 0.211 0.523 0.833 0.962 0.998 1.000 1.000 1.000 (i > 1) SUR
LM SARMA
0.064 0.137 0.284 0.515 0.751 0.909 0.983 0.999 1.000 (i > 1) SUR
LM SARMA
0.665 0.993 1.000 1.000 1.000 1.000 1.000 1.000 1.000
Table 4: Empirical power T=4. R=81. Percentage of rejections of the null hypothesis, 5% sig level. SEM Model
SLM Model
Case 1: Temporal independece (Σ1 = I T ) and stability in spatial dependence ρi = ρ j (∀i, j) SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.039 0.059 0.084 0.185 0.221 0.394 0.510 0.971 1.000
0.045 0.056 0.044 0.041 0.050 0.037 0.047 0.060 0.052
SUR
* SUR
LM SLM LM SLM
0.076 0.228 0.671 0.949 0.972 0.999 1.000 1.000 1.000
0.079 0.220 0.624 0.917 0.952 0.996 1.000 1.000 1.000
SUR
LM SARMA
λ
0.065 0.164 0.524 0.899 0.928 0.994 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
SUR
LM SEM
LM SEM
LM SLM
0.084 0.373 0.817 0.984 1.000 1.000 1.000 1.000 1.000
0.084 0.298 0.707 0.950 0.997 1.000 1.000 1.000 1.000
0.065 0.108 0.229 0.451 0.696 0.924 0.991 1.000 1.000
* SUR
LM SLM
SUR
LM SARMA
0.054 0.060 0.082 0.069 0.098 0.163 0.166 0.234 0.335
0.076 0.285 0.715 0.956 1.000 1.000 1.000 1.000 1.000
Case 2: Extreme temporal correlation (Σ3 ) and estability in spatial dependence ρi = ρ j (∀i, j) SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.049 0.099 0.168 0.139 0.292 0.837 0926 0.983 1.000
0.059 0.084 0.094 0.141 0.088 0.096 0.099 0.105 0.120
SUR
* SUR
LM SLM LM SLM
0.990 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.988 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SUR
LM SARMA
λ
0.963 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
SUR
LM SEM
LM SEM
LM SLM
0.090 0.359 0.817 0.987 1.000 1.000 1.000 1.000 1.000
0.099 0.349 0.814 0.987 1.000 1.000 1.000 1.000 1.000
0.051 0.070 0.073 0.102 0.122 0.175 0.235 0.306 0.487
* SUR
LM SLM
0.055 0.069 0.065 0.089 0.125 0.146 0.208 0.268 0.502
Case 3: Temporal independence (Σ1 = IT ) and instability in spatial dependence ρ1 ≠ 0; ρi = 0 SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.048 0.052 0.055 0.065 0.091 0.124 0.218 0.340 0.577
0.050 0.046 0.042 0.042 0.054 0.054 0.055 0.044 0.058
SUR
* SUR
LM SLM LM SLM
0.053 0.112 0.180 0.277 0.268 0.702 0.885 1.000 1.000
0.050 0.107 0.161 0.228 0.229 0.621 0.752 0.881 0.991
SUR
LM SARMA
λ
0.047 0.088 0.119 0.186 0.201 0.558 0.722 0.859 0.910
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
SUR
LM SEM
LM SEM
LM SLM
0.061 0.127 0.259 0.506 0.770 0.898 0.984 0.998 1.000
0.059 0.104 0.200 0.415 0.547 0.775 0.952 0.979 0.986
0.059 0.065 0.088 0.126 0.348 0.396 0.477 0.848 0.968
* SUR
LM SLM
0.053 0.051 0.049 0.056 0.062 0.095 0.097 0.099 0.245
Case 4: Extreme temporal correlation (Σ3 ) and instability in spatial dependence ρ1 ≠ 0; ρi = 0 ρ 0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
SUR
* SUR
LM SEM LM SEM
0.066 0.247 0.327 0.577 0.521 0.705 0.888 0.921 1.000
0.051 0.051 0.075 0.046 0.060 0.055 0.058 0.049 0.061
SUR
* SUR
LM SLM LM SLM
0.483 0.988 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.458 0.983 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SUR
LM SARMA
λ
0.362 0.960 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
20
SUR
* SUR
SUR
LM SEM
LM SEM
LM SLM
0.641 0.996 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.554 0.989 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.140 0.275 0.357 0.565 0.578 0.771 0.875 0.976 0.982
* SUR
LM SLM
0.054 0.086 0.091 0.102 0.126 0.103 0.143 0.072 0.111
SUR
LM SARMA
0.083 0.291 0.711 0.967 0.998 1.000 1.000 1.000 1.000 (i > 1) SUR
LM SARMA
0.059 0.100 0.185 0.398 0.646 0.850 0.959 0.995 1.000 (i > 1) SUR
LM SARMA
0.514 0.981 1.000 1.000 1.000 1.000 1.000 1.000 1.000
Table 5: Empirical power T=10. R=81. Percentage of rejections of the null hypothesis, 5% sig level. SEM Model
SLM Model
Case 1: Temporal independence (Σ1 = IT ) and stability in spatial dependence ρi = ρ j (∀i, j) SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.033 0.044 0.090 0.194 0.291 0.570 0.681 0.882 1.000
0.041 0.044 0.045 0.041 0.052 0.035 0.045 0.051 0.086
SUR
* SUR
LM SLM LM SLM
0.075 0.340 0.861 1.000 1.000 1.000 1.000 1.000 1.000
0.082 0.338 0.827 0.997 1.000 1.000 1.000 1.000 1.000
SUR
LM SARMA
λ
0.051 0.241 0.718 0.996 0.999 1.000 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
SUR
LM SEM
LM SEM
LM SLM
0.098 0.536 0.963 1.000 1.000 1.000 1.000 1.000 1.000
0.083 0.419 0.911 0.999 1.000 1.000 1.000 1.000 1.000
0.073 0.120 0.273 0.553 0.857 0.978 0.999 1.000 1.000
* SUR
LM SLM
SUR
LM SARMA
0.057 0.058 0.079 0.087 0.125 0.138 0.218 0.225 0.320
0.079 0.398 0.897 0.998 1.000 1.000 1.000 1.000 1.000
Case 2: Extreme temporal correlation (Σ3 ) and estability in spatial dependence ρi = ρ j (∀i, j) SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.057 0.149 0.299 0.422 0.530 0.626 0.810 0.926 1.000
0.055 0.105 0.159 0.129 0.133 0.148 0.267 0.441 0.908
SUR
* SUR
LM SLM LM SLM
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SUR
LM SARMA
λ
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
SUR
LM SEM
LM SEM
LM SLM
0.087 0.523 0.952 1.000 1.000 1.000 1.000 1.000 1.000
0.086 0.519 0.948 1.000 1.000 1.000 1.000 1.000 1.000
0.076 0.064 0.094 0.094 0.159 0.199 0.232 0.361 0.388
* SUR
LM SLM
0.068 0.055 0.093 0.079 0.156 0.203 0.209 0.353 0.329
Case 3: Temporal independence (Σ1 = IT ) and instability in spatial dependence ρ1 ≠ 0; ρi = 0 SUR
ρ
* SUR
LM SEM LM SEM
0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
0.046 0.042 0.048 0.057 0.065 0.116 0.189 0.263 0.670
0.042 0.046 0.038 0.047 0.045 0.046 0.048 0.053 0.049
SUR
* SUR
LM SLM LM SLM
0.062 0.091 0.125 0.196 0.217 0.556 0.622 0.751 1.000
0.057 0.098 0.111 0.195 0.193 0.470 0.684 0.754 1.000
SUR
LM SARMA
λ
0.050 0.078 0.090 0.144 0.151 0.394 0.625 0.699 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
SUR
* SUR
SUR
LM SEM
LM SEM
LM SLM
0.059 0.086 0.160 0.305 0.550 0.770 0.944 0.989 1.000
0.050 0.075 0.136 0.252 0.445 0.691 0.822 0.932 0.944
0.059 0.059 0.077 0.101 0.145 0.160 0.454 0.643 0.947
* SUR
LM SLM
0.055 0.049 0.065 0.066 0.059 0.067 0.069 0.104 0.107
Case 4: Extreme temporal correlation (Σ3 ) and instability in spatial dependence ρ1 ≠ 0; ρi = 0 ρ 0.02 0.06 0.10 0.14 0.16 0.20 0.30 0.40 0.90
SUR
* SUR
LM SEM LM SEM
0.059 0.120 0.243 0.396 0.241 0.440 0.517 0.598 0.872
0.050 0.048 0.055 0.038 0.065 0.036 0.066 0.061 0.049
SUR
* SUR
LM SLM LM SLM
0.170 0.985 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.162 0.979 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SUR
LM SARMA
λ
0.136 0.939 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
21
SUR
* SUR
SUR
LM SEM
LM SEM
LM SLM
0.424 0.950 0.998 1.000 1.000 1.000 1.000 1.000 1.000
0.362 0.880 0.994 1.000 1.000 1.000 1.000 1.000 1.000
0.137 0.325 0.296 0.476 0.492 0.610 0.762 0.916 0.929
* SUR
LM SLM
0.082 0.079 0.062 0.090 0.112 0.108 0.102 0.077 0.093
SUR
LM SARMA
0.084 0.397 0.894 0.999 1.000 1.000 1.000 1.000 1.000 (i > 1) SUR
LM SARMA
0.054 0.077 0.135 0.239 0.422 0.658 0.875 0.973 0.999 (i > 1) SUR
LM SARMA
0.348 0.882 0.992 1.000 1.000 1.000 1.000 1.000 1.000
Anselin et al (1996) study the behaviour of the Lagrange Multiplier for an omitted spatial lag of the endogenous variable, using one single cross-section, referred to as LM φ . This test is equivalent to our LM SUR SEM . Anselin et al (1996) report an empirical power of 0.463 for the LM φ with ρ=0.1 while in our experiment, and for T=10, Case 1, the empirical power of the same test increases up to 0.861 (0.543 with T=2). This increment is exclusively due to the use of several cross-sections, not only one. Moreover, the power of the test increases more rapidly when there is time dependence among the equations. For example, in Case 2 and using 10 cross-sections, the power function attains its maximum value of 1 with a very low value of the parameter, 0.02 when T=10. The estimated power of the LM SUR SEM test also improves with the number of equations (T). Figure 2 represents the estimated power curves for the different cross-sections (T=2, 4, 10; Case 1). We add the results obtained by Anselin et al. (1996) for the same test, using only one equation. The improvement is evident, especially for low values of the spatial dependence coefficient. However contrary to what happened in the case of the test LM SUR SLM , the use of a stronger temporal correlation between the SUR equations does not help to increase in the power of the test. Figure 2: Estimated power of the LM SUR SEM test, Case 1. 1,0
0,8
0,6
0,4
0,2
0,0 0
0,1
0,2
0,3
0,4
0,5
T=1 (Anselin et al 1996)
22
0,6 T=2
0,7 T=4
0,8 T=10
0,9
1
SUR As highlighted by Anselin et al. (1996), the LM SUR SLM and LM SEM tests are not robust to the
specification of the alternative hypothesis. This is the reason for the existence of the robust tests, developed in section 3.1. SUR The performance of the two tests, LM *SUR SLM and LM *SEM , are very similar. They are
scarcely affected by the number of equations (our results, with different T’s are in line with the findings of Anselin et al, 1996). The intensity of the correlation among the SUR equations has a greater impact, especially in the case of a very high temporal dependence (Case 2). The ‘robustness adjustment’ does not work quite as well which is evident in the case of the LM *SUR SEM . For the case of the LM *SUR SLM test we obtain rejection frequencies that are slightly higher than the nominal value of 5%. This tendency is more severe as the SUR structure among the equations becomes stronger. The same description applies in relation to the LM *SUR SEM : good behaviour for moderate to low relations of temporal dependence in the SUR system and a sharp deterioration in the case of a very strong time dependence. 4.3. Testing for the instability of the spatial dependence coefficients in SUR models.
