Lecture 7: GMM Estimation of Spatial Models1 Estimation of SLM: Spatial Two Stage Estimation (S2SLS) Introduction Instruments Assumptions EstimatorandAsymptoticDistribution 2 Estimation

Lecture 7: GMM Estimation of Spatial Models

Mauricio Sarrias

Universidad Católica del Norte

November 22, 2017

1 Estimation of SLM: Spatial Two Stage Estimation (S2SLS)IntroductionInstrumentsAssumptionsEstimator and Asymptotic Distribution

2 Estimation of SEM: Method of Moment Estimation and FGLSPreliminariesSpatially Weighted Least Squares and FGLSMoment ConditionsAsymptotic Properties

3 Estimation of SAC Model: The GS2SLS ProcedureIntuition behind the procedureMoment Conditions RevisedAssumptionsEstimator and Estimation Procedure in a Nutshell




Mandatory Reading

(AR)-Chapter 7, 9 and 11Kim, C. W., Phipps, T. T., & Anselin, L. (2003). Measuring the benefitsof air quality improvement: a spatial hedonic approach. Journal ofenvironmental economics and management, 45(1), 24-39.Kelejian, H. H., & Prucha, I. R. (1999). A generalized moments estimatorfor the autoregressive parameter in a spatial model. Internationaleconomic review, 40(2), 509-533.Messner, S. F., & Anselin, L. (2004). Spatial Analyses of Homicide withAreal Data. Spatially integrated social science, 12.Kelejian, H. H., & Prucha, I. R. (1998). A generalized spatial two-stageleast squares procedure for estimating a spatial autoregressive model withautoregressive disturbances. The Journal of Real Estate Finance andEconomics, 17(1), 99-121.Arraiz, I., Drukker, D. M., Kelejian, H. H., & Prucha, I. R. (2010). ASpatial Cliff-Ord-Type Model with Heteroskedastic Innovations: Smalland Large Sample Results. Journal of Regional Science, 50(2), 592-614.

S2SLS

Recall that Spatial Lag Model (SLM) is given by:

y = ρWy + Xβ + ε.

A more concise way to express the model is as:

y = Zδ + ε,

where Z = [X,Wy] and the (K + 1)× 1 coefficient column vector isrearranged as δ = (β>, ρ)>.

As we know, the presence of the spatially lagged dependent variable on theRHS induces endogeneity or simultaneous equation bias.

S2SLS

Instead of applying QML or ML, we might rely on the IV approach todeal with the endogeneity problem.Principle: Find a set of instruments H that are strongly correlated withZ, by asymptotically uncorrelated with ε.Important point: We just assume that Wy is the only endogenousvariable.

H should contain al the predetermined variables, X, and the instrument(s)for Wy.We will relax this later.




S2SLSInstruments

What is the best instrument(s) for Wy?Optimal instrumental variables: the ‘best instruments’ for the r.h.s variablesare the conditional means. Thus, the ideal instruments are:

E (Z|X) = [E (X|X) ,E (Wy|X)]= [X,WE (y|X)] since W is non-stochastic.

The best instruments for X is X.The best instruments for Wy is WE (y|X).

S2SLSInstruments

Given that the roots of ρWn are less than one in absolute value, theconditional expectation can also be written as:

E(Wy|X) = WE (y|X)= W (In − ρW)−1 Xβ= W

[In + ρW + ρ2W2 + ρ3W3 + ...

]Xβ0

= W[ ∞∑l=1

ρl0Wl

]Xβ0

= WXβ + W2X(ρβ) + W3X(ρ2β) + W4X(ρ3β) + ...

This reveals that E(y|X) is linear in X,WX,W2X... This observationmotivated Kelejian and Prucha (1998) to select a set of instruments based on,at least, the linearly independent columns of (X,WX,W2X).

S2SLSInstruments

Computational issues with the inverse of the n× n matrix (In − ρ0W).Kelejian and Prucha (1998, 1999) suggest the use of an approximationof the best instruments:

They suggest H which contains, say, X,WX,W2X, ...,WlX, and tocompute approximations of the best instruments from a regression of therhs variables against H, where l is a pre-selected finite constant and isgenerally set to 2 in applied studies.

Thus, in general we can write the instruments as:

H = (X,WX,W2X)

S2SLSIntuition of Instruments

The intuition behind the instruments is the following:Since X determines y, then it must be true that WX,W2X, ...determines Wy.Furthermore, since X is uncorrelated with ε, then WX must be alsouncorrelated with ε.