The (possible) time instability of the parameter of spatial dependence is a factor that clearly affects the power of the tests. Cases 3 and 4 presented in Tables 4 to 6 show that this instability is a SUR potential risk factor that especially affects the LM SUR SLM and LM SEM tests.
It is interesting to note that the two tests react differently, depending on the level of correlation between the SUR equations. As such, for processes of substantive spatial dependence, DGP1, the LM SUR SLM test suffers severe power losses under time instability. On the other hand, for a SEM type process, the time dependence benefits the power of LM SUR SEM test. Neither of the two robust tests appears to be affected by the (possible) time stability of the spatial parameter and, in some cases, they perform better than their corresponding raw Multipliers. Given this apparently puzzling situation, we believe that it is really important to check for the constancy of the coefficients of spatial dependence in SUR specifications. The tests developed 23
in Section 3.2 may be useful in this aspect. Tables 6 and 7 present the results obtained for the two SUR Multipliers, LM SUR SEM ( λ ) and LM SLM (ρ ) .
Table 6: Empirical size of the time instability spatial dependence tests. Percentage of rejections of the null hypothesis, 5% significance level. SLM: LM SUR SEM: LM SUR SLM (ρ ) SEM ( λ ) R Σ1
49 81
Σ2
49 81 49
Σ3 81
ρ/λ
T=2
T = 4 T = 10 T = 2
T = 4 T = 10
0.2 0.5 0.9 0.2 0.5 0.9 0.2 0.5 0.9 0.2 0.5 0.9 0.2 0.5 0.9 0.2 0.5 0.9
0.048 0.052 0.044 0.064 0.064 0.053 0.048 0.059 0.042 0.051 0.043 0.036 0.050 0.046 0.025 0.060 0.054 0.048
0.054 0.047 0.053 0.045 0.053 0.049 0.047 0.045 0.050 0.048 0.054 0.049 0.055 0.053 0.026 0.046 0.047 0.045
0.032 0.034 0.049 0.030 0.050 0.037 0.036 0.036 0.036 0.042 0.045 0.034 0.044 0.034 0.051 0.043 0.041 0.049
0.049 0.045 0.041 0.051 0.053 0.044 0.047 0.048 0.046 0.056 0.051 0.054 0.042 0.044 0.037 0.046 0.043 0.027
0.038 0.035 0.031 0.044 0.040 0.041 0.045 0.031 0.025 0.041 0.048 0.039 0.027 0.028 0.025 0.044 0.047 0.030
0.026 0.036 0.044 0.044 0.032 0.055 0.034 0.035 0.050 0.027 0.060 0.050 0.031 0.039 0.079 0.037 0.048 0.076
It is encouraging that neither of the two tests suffers with size problems: as shown in Table 6, the empirical size of both tests is always close to the nominal 5%. In order to evaluate the power of both tests, we have repeated the same design as in Cases 3 and 4 (that is, there is spatial correlation in the first cross-section, whereas the remaining crosssections are characterized by the spatial independence). The results are shown in Table 7. Both tests react sharply to the value of the spatial dependence parameter of the first cross-section, especially in the case of an SLM process. The power of both is smaller when the number of cross-sections (T) is high. As expected, the estimated power function increases rapidly as we introduce a stronger time correlation structure between the SUR equations.
24
Table 7: Estimated power of the time instability spatial dependence tests. Percentage of rejections of the null hypothesis, 5% significance level. R=81. SLM: LM SUR SLM (ρ ) ρ1
T=2
T=4
T = 10
Σ1
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0.190 0.601 0.905 0.998 1.000 1.000 1.000 1.000 1.000
0.201 0.663 0.977 0.996 1.000 1.000 1.000 1.000 1.000
0.114 0.431 0.845 0.959 1.000 1.000 1.000 1.000 1.000
ρ1
T=2
T=4
Σ3
0.02 0.06 0.10 0.14 0.16 0.18 0.20 0.30 0.40 0.90
0.371 0.963 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.405 0.992 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
SEM: LM SUR SEM ( λ ) λ1
T=2
T=4
T=10
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0.082 0.143 0.273 0.468 0.621 0.838 0.955 0.992 1.000
0.090 0.136 0.252 0.436 0.693 0.866 0.972 0.995 0.999
0.057 0.100 0.192 0.347 0.562 0.777 0.928 0.988 0.999
T = 10
λ1
T=2
T=4
T=10
0.141 0.965 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.02 0.06 0.10 0.14 0.16 0.18 0.20 0.30 0.40 0.90
0.125 0.279 0.493 0.706 0.859 0.954 0.995 1.000 1.000 1.000
0.103 0.179 0.324 0.532 0.722 0.907 0.976 0.993 1.000 0.995
0.070 0.125 0.196 0.319 0.470 1.000 1.000 1.000 1.000 0.960
5. Final Conclusions Multivariate models are a very common, useful technique to simultaneously deal with temporal and spatial structures. The seemingly unrelated regressions, SUR models, are one of these techniques, which are very popular due to the fact that they require few assumptions and computations. As far as we know, since the seminal works of Anselin (1988a and b), very little has been commented on about the possibilities of combining SUR models with spatial processes. We think that it is time to fill the gap. Specifically, in this paper we develop a battery of tests based on the principle of the Lagrange Multiplier that are, firstly, efficient in order to detect the presence of spatial elements in SUR models and, secondly, useful to discriminate the type of spatial structure, SLM or SEM, that highlights the data. The solved Monte Carlo contains some results that are worth highlighting. Firstly, the tests of diagonality (specifically, time independence between the different cross-sections) tend to be very uncertain under the presence of spatial dependence. In general, they tend to unduly reject the null 25
hypotheses. Secondly, the performance of the robust Multipliers, whose aim is to discriminate between SLM and SEM processes, deteriorates much more so as the symptoms of time instability increase. For these reasons, and because of the difficulties of dealing simultaneously with several cross-sections, it is advisable to follow a conservative model specification strategy when dealing with SUR models. General-to-Specific approaches appear to adapt better to this scheme, although these comments pertain to a future research agenda.