S2SLSOptimal Instruments

Using the conditional expectation, Lee (2003) suggested the instrumentmatrix:

H =[X,W(I− ρW)−1Xβ

],

which requires the use of consistent first round estimates for ρ and β. InKelejian et al. (2004), a similar approach is outlined where the matrix inverseis replaced by the power expansion. This yield an instruments matrix as:

H =[

X,W( ∞∑l=1

ρl0Wl

)Xβ].

In any practical implementation, the power expansion must be truncated atsome point.




S2SLSAssumptions

Heterokedastic Errors (Kelejian and Prucha, 2010)

The errors εi,n, 1 ≤ i ≤ n, n ≥ 1 satisfy E(εi,n) = 0, E(ε2i,n) = σ2i,n, with

0 < aσ ≤ σ2i,n ≤ aσ <∞. Additionally the errors are assumed to possess

fourth moments, that is sup1≤i≤n,n≥1 E |εi,n|4+η for some η > 0. Furthermore,

for each n ≥ 1 the random variables ε1,n, ..., εn,n are totally independent.

Remark: Triangular arrays.Note: Kelejian and Prucha (1998) derived the asymptotic propertiesassuming that the errors are homokedastic. But, Kelejian and Prucha(2010) extend the model by assuming heteroskedasticity.

S2SLSAssumptions

Now, we state some assumptions about the behavior of the spatial weightmatrix W.

Diagonal elements of Wn (Kelejian and Prucha, 1998)

All diagonal elements of the spatial weighting matrix Wn are zero

This assumption is a normalization of the model and it also implies that nospatial unit is viewed as its own neighbor.

S2SLSAssumptions

Nonsingularity (Kelejian and Prucha, 1998)

The matrix (In − ρ0Wn) is nonsingular with |ρ0| < 1.

Under Nonsigularity Assumption, we can write the reduced form of the truemodel as:

yn = (In − ρ0Wn)−1Xnβ0 + (In − ρ0Wn)−1εn.

Heterokedastic-Errors Assumption implies further that the populationvariance-covariance matrix of yn is equal to

E(yny>n ) = Ωyn = (In − ρ0Wn)−1Σn(In − ρ0W>n )−1, (1)

where Σ = diag(σ2i,n). If we assume homokedasticity, then the

variance-covariance matrix of y reduces to:

E(yny>n ) = Ωyn = σ2ε (In − ρ0Wn)−1(In − ρ0W>

n )−1.

S2SLSAssumptions

Bounded matrices (Kelejian and Prucha, 1998)

The row and column sums of the matrices Wn and (In − ρ0Wn) are boundeduniformly in absolute value.

Thus, the variance of yn in Equation (1), which depend on Wn and(In − ρ0Wn), are uniformly bounded in absolute value as n goes toinfinity, thus limiting the degree of correlation between, respectively, theelements of εn and yn.Applied to Wn, this assumption means that each cross-sectional unit canonly have a limited number of neighbors.Applied to (In − ρWn) limits the degree of correlation.

S2SLSAssumptions

No Perfect Multicolinearity (Kelejian and Prucha, 1998)

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

S2SLSAssumptions

Now we state some assumptions about the instruments.

Rank Instruments, (Kelejian and Prucha, 1998)

The instrument matrices Hn have full column rank L ≥ K + 1 for all n largeenough. Furthermore, the elements of the matrices Hn are uniformly boundedin absolute value. They are composed of a subset of the linearly independentcolumns of (X,WX,W2X, ...).

Limits of Instruments (Kelejian and Prucha, 1998)

Let Hn be a matrix of instruments, then:1 limn→∞ n−1H>nHn = QHH where QHH is finite and nonsingular (full

rank).2 plimn→∞ n−1H>nZn = QHZ where QHZ is finite and has full column

rank.

S2SLSAssumptions

limn→∞ n−1H>nHn = QHH implies that WnXn and Xn cannot belinearly dependent.

This condition would be violated if for example WnXn include a spatiallag for the constant term or the model is the pure SLM.

plimn→∞ n−1H>nZn = QHZ requires a non-null correlation between theinstruments and the original variables.




S2SLSDefinition

To operationalize the S2SLS we first need the predicted values for Zn based onthe OLS regression of Zn on Hn in the first stage.