Acknowledgements: This work has been carried out with the financial support of project ECO2009-10534/ECON of the Ministerio de Ciencia y Tecnología of the Reino de España and the proyect 11897/PHCS/09 of the Fundación Séneca – Agencia de Ciencia y Tecnología de la Región de Murcia. References Acosta, P., De Mattos E. and A. Fava (2006): Monte Carlo properties of spatial dependence LM tests, Applied Economics Letters, 13, 905–910 Anselin, L. (1988a): Spatial Econometrics: Methods and Models. Dordrecht: Kluwer. Anselin, L. (1988b): A test for Spatial Autocorrelation in Seemingly Unrelated Regressions, Economics Letters, 28, 335-341. Anselin, L. (1990): Spatial dependence and spatial structural instability in applied regression analysis. Journal of Regional Science 30, 185–207. Anselin, L, A. Bera, R. Florax and M. Yoon, (1996): Simple Diagnostic Tests for Spatial Dependence. Regional Science and Urban Economics, 26, 77-104. Anselin, L, LeGallo, J. and H. Jayet (2007): Spatial panel econometrics. In Matyas L. and Sevestre P. (Eds.): The Econometrics of Panel Data. Dordrecht: Kluwer Academic Publishers. Arora, S. and M. Brown (1977): Alternative Approaches to Spatial Autocorrelation: An Improvement over Current Practice. International Regional Science Review, 2, 67-78. Badinger, H., W. Müller and G. Tondl (2004): Regional Convergence in the European Union, 19851999: A Spatial Dynamic Panel Analysis. Regional Studies, 38, 241–253. Baltagi, B. (2008): Econometric Analysis of Panel Data, 4th edition. New York: John Wiley and Sons. Bera, A. and M. Yoon, (1993): Specification Testing with Locally Misspecified Alternatives. Econometric Theory 9, 649-658. Beenstock, M. and D. Felsenstein (2007): Spatial Vector Autoregressions. Spatial Economic Analysis 2, 167-196. Beenstock, M. and D. Felsenstein (2008): Regional Heterogeneity, Conditional Convergence and Regional Inequality. Regional Studies 42, 475-488. Breusch, T. and Pagan, A. (1980): The Lagrange multiplier test and its applications to model specification in econometrics. Review of Economic Studies 47, 239–254. Burridge, P. (1981): Testing for a Common Factor in a Spatial Autoregression Model. Environment and Planning A, 13, 795-800. Carlino, G. and R. DeFina (1999): The Differential Regional Effects of Monetary Policy: Evidence from the U.S. States. Journal of Regional Science 39, 339-358. Chen, X. and T. Conley (2001): A New Semiparametric Spatial Model for Panel Time Series. Journal of Econometrics 105, 59-83. 26
Coakley, J., A. Fuentes and R. Smith (2002): A Principal Components Approach to Cross-section Dependence in Panels. Working Paper, Department of Economics, Birbeck College. University of London, London. Conley, T. (1999): GMM Estimation with Cross-sectional Dependence. Journal of Econometrics 92, 1-45. Conley, T. and F. Molinari (2008): Spatial Correlation Robust Inference with Errors in Location or Distance. Journal of Econometrics 40, 76-96. Davidson, J. (2000). Econometric Theory. Oxford: Blackwell. Di Giacinto, V. (2003): Differential Regional Effects of Monetary Policy: A Geographical SVAR Approach. International Regional Science Review 26, 313-341. Di Giacinto, V. (2006): A Generalized Space-Time Model with an Application to Regional Unemployment Analysis in Italy. International Regional Science Review, 29, 159-198. Driscoll, J. and A. Kraay (1998): Consistent Covariance Matrix Estimation with Spatially Dependent Panel Data. The Review of Economics and Statistics 80, 549-560. Dufour, J. and L. Khalaf (2002): Exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions. Journal of Econometrics 106, 143–170. Dwivedi, T. and K. Srivastava (1978): Optimality of Least Squares in Seemingly Unrelated Regressions Models. Journal of Econometrics, 7, 391-395. Elhorst, P. (2001): Dynamic Models in Space and Time. Geographical Analysis, 33, 244-268. Elhorst, P. (2003): Specification and Estimation of Spatial Panel Data Models. International Regional Science Review, 26, 244-268. Elhorst, P. (2004): Serial and Spatial Error Dependence in Space-Time Models. In Getis A., J. Mur and H. Zoller (Eds.): Spatial Econometrics and Spatial Statistics, pp 176-193. Basingstoke: Palgrave Macmillan. Elhorst, P. (2005): Unconditional Maximum Likelihood Estimation of Linear and Log-linear Dynamic Models for Spatial Panels. Geographical Analysis, 37, 85-106. Elhorst, P. (2008): Serial and spatial error correlation. Economics Letters 100 422–424. Elhorst, P., Blien, U. and K. Wolf (2006): New evidence on the wage curve: A spatial panel approach. International Regional Science Review 30: 173-191. Egger, P. and M. Pfaffermayr (2004): Distance, Trade and FDI: A Hausman-Taylor SUR Approach. Journal of Applied Econometrics 16, 227-246. Fingleton, B. (2001): Theoretical Economic Geography and Spatial Econometrics: Dynamic Perspectives. Journal of Economic Geography 1, 201-225. Fingleton, B. (2007): Multi-equation spatial econometric model, with application to EU manufacturing productivity growth. Journal of Geographical Systems 9, 119-144. Hordijk, L. (1979): Problems in Estimating Econometric Relations in Space. Papers, Regional Science Association, 42, 99-115. Hordijk, L. and P. Nijkamp (1977): Dynamic Models of Spatial Autocorrelation. Environment and Planning A, 9, 505-519. Kosfeld, R. and D. Christian (2006): Thresholds for employment and unemployment: A spatial analysis of German regional labour markets, 1992–2000. Papers in Regional Science, 85, 523542 Kurata, H. (2001): One-sided tests for independence of seemingly unrelated regression equations, Journal of Multivariate Analysis 90, 393–406 Lee, L. (2004): Asymptotic Distributions of Maximum Likelihood Estimators for Spatial Autoregressive Models. Econometrica 72, 1899-1925 LeGallo, J. and S. Dall’erba (2006): Evaluating the temporal and spatial heterogeneity of the European convergence process: 1980-1999, Journal of Regional Science, 46, 269-288. LeGallo, J. and C. Chasco (2008): Spatial Analysis of Urban Growth in Spain, 1900-2001. Empirical Economics, 34, 59-80. Lauridsen, J. Bech, M. López, F. and M. Maté (2008): A spatiotemporal analysis of public pharmaceutical expenditures. Annals of Regional Science, DOI 10.1007/s00168-008-0259-2. 27