First stageConsider this first stage as the regression Zn = Hnθ + ξn, so thatθ = (H>nHn)−1H>nZn

The predicted values are computed as:

Zn = Hnθn = Hn(H>nHn)−1H>nZn = PH,nZn (2)

where PH,n is the projection matrix. The second stage uses the predictedvalues of Zn:

Second Stage

δS2SLS =(

Z>n Zn)−1

Z>nyn

S2SLSDefinition

Definition (Spatial Two Stage Least Square Estimator)Let Hn be the matrix (n× L) of instruments. Then the S2SLS is given by:

δS2SLS =(

Z>n Zn)−1

Z>nyn (3)

where:

Zn = Hnθn = Hn(H>nHn)−1H>nZn = PH,nZn (4)

Note also that Hn is a n× L matrix, which also includes the exogenousvariables Xn. It is also important to note that the projection matrix does notaffect Xn, but it does affect the endogenous variable Wnyn:

PH,nZn = [Xn,PH,nWnyn] =[Xn,Wnyn

](5)

S2SLS Estimator as GMM

Recall that the GMM estimator is defined as the solution of the minimization problem:

δGMM︸︷︷︸K×1

= arg minβ

gn(β)>︸︷︷︸1×L

Υ−1n︸︷︷︸

L×L

gn(β)︸︷︷︸L×1

,

where

gn =1n

H>ε =1n

H> (y− Zδ)

The matrix Υ−1n is the optimal weight matrix, which correspond to the inverse of the covariance

matrix of the sample moments:Υ =

1nσ

2εH>H

Then, the function to minimize is:

J =1nσ2

[H>y−H>Zδ

]> (H>H

)−1 [H>y−H>Zδ

]Obtaining the first order conditions and solving for δ, we obtain:

δGMM =(

Z>PHZ)−1

Z>PHy (6)

S2SLS Asymptotic distribution

Theorem (Spatial 2SLS Estimator for SLM)Suppose that Assumptions hold. Then the S2SLS estimator defined as

δn =(

Z>n Zn)−1

Z>n yn (7)

is consistent, and its asymptotic distribution is:

√n(δn − δ0) d−→ N(0,Ωn) (8)

where

Ωn = P>H>ΣHP (9)

Inference on δ is then based on the asymptotic variance-covariance matrix:

Var(δ2SLS) =[

Z>H(

H>H)−1

H>Z]−1

×[

Z>H(

H>H)−1 (

H>ΣH)(

H>H)−1

H>Z]

×[

Z>H(

H>H)−1

H>Z]−1

=(

Z>Z)−1 (

Z>ΣZ)(

Z>Z)−1

(10)

Some basics on Asypmtotics

Definition (Order of Magnitude in Probability)1 The sequence Xn is Op(nk) iff one can always find a finite interval

within which the outcome (1/nk)Xn will occur with probability arbitraryclose to 1 for each term in the sequence.

2 If a random sequence is Op(1), the random sequence is said to bebounded in probability.

3 The sequence Xn is op(nk) iff (1/nk)Xn converges in probability tozero.

Any random variable X with cdf F is Op(1) (White, 2014, pag.28)


Example (Order of Magnitude in Probability)Let Xn be such that Xi ∼ N(0, 1),∀i, with all the terms in the sequencebeing independent random variables. Define Zn as Zn =

∑ni=1Xi. Then

Xn itself is Op(1), i.e., is bounded in probability, and Zn is Op(n1/2).Note that n−1/2Zn = n−1/2∑n

i=1Xi ∼ N(0, 1). Finally, in the sequencedefined by Yn = n−1/2(Xn + Zn), note that n−1/2Xn is op(1), while n−1/2Znis Op(1), implying that as n→∞, n−1/2Zn is the dominant term in thedefinition of Yn while n−1/2Xn is the stochastically irrelevant as n→∞.


Theorem (Sufficient Conditions for Consistency)Chebyshev’s inequality implies that a sufficient conditions for an estimatorbase don a sample of size n, say θn, say to be consistent for θ are:

E(θn

)= θ0

limn→∞

Var(θn

)= 0

(11)

Slightly weaker conditions for consistency are:

limn→∞

E(θn

)= θ0

limn→∞

Var(θn

)= 0

(12)

Theorem (CLT for Vectors of Linear Quadratic Forms withHeterokedastic Innovations)

Assume the following:1 For r = 1, ...,m let Ar,n with elements (aijr)i,j=1,...,n be an n× n

non-stochastic symmetric real matrix with sup1≤j≤n,n≥1∑ni=1 |aijr| <∞,

2 and let ar = (air, ..., anr)> be a n× 1 non-stochastic real vector withsupn

∑n

i=1|air|δ1

n <∞ for some δ1 > 2.3 Let ε = (ε1, ..., εn)> be an n× 1 random vector with the εi distributed

totally independent with E [εi] = 0,E[ε2i], and sup1≤i≤n,n≥1 E |εi|

δ2 <∞for some δ2 > 4.