Lundberg, J. (2006): Spatial interaction model of spillovers from locally provided public services Regional Studies, 40, 631-644. Malinvaud, E. (1970): Statistical Methods of Econometrics. Amsterdam: North Holland. Moscone, F., Tosetti, E. and M. Knapp (2007): SUR model with spatial effects: An application to mental health expenditure. Health Economics, 16, 1403-1408. Mur, J. and A. Angulo (2006): The Spatial Durbin Model and the Common Factor Tests. Spatial Economic Analysis 2, 207-226. Mur, J. and A. Angulo (2009): Model selection strategies in a spatial setting: some additional results. Regional Science and Urban Economics, 39, 200-213 Pesaran, H. (2005): Estimation and Inference in Large Heterogeneous Panels with Cross Section Dependence. Technical Report, Department of Economics. University of Cambridge, Cambridge. Rey, S. and B. Montouri (1999): US Regional Income Convergence: A Spatial Econometrics Perspective. Regional Studies, 33, 143-156. Schmidt, P. (1976): Econometrics. New York: Marcel Dekker. Shiba, T. and H. Tsurumi (1988): Bayesian and Non-Bayesian Test of independence in seemingly unrelated regressions International economic review, 29, 2 377-395. Theil, H. (1971): Principles of Econometrics. New York: John Wiley and Sons. Tsay, W. (2004): Testing for contemporaneous correlation of disturbances in seemingly unrelated regressions with serial dependence. Economics Letters 83 69-76. Wang, X. and K. Kockelman (2007): Specification and Estimation of a Spatially and Temporally Autocorrelated Seemingly Unrelated Regression Model: Application to Crash Rates in China. Transportation, 34, 281-300. White, E. and G. Hewings (1982): Space-Time Employment Modeling: Some Results Using Seemingly Unrelated Regression Estimators. Journal of Regional Science, 22, 283-302. Zellner, A. (1962): An Efficient Method of Estimating Seemingly Unrelated Regressions and Test of Aggregation Bias. Journal of the American Statistical Association, 57, 348-68.
28
Apendix: Results of the ML estimation of SUR models with spatial effects This Appendix contains additional results on the ML estimation of the different SUR models introduced in sections 2 and 3. Specifically, we focus on the expressions of the score vector and of the information matrix. We include also details in relation to the obtaining of the Lagrange Multipliers. Section A.I. Score vector and information matrix of the SUR-SARAR model. As indicated, the compact expression of this model is: Ay = Xβ+ u ⎫⎪ Bu = ε ⎬ ε N(0, Ω) ⇒ Ω = Σ⊗ IR ⎪⎭
(A1)
A = ITR − Λ⊗ W B = ITR − ϒ⊗ W The logarithm of the likelihood function of the model of (A1) is the following: −1
l(y; θ) = −
( Ay − Xβ) ' B ' ( Σ ⊗ IR ) B ( Ay − Xβ ) RT R ln ( 2π ) − ln Σ + ∑ Tt =1 ln B t + ∑ Tt =1 ln A t − 2 2 2
(A2)
where θ ' = ⎡⎣β '; λ1;L; λ T; ρ1;L; ρT; vech(Σ) '⎤⎦ is the vector of parameters of the model, of order (k+2T+T(T+1)/2)x1. The score vector has the following structure: ⎡ ⎤ ⎢ ∂l ⎥ ⎢ ⎥ ⎤ ⎢ ∂β ⎥ ⎡ X ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ B ( Ay − Xβ ) ∂ l ⎢ ⎥ ⎢ ⎥ tt −1W ⎤ + Ay − Xβ ' B ' ⎡ −1 ⊗ ⎡ ⎤ B E W tr y − ⊗ ⎢ ⎥ ( ) A Σ I ⎢ ∂λ t ⎥ R⎦ ⎣ t ⎦ ⎣ ⎥ ⎢ t =1,...T ⎥ ⎢ g(θ) = ⎢ ∂l ⎥ = ⎢ − tr ⎡⎣B −t 1W ⎤⎦ + ( Ay − Xβ ) ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E tt ⊗ W ( Ay − Xβ ) ⎥ ⎥ ⎢ ⎥ ⎢ Ay − Xβ ) ' B ' ⎡⎢ Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( Ay − Xβ ) ⎥ ( ⎢ ∂ρ t ⎥ ⎢ R ⎣ ⎦ −1 ij ⎥ ⎢ t =1,...T ⎥ ⎢ − tr ⎡⎣ Σ E ⎤⎦ + 2 ⎦ ⎢ ∂l ⎥ ⎣ 2 ⎢ ∂ σ ij ⎥ ⎢i, j=1,...T ⎥ ⎣ ⎦
(
(
(
)
)
)
(A3)
{
}
vech(Σ) is a T(T+1)x1 vector of variances and covariances, vech(Σ) = σij; j ≤ i,i = 1, 2,..., T . In
(A3)
∂l means the first partial derivative of the function l with respect to (i,j)-th element of ∂ σij
vech(Σ) ; the same applies in relation to
∂l ∂l and : both refer to the first derivative of function l ∂ρ t ∂λ t
with respect the t-th spatial dependence coefficient. Moreover, Eij (analogously Ett ) is a matrix of order (TxT) whose elements are all zero except the (i,j) and the (j,i) which have the value 1. The set of second derivatives is:
29
∂ 2l = − X ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ BX ∂β∂β ' ∂ 2l = − X ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ B E tt ⊗ W y ∂β∂ λ t ∂ 2l = −X ' ⎡B ' Σ −1E tt ⊗ W + E tt Σ −1 ⊗ W ' B ⎤ ( Ay − Xβ ) ⎥⎦ ⎣⎢ ∂β∂ρ t 2 ∂ l = −X ' B ' ⎡⎢ Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( Ay − Xβ ) ⎣ ⎦ ∂β∂ σij
(
{(
} {(
)
(
∂ 2l ∂λ 2t 2
)
(A4.a)
}
)
)
(
)
(
)
= − tr ⎡⎣ A −t 1W A −t 1W ⎤⎦ − y ' E tt ⊗ W ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ B E tt ⊗ W y
(
) ( (
(
)
∂ l = − y ' Ess ⊗ W ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ B E tt ⊗ W y ∂ λ t∂ λ s ∂ 2l = − ( Ay − Xβ ) ' ⎡⎢ E tt ⊗ W ' Σ −1 ⊗ I R B + B' Σ −1 ⊗ I R E tt ⊗ W ⎤⎥ E tt ⊗ W y ⎣ ⎦ ∂ λ t∂ρ t 2 ∂ l = − ( Ay − Xβ ) ' ⎡⎢ Ess ⊗ W ' Σ −1 ⊗ I R B + B' Σ −1 ⊗ I R Ess ⊗ W ⎤⎥ E tt ⊗ W y ⎣ ⎦ ∂ λ t∂ρs ∂ 2l = − ( Ay − Xβ ) ' B ' ⎡⎢ Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B E tt ⊗ W y ⎣ ⎦ ∂ λ t∂ σij
)( )(
(
∂ 2l ∂ρ2t 2
)
(
)(
)
(
)(
)
(
(
)
)( )(
) )
(A4.b)
)
(
)
= − tr ⎡⎣ B −t 1W B −t 1W ⎤⎦ − ( Ay − Xβ ) ' E tt ⊗ W ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E tt ⊗ W ( Ay − Xβ )
( (
) ⎡⎣Σ −1 ⊗ IR ⎤⎦ ( Ett ⊗ W ) ( Ay − Xβ) ) ⎡⎢⎣( Σ−1Eij Σ−1) ⊗ IR ⎤⎥⎦ B ( Ay − Xβ)
∂ l = − ( Ay − Xβ ) ' Ess ⊗ W ' ∂ρ t∂ρs ∂ 2l = − ( Ay − Xβ ) ' E tt ⊗ W ' ∂ρ t∂ σij
∂ 2l R ⎡ −1 ij −1 ij ⎤ = tr Σ E Σ E − ( Ay − Xβ ) ' B ' ⎡⎢ Σ −1Eij Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( Ay − Xβ ) 2 ⎦ ⎣ ⎦ ∂ σij 2 ⎣
(
)
R ∂ 2l = tr ⎡ Σ −1Esr Σ −1Eij ⎤ − ( Ay − Xβ ) ' B ' ⎡⎢ Σ −1Esr Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( Ay − Xβ ) ⎦ ⎣ ⎦ ∂ σij∂ σsr 2 ⎣
(
)
(A4.c)
(A4.d)
The expected value of these terms, changing the sign, is (for brevity’s sake, we use the ⎡ 2 ⎤ notation I ηγ = −E ⎢ ∂ l ⎥ ): ⎢⎣ ∂η∂γ ' ⎥⎦
Iββ = X ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ BX
(
)
tt Iβλ t = X ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E ⊗ W B A −1Xβ; t = 1, 2,...., T
Iβρ t = 0; t = 1, 2,...., T Iβσij = 0; i, j = 1, 2,...., T
30
(A5.a)
tt tt Xβ + tr H SARAR I λ t λt = tr ⎡⎣ A −t 1WA −t 1W⎤⎦ + β 'X ' H SARAR B '−1[ Σ ⊗ I R ] B −1; t = 1, 2,....,T ts ts Xβ + tr H SARAR I λ t λs = β 'X ' H SARAR B '−1[ Σ ⊗ I R ] B −1; t,s = 1, 2,....,T
( ) { = tr {( Σ ⊗ I ) B ⎡⎢( E Σ ) ⊗ W '⎤⎥ B ⎡E ⎣ ⎦ ⎣
}
−1 tt tt tt I λ t ρt = tr ( Σ ⊗ I R ) B '−1 ⎡⎢ E Σ −1 ⊗ W '⎤⎥ B ⎡⎣E ⊗ W⎤⎦ + ⎡⎣E ⊗ (WW)⎤⎦ ( BA ) ; t = 1, 2,....,T ⎣ ⎦
I λ t ρs
R
(
'−1
ss
−1
)
(
tt
⊗ W⎤ ( BA ) ⎦
};
−1
(A5.b)
t,s = 1, 2,....,T
)
−1 tt ij I λ t σij = tr ⎡⎢B E ⊗ W ( BA ) ⎤⎥ ⎡⎢ E Σ −1 ⊗ I R ⎤⎥ ; t;i, j = 1, 2,....,T ⎦ ⎣ ⎦⎣ tt Iρt ρt = tr ⎡⎣B −t 1W B −t 1W ⎤⎦ + σ tttr ⎡⎣E ⊗ ( W ' W ) ⎤⎦ ⎡⎣ B −1 ( Σ ⊗ I R ) B '−1⎤⎦ ; t = 1, 2,...., T
tt ss Iρt ρs = tr ⎡⎣(E Σ −1E ) ⊗ ( W ' W ) ⎤⎦ ⎡⎣ B −1 ( Σ ⊗ I R ) B '−1⎤⎦ ; t,s = 1, 2,...., T
(
(A5.c)
)
tt ij Iρt σij = tr ⎡⎢ E Σ −1E ⊗ W ⎤⎥ B '−1; t;i, j = 1, 2,...., T ⎣ ⎦
Iσijσsr =
R ⎡ −1 ij −1 sr ⎤ tr Σ E Σ E ; i, j,s,r = 1,2,....,T ⎦ 2 ⎣
(
)
(A5.d)
(
)
ts where H SARAR = A '−1 Ess ⊗ W ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ B E tt ⊗ W A −1 and σ st is the element (t,s) of the
matrix Σ −1 . We introduce the following ordering of the information matrix: ⎡ I ββ ⎤ I βλ I βρ I βσ ⎢ (kxk) (kxT) (kxT) (kx(T(T + 1) / 2)) ⎥ ⎢ ⎥ I λρ I λλ I λσ ⎢ (TxT) (TxT) (Tx(T(T + 1) / 2)) ⎥ T,R (θ) = ⎢ ⎥ I I ρρ I ρσ ⎢ ⎥ (TxT) (Tx(T(T + 1) / 2)) ⎥ ⎢ I σσ ⎢ ⎥ ((T(T + 1) / 2) ⎥ ⎢ x(T(T + 1) / 2)) ⎥⎦ ⎢⎣
(A6)
Section A.II. Testing for spatial effects in the SUR-SARAR model. The null hypothesis is that there are no spatial effects in the SUR model of (A1):
}
H0 : λ t = ρ t = 0 HA : No H0
Under H0 ⇒ A = B = ITR
(A7)
y = Xβ + u ⎪⎫ ⇒u =ε ⎬ ε N(0, Ω) ⇒ Ω = Σ ⊗ IR ⎪⎭ The score of (A3) becomes:
31
g(θ) H 0
⎡ ⎤ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂β ⎥ 0 ⎡ ⎤ ⎡ 0 ⎤ ⎡ ⎢ ∂l ⎥ ⎡g (β) ⎤ ⎢ ⎥ tt − 1 H0 ⎡ ⎤ ⎢ ∂λ ⎥ ⎢ ⎥ ⎢ uˆ ' Σ E ⊗ W y ⎥ ⎢ ∑ T σ st uˆ s' W y ⎥ ⎢ ⎡ −1 o ⎦ t t ⎥ ⎢⎣Σ ⎢ s=1 ⎢ t =1,...T ⎥ ⎢g (λ ) H ⎥ ⎢ ⎣ t =1,...T ⎥ t 1,...T = 0 ⎥=⎢ = ⎢ ∂l ⎥ = ⎢ = ⎢ T st ' ⎥=⎢ ⎥ g −1 tt ⎢ ⎥ ⎢ (ρ) H 0 ⎥ ⎢ uˆ ' ⎣⎡ Σ E ⊗ W ⎤⎦ uˆ ⎥ ⎢ ∑ s=1 σ uˆ sW uˆ t ⎥ ⎢ ⎡ Σ −1 o t =1,...T ⎢ ⎥ ⎢⎣ ⎢ ∂ρ t ⎥ ⎢g ⎥ ⎢ t =1,...T ⎥ ⎢ ⎥⎦ ⎢⎣ 0 ⎢ t =1,...T ⎥ ⎢⎣ (Σ) H 0 ⎥⎦ ⎢ ⎣ 0 ⎣ ⎦⎥ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂ σ ij ⎥ ⎣⎢i, j=1,...T ⎦⎥
( (
0 ⎤ ⎥ (A8) ˆ ⎤ U ' YL ' τ ⎥ ⎦ ⎥ ˆ 'U ˆ ⎤ ' τ⎥ U L ⎦ ⎥ ⎥⎦ 0
)
)
uˆ is the (TRx1) vector of residuals of the SUR model, estimated in the absence of spatial effects. The score is made up of two sub-vectors of zeros, of orders (kx1) and (T(T+1)/2)x1, respectively, and another two non-zero sub-vectors, both of order (Tx1), as appears in (A8). Using the Hadamard product, o , the first non-zero subvector can be expressed as: g (λ ) second as g (ρ)
H0
(
H0
(
)
ˆ ' Y ⎤ ' τ and the = ⎡ Σ −1 o U L ⎦ ⎣
)
ˆ is an (RxT) ˆ 'U ˆ ⎤ ' τ . In both cases, τ is a (Tx1) vector of ones, U = ⎡ Σ −1 o U L ⎦ ⎣
matrix whose columns contains the SUR residuals corresponding to the different cross-sections: ˆ = [ uˆ uˆ L uˆ ] , and U ˆ = [ Wuˆ Wuˆ L Wuˆ ] ; YL is specified ˆ is its spatial lag U U 1 2 T 1 2 T L L analogously. Furthermore, the elements of the information matrix, also under the null hypothesis, become: Iββ = X ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ X tt Iβλ t = X ' ⎡⎣ Σ −1E ⊗ W ⎤⎦ Xβ; t = 1, 2,...., T
(A9.a)
Iβρt = 0; t = 1, 2,...., T Iβσij = 0; i, j = 1, 2,...., T
(
)
tt 2 I λ t λ t = σ tt β 'X ' ⎡⎣E ⊗ ( W ' W) ⎤⎦ Xβ + σ tttr( W ' W) + trW ; t = 1, 2,...., T ts I λ t λs = σ tsβ 'X ' ⎡⎣E ⊗ ( W ' W) ⎤⎦ Xβ + σ tsσ tstr( W ' W); t,s = 1, 2,...., T
I λ t ρt = σ tt σ tttr( W ' W) + tr( WW); t = 1, 2,...., T
(A9.b)
I λ t ρs = σ tsσ ts ( trW ' W ) ; t,s = 1, 2,...., T I λ t σij = 0; t;i, j = 1, 2,...., T
32
(
)
Iρt ρt = tr( W ' W ) σ tt σ tt + 1 ; t = 1, 2,...., T Iρt ρs = σ tsσ tstr( W ' W); t,s = 1, 2,...., T
A9.c)
Iρt σij = 0; t;i, j = 1, 2,...., T I σijσsr =
R ⎡ −1 ij −1 sr ⎤ tr Σ E Σ E ; i, j,s, r = 1, 2,...., T ⎦ 2 ⎣
(A9.d)
All these elements have been defined previously except σts which refers to the element (t,s) of matrix Σ. To sum up, the information matrix becomes:
I T,R (θ) H 0
⎡ I ββ 0 0 ⎤ I βλ ⎢ (kxk) (kxT) (kxT) (kx(T(T + 1) / 2)) ⎥ ⎢ ⎥ 0 I λρ I λλ ⎢ (TxT) (TxT) (Tx(T(T + 1) / 2)) ⎥ ⎥ =⎢ 0 I ρρ ⎢ ⎥ (TxT) (Tx(T(T + 1) / 2)) ⎥ ⎢ I σσ ⎢ ⎥ ((T(T + 1) / 2) ⎥ ⎢ x(T(T + 1) / 2)) ⎦⎥ ⎣⎢
(A10)
This matrix is block-diagonal:
⎡ I T,R (θ) H0 = ⎢ M11 ⎣ M 21
⎧ ⎡ I ββ 0 ⎤ I βλ ⎪ ⎢ (kxk) (kxT) (kxT) ⎥ ⎪ ⎢ I λρ ⎥ I λλ ⎪M11 = ⎢ (TxT) (TxT) ⎥ ⎪ ⎢ ⎥ I ρρ ⎥ ⎪ ⎢ (TxT) ⎦ ⎣ ⎪ ⎪ ⎡ ⎤ I M12 ⎤ ⇒ ⎪M = ⎢ ((T(T −σσ 1) / 2) ⎥ ⎨ 22 M 22 ⎥⎦ ⎪ ⎢⎣ x(T(T − 1) / 2)) ⎥⎦ ⎪ ⎡0 ⎤ ⎪M12 = ⎢0⎥ ⎪ ⎢⎣0⎥⎦ ⎪ 0 0 0] = ⎪M12 [ ⎪ ⎪⎩
(A11)
Sub-matrix Iλρ is diagonal, which implies that the ML estimators of ρt and of λt, under the null hypothesis, are correlated only for the same cross-section but they are independent for different cross-sections; that is, Cov(ρ t , λ s ) = 0 if t ≠ s and Cov(ρ t , λ s ) ≠ 0 if t=s. The Lagrange Multiplier, for the hypothesis of (A7), is the quadratic form of the score evaluated in the null hypothesis (as in A8), on the inverse of the information matrix, also evaluated in the null hypothesis (as in A9 and A10). The result that we obtain is that:
33
⎡ ⎤ ⎡ T,R (θ) ⎤ LM SUR SARAR = ⎣ g(θ) H 0 ⎦ ' ⎣⎢ I H0 ⎦⎥
⎡ ⇒ LM SUR SARAR = ⎢ g ' ( λ ) H 0 ⎣
−1
⎡g(θ) ⎤ χ 2(2T) H 0 ⎦ as ⎣
−1 ⎡ I λλ − I λβI ββ I βλ I λρ ⎤ g ' (ρ) ⎤⎥ ⎢ ⎥ H0 ⎦ I ρλ I ρρ ⎥⎦ ⎣⎢
(A12)
−1 ⎡ g
(λ ) H ⎤ 0 ⎢ ⎥ χ 2(2T) g ⎢ (ρ) H ⎥ as 0⎦ ⎣
Section A.III. Score vector and information matrix of the SUR-SLM model. The model that corresponds to this case is: y t = λ tW y t + x tβ + ε t ⇒ A t y t = x tβ + ε t A t = I R − λ tW All the elements that intervene in this specification are known. More compact:
Ay = Xβ+ ε ε N(0, Ω) ⇒Ω = Σ⊗ IR
}
(A13)
(A14)
A = ITR −Λ⊗ W
Matrix Ω is the same as that indicated in (A1). The logarithm of the likelihood function, introducing the SLM structure of (A14), is: l(y; θ) = −
where
( Ay − Xβ) ' ( Σ ⊗ IR ) RT R ln ( 2π ) − ln Σ + ∑ Tt =1 ln A t − 2 2 2
θ ' = [β '; λ1;L; λ T; vech(Σ) ']
−1
( Ay − Xβ)
(A15)
is the vector of parameters of the model, of order
(k+T+T(T+1)/2)x1. The score vector has the following structure: ⎡ ⎤ ⎤ ⎢ ∂l ⎥ ⎡ ⎢ ⎥ ⎢ ⎥ X ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ ( Ay − Xβ ) ⎢ ⎥ ∂β ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ∂l ⎥ tt −1W ⎤ + Ay − Xβ ' ⎡ −1 ⊗ ⎡ ⎤ = − ⊗ g(θ) = ⎢ tr y E W ⎢ ⎥ ( ) I A Σ R ⎣ t ⎦ ⎣ ⎦ ∂λ t ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ t =1,...T ⎥ ⎢ ij −1 − 1 ⎢ ⎡ ⎤ ∂ l ⎢ ⎥ ( Ay − Xβ ) ' ⎣⎢ Σ E Σ ⊗ I R ⎦⎥ ( Ay − Xβ ) ⎥⎥ R ⎢ ij − 1 ⎢ ∂ σ ij ⎥ − tr ⎡ Σ E ⎤ + ⎥⎦ ⎦ 2 ⎢i, j=1,...T ⎥ ⎢⎣ 2 ⎣ ⎣ ⎦
(
(
)
(A16)
)
The second derivatives corresponding to this case are: ∂ 2l = − X ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ X ∂β∂β ' ∂ 2l = − X ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E tt ⊗ W y ∂β∂ λ t ∂ 2l = − X ' ⎡⎢ Σ −1Eij Σ −1 ⊗ I R ⎤⎥ ( Ay − Xβ ) ⎣ ⎦ ∂β∂ σij
(
(
)
(A17.a)
)
34
∂ 2l ∂λ 2t 2
(
)
(
)
= − tr ⎡⎣ A −t 1W A −t 1W ⎤⎦ − y ' E tt ⊗ W ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E tt ⊗ W y
(
) (
(
)
∂ l = − y ' Ess ⊗ W ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E tt ⊗ W y ∂ λ t∂ λ s ∂ 2l = − ( Ay − Xβ ) ' ⎡⎢ Σ −1Eij Σ −1 ⊗ I R ⎤⎥ E tt ⊗ W y ⎣ ⎦ ∂ λ t∂ σij
)
(
(A17.b)
)
∂ 2l R ⎡ −1 ij −1 ij ⎤ = tr Σ E Σ E − ( Ay − Xβ ) ' ⎡⎢ Σ −1Eij Σ −1Eij Σ −1 ⊗ I R ⎤⎥ ( Ay − Xβ ) ⎦ ⎣ ⎦ ∂ σij2 2 ⎣
(
)
2
(
)
∂ l R = tr ⎡ Σ −1Esr Σ −1Eij ⎤ − ( Ay − Xβ ) ' ⎢⎡ Σ −1Esr Σ −1Eij Σ −1 ⊗ I R ⎥⎤ ( Ay − Xβ ) ⎦ ⎣ ⎦ ∂ σij∂ σsr 2 ⎣
(A17.c)
The expected values, changing their sign, are: Iββ = X ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ X
(
)
tt Iβλ t = X ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E ⊗ W A −1Xβ; t = 1, 2,...., T
(A18.a)
Iβσij = 0; i, j = 1, 2,...., T tt Xβ + tr tt I λ t λ t = tr ⎡⎣ A −t 1W A −t 1W ⎤⎦ + β 'X ' H SLM H SLM [ Σ ⊗ I R ] ; t = 1, 2,...., T ts Xβ + tr ts Σ ⊗ I λ t λs = β 'X ' H SLM I R ] ; t,s = 1, 2,...., T H SLM [
(
)
(
(A18.b)
)
tt ij I λ t σij = tr ⎡⎢ E ⊗ W A −1⎤⎥ ⎡⎢ E Σ −1 ⊗ I R ⎤⎥ ; t;i, j = 1, 2,...., T ⎣ ⎦⎣ ⎦
I σijσsr =
R ⎡ −1 ij −1 sr ⎤ tr Σ E Σ E ; i, j,s, r = 1, 2,...., T ⎦ 2 ⎣
(
)
(
(A18.c)
)
tt st ts = '−1 Ess ⊗ W ' ⎡ −1 ⊗ where H SLM A I R ⎤⎦ E ⊗ W A −1 and σ , as before, is the element (t,s) of ⎣Σ
matrix Σ −1 . Now we use the following ordering of the information matrix: ⎡ ⎤ I βλ I βσ ⎢ I ββ ⎥ ⎢ (kxk) (kxT) (kx(T(T + 1) / 2)) ⎥ ⎢ ⎥ I λλ I λσ I T,R (θ) = ⎢ (TxT) (Tx(T(T + 1) / 2)) ⎥ ⎢ ⎥ I σσ ⎢ ((T(T + 1) / 2) ⎥ ⎢⎣ x(T(T + 1) / 2)) ⎥⎦
Section A.IV. Testing for spatial effects in the SUR-SLM model. The null hypothesis in which we are interested is that the spatial lag of the endogenous variable included in the right-hand side of the SUR model of (A13) is not relevant:
}
H0 : λ t = 0 HA : No H0
Under H0 ⇒ A = ITR
⇒ y = Xβ + ε ε N(0, Ω) ⇒ Ω = Σ ⊗ IR
35
}
(A19)
The score of (A16) becomes:
g(θ) H0
⎡ ⎤ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂β ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎡g (β) ⎤ ⎢ 0 ⎡ ⎥ 0 ⎢ ⎥ ⎢ ⎥ H0 ⎢ ∂l ⎥ ⎢⎡ −1 ⎢ ˆ ⎡ −1 tt ⎥ ⎢ T st ' ⎢ ⎥ ⎥ ⎤ =⎢ ⎥ = g (λ) = ⎢u ' ⎣Σ E ⊗ W⎦ y⎥ = ∑s=1σ uˆ sWy t = ⎢⎣Σ o H0 ⎥ ⎢ ⎥ ⎢ ∂λ t ⎥ ⎢ t =1,...T ⎢ ⎥ ⎢ t =1,...T ⎢ ⎥ ⎥ g ⎣⎢ = t 1,...T ⎢ ⎥ ⎣ (Σ) H0 ⎦ ⎢ 0 ⎥ ⎣ ⎦ 0 ⎣ ⎦ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂σij ⎥ ⎣⎢i,j=1,...T⎦⎥
(
0 ⎤ ⎥ ˆ ⎤ U'YL ' τ⎥ ⎦ ⎥⎦ 0
)
(A20)
As before, uˆ is the (TRx1) vector of residuals of the SUR model in the absence of spatial effects. The other terms are known. The elements of the information matrix, also under the null hypothesis, are: Iββ = X ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ X −1 tt Iβλ t = X ' ⎡⎣( Σ E ) ⊗ W ⎤⎦ Xβ; t = 1, 2,...., T
(A21.a)
Iβσij = 0; i, j = 1, 2,...., T
(
)
(
)
tt 2 I λ t λ t = σ tt ⎡⎢β ' X ' E ⊗ ( W ' W) Xβ + σ tttr( W ' W) ⎤⎥ + trW ; t = 1, 2,...., T ⎣ ⎦ ts I λ t λs = σ ts ⎢⎡β 'X ' E ⊗ ( W ' W) Xβ + σ tstr( W ' W) ⎥⎤ ; t,s = 1, 2,...., T ⎣ ⎦
(A21.b)
I λ t σij = 0; t;i, j = 1, 2,...., T I σijσsr =
R ⎡ −1 ij −1 sr ⎤ tr Σ E Σ E ; i, j,s, r = 1, 2,...., T ⎦ 2 ⎣
(A21.c)
This matrix has a block-diagonal structure:
I T,R (θ) H0
⎡ ⎤ 0 I βλ ⎢ I ββ ⎥ (kxT) ⎢ (kxk) (kxT) ⎥ ⎢ ⎥ ⎡ M11 0 I λλ =⎢ = (TxT) (Tx(T(T − 1) / 2)) ⎥ ⎢⎣ M 21 ⎢ ⎥ I σσ ⎢ ((T(T − 1) / 2) ⎥ ⎢⎣ x(T(T − 1) / 2)) ⎥⎦
Finally, the Lagrange Multiplier, emerges as:
36
⎧ ⎡ I ββ I βλ ⎤ ⎪ ⎢ (kxk) (kxT) ⎥ ⎪M11 = ⎢ I λλ ⎥ ⎪ ⎢⎣ (TxT) ⎥⎦ ⎪ ⎡ ⎤ I M12 ⎤ ⎪M = ⎢ ((T(T −σσ 1) / 2) ⎥ ⎨ M 22 ⎥⎦ ⎪ 22 ⎢ x(T(T − 1) / 2)) ⎥ ⎣ ⎦ ⎪ 0 ⎡ ⎤ ⎪M12 = ⎢⎣0 ⎥⎦ ⎪M = [ 0 0] ⎪ 12 ⎩
(A22)
⎡ ⎤ ⎡ T,R (θ) ⎤ LM SUR SLM = ⎣ g(θ) H 0 ⎦ ' ⎢⎣ I H0 ⎦⎥
−1
⎡g(θ) ⎤ χ 2(T) H 0 ⎦ as ⎣ (A23)
⇒ LM SUR SLM = g ' ( λ )
⎡ I λλ − I λβI −1 I βλ ⎤ ββ ⎦ H0 ⎣
−1
g (λ )
H0
as
χ 2(T)
The terms of the score associated with parameters λ can also be expressed as: g (λ )
H0
(
)
ˆ 'Y ⎤ 'τ = ⎡ Σ −1 o U L ⎦ ⎣
(A24)
Section A.V. Score vector and information matrix of the SUR-SEM model. The specification corresponding to this model is: y = Xβ + u ⎫⎪ Bu = ε ⎬ ε N(0, Ω) ⇒ Ω = Σ ⊗ IR ⎪⎭
(A25)
B = ITR − ϒ ⊗ W
The logarithm of the likelihood function is: −1
l(y; θ) = −
( y − Xβ) B'( Σ ⊗ IR ) B ( y − Xβ) RT R ln ( 2π ) − ln Σ + ∑ Tt =1 ln B t − 2 2 2
(A26)
where θ ' = ⎡⎣β '; ρ1;L; ρ T; vech(Σ) '⎤⎦ is a vector of parameters, of order (k+T+T(T+1)/2)x1. The score vector has the following composition: ⎡ ⎤ ⎤ ⎢ ∂l ⎥ ⎡ ⎢ ⎥ ⎢ ⎥ X ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ B ( y − Xβ ) ∂β ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ∂l ⎥ ⎢ ⎡ −1 ⎤ g(θ) = ⎢ = ⎢ − tr ⎣B t W ⎦ + ( y − Xβ ) ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ Ett ⊗ W ( y − Xβ ) ⎥ ⎥ ∂ρ t ⎥ ⎢ ⎥ ⎢ = t y − Xβ ) ' B ' ⎡⎢ Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( y − Xβ ) ⎥ ( ⎢ 1,...T ⎥ ⎢ R ⎣ ⎦ ij ⎥ ⎢ ∂l ⎥ ⎢ − tr ⎡⎣ Σ −1E ⎤⎦ + 2 2 ⎢ ⎥ ⎢ ∂ σ ij ⎥ ⎢ ⎥⎦ ⎣ ⎢i,j=1,...T ⎥ ⎣ ⎦
(
(
)
)
(A27)
The second derivatives of the logarithm of the likelihood function are: ∂ 2l = − X ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ BX ∂β∂β ' ∂ 2l = − X ' ⎡ B ' Σ −1E tt ⊗ W + E tt Σ −1 ⊗ W ' B ⎤ ( y − Xβ ) ⎢⎣ ⎥⎦ ∂β∂ρ t ∂ 2l = − X ' B ' ⎡⎢ Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( y − Xβ ) ⎣ ⎦ ∂β∂ σij
{( (
} {(
)
)
}
)
37
(A28.a)
∂ 2l ∂ρ2t 2
(
)
(
)
= − tr ⎡⎣B −t 1W B −t 1W ⎤⎦ − ( y − Xβ ) ' E tt ⊗ W ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E tt ⊗ W ( y − Xβ )
( (
) )(
( )
)
∂ l = − ( y − Xβ ) ' Ess ⊗ W ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ E tt ⊗ W ( y − Xβ ) ∂ρ t∂ρs ∂ 2l = − ( y − Xβ ) ' E tt ⊗ W ⎡⎢ Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( y − Xβ ) ⎣ ⎦ ∂ρ t∂ σij
(A28.b)
∂ 2l R ⎡ −1 ij −1 ij ⎤ = tr Σ E Σ E − ( y − Xβ ) ' B ' ⎡⎢ Σ −1Eij Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( y − Xβ ) 2 ⎦ ⎣ ⎦ ∂ σij 2 ⎣
(
)
R ∂ 2l = tr ⎡ Σ −1Esr Σ −1Eij ⎤ − ( y − Xβ ) ' B ' ⎡⎢ Σ −1Esr Σ −1Eij Σ −1 ⊗ I R ⎤⎥ B ( y − Xβ ) ⎦ ⎣ ⎦ ∂ σij∂ σsr 2 ⎣
(
)
(A28.c)
Their expected value, after changing the sign is: Iββ = X ' B ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ BX Iβρt = 0; t = 1, 2,...., T
(A29.a)
Iβσij = 0; i, j = 1, 2,...., T
(
)
tt Iρt ρt = tr ⎡⎣ B −t 1W B −t 1W ⎤⎦ + σ tttr E ⊗ ( W ' W ) ⎡⎣ B −1 ( Σ ⊗ I R ) B '−1⎤⎦ ; t = 1, 2,...., T
(
)
(
)
tt ss Iρt ρs = tr ⎢⎡ E Σ −1E ⊗ ( W ' W ) ⎤⎥ ⎡⎣ B −1 ( Σ ⊗ I R ) B '−1⎤⎦ ; t,s = 1, 2,...., T ⎣ ⎦
(A29.b)
tt ij Iρt σij = tr ⎡⎢ E Σ −1E ⊗ W ⎤⎥ B '−1; t;i, j = 1, 2,...., T ⎣ ⎦
I σijσsr =
R ⎡ −1 ij −1 sr ⎤ tr Σ E Σ E ; i, j,s, r = 1, 2,...., T ⎦ 2 ⎣
(A29.c)
We use the following ordering of the information matrix: ⎡ ⎤ I βρ I βσ ⎢ I ββ ⎥ ⎢ (kxk) (kxT) (kx(T(T + 1) / 2)) ⎥ ⎢ ⎥ I ρρ I ρσ I T,R (θ) = ⎢ (TxT) (Tx(T(T + 1) / 2)) ⎥ ⎢ ⎥ I σσ ⎢ ((T(T + 1) / 2) ⎥ ⎢ x(T(T + 1) / 2)) ⎥⎦ ⎣
(A30)
Section A.VI. Testing for spatial effects in the SUR-SEM model. The null hypothesis that we want to test is that there is no SEM structure in the error terms of the SUR, which means that:
}
H0 : ρ t = 0 ⇒ Under H0 ⇒ B = ITR ⇒ y = Xβ + ε ε N(0, Ω); Ω = Σ ⊗ IR HA : No H0
The score of (A27) under the null hypothesis of (A31) becomes:
38
}
(A31)
g(θ) H0
⎡ ⎤ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂β ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎡g (β) ⎤ ⎢ 0 0 ⎤ ⎥ ⎢ 0 ⎢ ⎥ ⎡ H0 ⎥ ⎢ ∂l ⎥ ⎢ ⎢ ⎥ tt T st ' ˆ ˆ ⎤ ' τ⎥ =⎢ ⎥ = ⎢g (ρ) ⎥ = ⎢uˆ ' ⎣⎡ Σ −1E ⊗ W⎦⎤ uˆ ⎥ = ⎢∑s=1 σ uˆ sWuˆ t ⎥ = ⎢ ⎡⎣ Σ −1 o U'U L ⎦ ⎥ ⎢ ⎥ H ⎢ ⎥ 0 ⎢ ∂ρ t ⎥ = t 1,...T ⎢ ⎥⎦ ⎢ ⎥ ⎢ t =1,...T 0 ⎥ ⎣ ⎢ t =1,...T ⎥ ⎢⎣g (Σ) H0 ⎥⎦ ⎢ 0 ⎥ ⎦ 0 ⎣ ⎦ ⎣ ⎢ ∂l ⎥ ⎢ ⎥ ⎢ ∂σij ⎥ ⎣⎢i,j=1,...T ⎦⎥
(
)
(A32)
uˆ is the (TRx1) vector of residuals of the SUR model without spatial effects. The elements of the information matrix, under the null hypothesis, are also simplified:
Iββ = X ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ X Iβρt = 0; t = 1, 2,...., T
(A33.a)
Iβσij = 0; i, j = 1, 2,...., T
(
)
Iρt ρt = trW ' W σ tt σ tt + 1 ; t = 1, 2,...., T Iρt ρs = σ tsσ tstrW ' W; t,s = 1, 2,...., T I ρ σij
(A33.b)
= 0; t;i, j = 1, 2,...., T
t
I σijσsr =
R ⎡ −1 ij −1 sr ⎤ tr Σ E Σ E ; i, j,s, r = 1, 2,...., T ⎦ 2 ⎣
(A33.c)
The structure of that matrix is, once again, block-diagonal:
I T,R (θ) H 0
⎡ ⎤ 0 0 ⎢ I ββ ⎥ (kxT) ⎢ (kxk) (kxT) ⎥ ⎢ ⎥ ⎡ M11 0 I ρρ =⎢ = (TxT) (Tx(T(T − 1) / 2)) ⎥ ⎣⎢ M 21 ⎢ ⎥ I σσ ⎢ ((T(T − 1) / 2) ⎥ ⎢ x(T(T − 1) / 2)) ⎥⎦ ⎣
⎧ ⎡ I ββ ⎤ 0 ⎪ ⎢ (kxk) ⎥ ⎪M11 = ⎢ ⎥ I σσ ⎪ ⎢ ((T(T − 1) / 2) ⎥ ⎪ ⎢⎣ x(T(T − 1) / 2)) ⎥⎦ ⎪ M12 ⎤ ⎪M = ⎡ I ρρ ⎤ M 22 ⎦⎥ ⎨⎪ 22 ⎢⎣ (TxT) ⎥⎦ ⎪M12 = ⎡ 0 ⎤ ⎣⎢ 0 ⎦⎥ ⎪ ⎪M12 = [ 0 0] ⎪ ⎪⎩
(A34)
This result facilitates the obtaining of the corresponding Multiplier: ⎡ ⎤ ⎡ T,R (θ) ⎤ LM SUR SEM = ⎣ g(θ) H 0 ⎦ ' ⎢⎣ I H0 ⎥⎦ ⇒ LM SUR SEM = g ' (ρ)
⎣⎡ I ρρ ⎤⎦ H0
−1
−1
= ⎡⎣ ∑ sT=1 σ st uˆ s' W uˆ t ⎤⎦ ' ⎡⎣ I ρρ ⎤⎦
⎡g(θ) ⎤ χ 2(T) H 0 ⎦ as ⎣
g (ρ)
−1
= H0
⎡ ∑ sT=1 σ st uˆ s' W uˆ t ⎤ = χ 2(T) ⎣ ⎦ as
39
(A35)
As in the previous cases, we can propose a more compact expression for the term of the score vector associated with the parameters ρ: g (ρ )
H0
(
)
ˆ 'U ˆ ⎤ 'τ = ⎡ Σ −1 o U L ⎦ ⎣
(A36)
Section A.VII. Testing for the time-constancy of the spatial dependence coefficients. The log-likelihood function of the SARAR model specified in expression (20) is similar to that of expression (2). The first corresponds to the restricted specification and the second to the unrestricted version of the model. The likelihood function of the first one is: l(y; θ) = −
( Ay − Xβ) ' ⎣⎡ Σ RT R ˆ − ln ( 2π ) − ln Σ + T ln Bˆ + T ln A 2 2
⎦ ( Ay − Xβ )
−1 ⊗ (B ˆ ' Bˆ ) ⎤
(A37)
2
The associated score vector is more compact in the restricted model: ⎡ ⎤ ⎢ ∂l ⎥ ⎢ ⎥ ⎤ ⎢ ∂β ⎥ ⎡ X ' ⎡ Σ −1 ⊗ (Bˆ ' Bˆ ) ⎤ ( Ay − Xβ ) ⎦ ⎥ ⎢ ⎥ ⎢ ⎣ ⎥ ⎢ ⎥ ⎢ −Ttr ⎡ ˆ −1W ⎤ + Ay − Xβ ' ⎡ −1 ⊗ (Bˆ ' Bˆ ) ⎤ ) ⎣Σ ⎦ ( IT ⊗ W) y ⎥ ⎢ ∂l ⎥ ⎢ ⎣A ⎦ ( ⎥ ⎢ ∂λ ⎥ ⎢ −1 g(θ) = ⎢ ⎥ ⎥ = ⎢−Ttr ⎡⎣Bˆ W ⎤⎦ + ( Ay − Xβ ) ' ⎡⎣ Σ −1 ⊗ (Bˆ ' W) ⎤⎦ ( Ay − Xβ ) ⎥ ⎢ ∂l ⎥ ⎢ ⎡ Σ −1Eij Σ −1 ⊗ (Bˆ ' Bˆ ) ⎤ ( Ay − Xβ ) ⎥ ⎢ ∂ρ ⎥ ⎢ A − β y X ' ( ) ⎢⎣ ⎥⎦ ⎥ ⎢ ⎥ ⎢ − R tr ⎡ −1Eij ⎤ + ⎥ ⎢ ⎥ ⎢ 2 ⎣Σ ⎦ 2 ⎢ ∂l ⎥ ⎣⎢ ⎦⎥ ⎢ ∂ ⎥ ⎢ σ ij ⎥ ⎢⎣i,j=1,...T ⎥⎦
(
(A38)
)
It is easy to test for the absence of spatial effects in the specification of (20). Now the null hypothesis only affects to two parameters:
}
H0 : λ = ρ = 0 ⇒ Under H ⇒ A = B = I 0 TR HA : No H0
(A39)
The associated Multiplier maintains a relatively complex structure: ⇒
SUR(cons) LM SARMA
= ⎡g '(λ ) ⎢⎣ H0
−1 ⎡ I λλ − I λβI ββ I βλ I λρ ⎤ g ' (ρ) ⎥⎤ ⎢ ⎥ H0 ⎦ I ρλ I ρρ ⎥⎦ ⎢⎣
−1 ⎡ g
(λ ) H ⎤ 0 ⎢ ⎥ χ 2(2) g ⎢ (ρ) H ⎥ as 0⎦ ⎣
(A40)
with: Iββ = X' ⎡⎣Σ−1⊗IR⎤⎦ X Iβλ = X' ⎣⎡Σ−1⊗W⎦⎤ Xβ Iλλ = T[ tr(W' W) + tr(WW)] +β'X' ⎡⎣Σ−1⊗(W' W)⎤⎦ Xβ Iρρ = T[ tr(W' W) + tr(WW)] Iρλ = T[ tr(W' W) + tr(WW)]
( (
) )
ˆ ⎤ g(λ) = τ' ⎡Σ−1o U'Y L ⎦τ ⎣ H0 ˆ ˆ ⎤τ g(ρ) = τ' ⎡Σ−1o U'U L ⎦ ⎣ H0
(A41)
The discussion for the case of the Spatial Lag Model of (21) is very similar. Now, the loglikelihood function and the score vector are as follows:
40
l(y; θ) = −
( Ay − Xβ) ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ ( Ay − Xβ ) RT R ˆ − ln ( 2π ) − ln Σ + T ln A 2 2 2
⎡ ⎤ ⎤ ⎢ ∂l ⎥ ⎡ ⎢ ⎥ ⎢ ⎥ X ' ⎡ −1 ⊗ I R ⎦⎤ ( Ay − Xβ ) ⎥ ⎢ ∂β ⎥ ⎢ ⎣ Σ ⎥ ⎢ ∂l ⎥ ⎢ ⎥ ˆ −1W ⎤ + ( Ay − Xβ ) ' ⎡⎣ Σ −1 ⊗ I R ⎤⎦ ( I T ⊗ W ) y g(θ) = ⎢ ⎥ = ⎢ −Ttr ⎡⎣ A ⎦ ∂λ ⎢ ⎥ ⎢ ⎥ ⎥ ij −1 − 1 ⎡ ⎤ ⎢ ∂l ⎥ ⎢ Ay − Xβ ) ' ⎢ Σ E Σ ⊗ I R ⎥ ( Ay − Xβ ) ⎥ ( ⎢ R ⎣ ⎦ ⎢ ∂σ ⎥ ⎡ −1 ij ⎤ ij ⎥ ⎢ − tr ⎣ Σ E ⎦ + ⎥ ⎢ 2 2 ⎣ ⎦ ⎣i, j=1,...T ⎦
(
(A42)
(A43)
)
The hypothesis that parameter λ is zero leads us back to the SUR without spatial effects:
}
H0 : λ = 0 ⇒ Under H0 ⇒ A = ITR HA : No H0
(A44)
and the Multiplier results to be: ⇒ LM SUR(cons) = g '(λ ) SLM
⎡ I λλ − I λβI −1 I βλ ⎤ ββ ⎦ H0 ⎣
−1
g (λ )
H0
as
χ 2(1)
(A45)
with:
(
)
ˆ ⎤ g(λ) = τ ' ⎡Σ−1o U'Y L ⎦τ ⎣ H0 ⎧Iββ = X' ⎡Σ−1⊗IR⎤ X ⎣ ⎦ ⎪ −1 ⎨Iβλ = X' ⎣⎡Σ ⊗W⎦⎤ Xβ ⎪ = T tr(W' W) + tr(WW) +β'X' ⎡ −1⊗(W' W)⎤ Xβ ] ⎣Σ ⎦ ⎩Iλλ [
(A46)
Finally, the log-likelihood function and the score vector for the Spatial Error Model of (22) are: l(y; θ) = −
( y − Xβ) ' ⎣⎡ Σ −1 ⊗ (Bˆ ' Bˆ )⎦⎤ ( y − Xβ ) RT R ln ( 2π ) − ln Σ + T ln Bˆ − 2 2 2
⎤ ⎡ ∂l ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ −1 ⎥ ⎢ ∂β ⎥ ⎢ X ' ⎣⎡ Σ ⊗ (Bˆ ' Bˆ ) ⎦⎤ ( y − Xβ ) ⎥ ⎢ ∂l ⎥ ⎢ ⎥ ⎥ = ⎢−Ttr ⎡Bˆ −1W ⎤ + ( y − Xβ ) ' ⎡ Σ −1 ⊗ (Bˆ ' W) ⎤ ( y − Xβ ) g(θ) = ⎢ ⎣ ⎦ ⎣ ⎦ ⎥ ⎢ ∂ρ t ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ij −1 − 1 ⎡ ⎤ ˆ ˆ ⎢ ∂l ⎥ ⎢ R ⎡ −1 ij ⎤ ( y − Xβ ) ' ⎣⎢ Σ E Σ ⊗ (B ' B) ⎦⎥ ( y − Xβ ) ⎥ ⎥ ⎢⎣ ∂σ ij ⎥⎦ ⎢⎣ − 2 tr ⎣ Σ E ⎦ + ⎦ 2
(
(A47)
(A48)
)
The null hypothesis of absence of spatial effects implies that parameter ρ is zero:
}
H0 : ρ = 0 ⇒ Under H0 ⇒ B = ITR HA : No H0 The expression of the Multiplier is relatively simple:
41
(A49)
SUR(cons)
LM SEM
−1g = g '(ρ) I ρρ (ρ) H0
(
H0
as
χ 2(1)
)
(A50)
ˆ 'U ˆ ⎤τ U = τ ' ⎡Σ L ⎦ ⎣ I ρρ = T [ tr( W ' W) + tr( WW)]
g (λ )
−1 o
H0
The problem addressed in Section 3.2 refers to testing the time-constancy of the parameters of spatial dependence introduced in the SUR specifications. In the case of the SARAR model associated to the hypothesis of (23), the Lagrange Multiplier appears in expression (24) and the information matrix that should be introduced in this expression is:
(
⎡ I ββ I βλ Iβρ ⎢ I λλ I λρ I T,R (θ) = ⎢ I ρρ ⎢ ⎢⎣
)
Iβσ ⎤ I λσ ⎥ ⎥ I ρσ ⎥ I σσ ⎥⎦
(A51)
where uˆ = ⎡I T ⊗ I R − λˆ W ⎤ y − Xβˆ and εˆ = ⎡⎣IT ⊗ ( I R − ρˆ W) ⎤⎦ uˆ . Furthermore: ⎣ ⎦ ˆ 'B ˆ )⎤ X ⎧Iββ = X' ⎡ Σ −1 ⊗ (B ⎣ ⎦ ⎪ ˆ ' WB ˆ ˆ −1)⎤ Xβ; t = 1, 2,....,T ⎪Iβλt = X' ⎡ Σ −1Ett ⊗ (B A ⎦⎥ ⎢ ⎣ ⎪ ⎪Iβρt = 0; t = 1, 2,....,T ⎪Iβσij = 0; i, j = 1, 2,....,T ⎪ ˆ ' BW ˆ ˆ −1⎤ Xβ ˆ ' −1W ' A ˆ −1W) + σ ttβ 'X' ⎡ A ˆ −1W ' B A ⎦ ⎪I λt λt = tr(A ⎣ ' ⎪ ⎡ ˆ −1 ˆ ˆ ˆ −1 ˆ ˆ −1⎤ + tr ⎢ A ⎪ ' W ' B ' BA W B ' B ⎥ ⎣ ⎦ ⎪ ts −1 ts ˆ ˆ ⎡ ˆ ' W ' B ' BWA ˆ −1)⎤ Xβ; t,s = 1, 2,....,T ⎪I λt λs = σ β 'X' E ⊗ (A ⎣ ⎦ ⎪ −1 −1⎤ −1 ˆ ⎪I ρ = σ tt σ tttr ⎡B ˆ ˆ −1B ˆ ˆ ˆ −1); t = 1, 2,....,T tr( + WBW WW A B ⎦ A ⎣ ' ⎨ λt t ˆ ˆ −1 ˆ −1⎤ ; t,s = 1, 2,....,T ⎪I ρ = σ tsσsstr ⎡B ˆ ' −1W ' BW A B ⎦ λ t s ⎣ ⎪ 0 if t ≠ i or t ≠ j ⎧ ⎪ = ⎨ tt I 1 − λ σ t ij ⎪ ˆ W) if t=i =j ⎩σ tr(A ⎪ − 1 − 1 ⎡ ˆ ' WB ˆ W⎤ + σ tt σ tttr ⎡B ˆ −1WW ' B ˆ −1⎤ ; t = 1, 2,....,T ⎪Iρt ρt = tr ⎣B ⎦ ⎣ ' ⎦ ⎪ ts tr ⎡ ˆ −1WW ' ˆ −1⎤ ; t,s = 1, 2,....,T = B ⎦ ⎪Iρt ρs σ σ ts ⎣B ' if t ≠ i and t ≠ j ⎪ (A52) ⎧⎪0 ⎪Iρt σij = ⎨ titr ⎡ ˆ −1W⎤ if t=i or t=j ⎪⎩σ ⎣B ' ⎦ ⎪ R ⎡ −1 ij −1 sr ⎤ ⎪ ⎪⎩I σijσsr = 2 tr ⎣ Σ E Σ E ⎦ ; i, j,s, r = 1, 2,....,T In the case of the SLM model to which refers the hypothesis of (26), we need the following
(
)
(
)
information matrix in order to solve for the expression of the Lagrange Multiplier that appears in (27):
42
⎡Iββ Iβλ Iβσ ⎤ I (θ) = ⎢ I λλ I λσ ⎥ ⎢ I σσ⎥⎦ ⎣ ⎧Iββ = X' ⎡ Σ −1 ⊗ I R ⎤ X ⎣ ⎦ ⎪ tt 1 − ⎡ ˆ −1 ⎤ Xβ; t = 1, 2,....,T ⎪Iβλt = X' ⎢ Σ E ⊗ WA ⎥⎦ ⎣ ⎪ = = 0; i, j 1, 2,....,T ⎪Iβσij ⎪I ˆ ' −1W ' A ˆ −1W) + σtt β't x 't ⎡ A ˆ −1W ' WA ˆ −1⎤ x tβ t = tr(A ⎣ ' ⎦ ⎪⎪ λt λt tt −1 −1⎤ ⎡ ˆ ˆ + σ σ tttr A ' W ' WA ; t = 1, 2,....,T ⎨ ⎣ ⎦ ⎪ ts ' ' ⎡ −1 −1⎤ ˆ ' W ' WA ˆ ˆ −1W ' WA ˆ −1⎤ ; t,s = 1, 2,....,T β + σts σ tstr ⎡ A ⎪I λ t λ s = σ β t x t ⎣ A ⎦ xs s ⎣ ' ⎦ ⎪ 0 if t ≠ i or t ≠ j ⎧ ⎪I λt σij = ⎨ tt ˆ −1W) if t=i=j ⎩σ tr(A ⎪ R ⎪ = tr ⎡ −1Eij Σ −1Esr ⎤ ; i, j,s, r = 1, 2,....,T ⎪⎩I σijσsr 2 ⎣ Σ ⎦ T,R
(
) (
)
(A53)
Finally, the same can be said with respect to the SEM model of (11). The null hypothesis of timeconstancy appears in (29), the Lagrange Multiplier is that of (30) in which intervenes the following information matrix: ⎡Iββ Iβρ Iβσ ⎤ I I ρρ I ρσ ⎥ ⎢ I σσ⎥⎦ ⎣ ˆ 'B ˆ )⎤ X ⎧Iββ = X' ⎡ Σ −1 ⊗ (B ⎣ ⎦ ⎪ ⎪Iβρt = 0; t = 1, 2,....,T ⎪Iβσij = 0; i, j = 1, 2,....,T ⎪ ⎡ ˆ ' −1WB ˆ −1W⎤ + σ tt σ tttr ⎡B ˆ −1W ' WB ˆ −1⎤ ; t = 1, 2,....,T ⎪Iρt ρt = tr ⎣B ⎦ ⎣ ' ⎦ −1 −1⎤ ⎪ ts ⎡ ˆ ' WW ' B ˆ ; t,s = 1, 2,....,T ⎪Iρt ρs = σ σ tstr ⎣B ⎦ ⎨ ⎧ ⎪ if t ≠ i, t ≠ j and i ≠ j ⎪0 ⎪ ⎪ ti ⎡ −1 ⎤ ˆ ' W if i ≠ j and t=i or t=j ⎪Iρt σij = ⎨σ tr ⎣B ⎦ ⎪ ⎪ tttr ⎡ ˆ −1W⎤ if t=i=j ⎪⎩σ ⎣B ' ⎪ ⎦ ⎪ R ⎡ −1 ij −1 sr ⎤ = tr Σ E Σ E ; i, j,s, r = 1, 2,....,T I ⎦ ⎩⎪ σijσsr 2 ⎣ T,R (θ) = ⎢
43
(A54)