Consider the m× 1 vector of linear quadratic forms vn = [Q1n, ..., Qmn]′ with:

Qrn = ε′Arε+ a′rε =n∑i=1

n∑j=1

aijrεiεj +n∑i=1

airεi. (13)

Theorem (CLT for Vectors of Linear Quadratic Forms withHeterokedastic Innovations, cont..)Let µv = E [vn] = [µQ1 , ..., µQ2 ]> and Σvn = [σQrs ]r,s=1,...,m denote the mean andVC matrix of vn, respectively, then:

µQr =n∑i=1

aiirσ2i

σQrs = 2n∑i=1

n∑j=1

aijraijsσ2i σ

2j +

n∑i=1

airaisσ2i

+n∑i=1

aiiraiis

[µ

(4)i − 3µ4

i

]+

n∑i=1

(airaiis + aisaiir)µ(3)i

with µ(3)i = E(ε3

i ) and µ(4)i = E(ε4

i ). Furthermore, given that n−1λmin(Σvn) ≥ c forsome c > 0, then

Σ−1/2vn (vn − µvn) d−→ N(0, Im)

and thus:

n−1/2(vn − µvn) a∼ N(0, n−1Σvn)

Asymptotic distribution

Sketch of Proof.As usual, we first write the estimator in terms of the population error term (for notationalconvenience we drop the sub indices):

δn = δ0 +(

Z>Z)−1

Z>ε

= δ0 +[(

H(H>H)−1H>Z)> (

H(H>H)−1H>Z)]−1 (

H(H>H)−1H>Z)>ε

= δ0 +[

Z>H(H>H)−1H>Z]−1

Z>H(H>H)−1H>ε

where we used Rank of Instruments Assumption. Solving for δn − δ0 and multiplying by√n

we obtain:

√n(δn − δ0) =

[(1n

H>Z)> ( 1

nH>H

)−1 ( 1n

H>Z)]−1 (

1n

H>Z)> ( 1

nH>H

)−1 1√n

H>ε

=1√n

P>H>ε,

(14)where

P =(

1n

H>H)−1 ( 1

nH>Z

)[(1n

H>Z)> ( 1

nH>H

)−1 ( 1n

H>Z)]−1

.


Sketch of Proof.From Limits of Instruments Assumption, we know that:

limn→∞

n−1H>nHn = QHH

plimn→∞ n−1H>nZn = QHZ .

Therefore, P p−→ P, where P is a finite matrix. Thus,

P−P = op(1) =⇒ P = P + op(1). (15)

Inserting (15) into (14), we get:

√n(δn − δ0) = 1√

n[P + op(1)]>H>ε

= P> 1√n

H>ε+ op(1)


Sketch of Proof.By the Rank of Instruments Assumption H is uniformly bounded inabsolute value. The Heterokedastic Errors Assumption implies that εi,nforms a triangular array of identically distributed random variables.Furthermore, we know from that assumption that E(ε) = 0 andVar(ε) = Σ = diag(σ2

i,n). By Limits of Instruments Assumption, weknow that limn→∞ n−1H>nHn = QHH is finite and is nonsingular. Thus,

E(

1√n

H>ε)

= 0

Var(

1√n

H>ε)

= 1n

H>ΣH(16)

Thus, by Chebyshev’s inequality n−1/2PH>ε = Op(1) (that is it convergesin distribution to something) and consequently:

√n(δn − δ0) = P> 1√

nHε+ op(1) = Op(1) + op(1) = Op(1) (17)


Sketch of Proof.Therefore using CLT for Linear Quadratic Forms,

1√n

H>n εnd−→ N

(0,H>nΣHn

)(18)

Finally :√n(δn − δ0) d−→ N(0,Ω) (19)

where

Ω = P>H>ΣHP (20)

Some Remarks

The estimator of Σ will based on HAC estimators.Under homokedasticity, the asymptotic variance-covariance matrixreduced to :

Var(δ2SLS) = σ2ε

(Q>HZQ−1

HHQHZ

)−1 (21)

A good estimator for the asymptotic variance will be:

Var(δ2SLS) = σ2ε

[Z>H(H>H)−1H>Z

]−1 (22)

where:

σ2 = ε>ε

n, ε = y− y (23)




Preliminaries

The SEM model is consistent but inefficient under OLS.ML approach estimate β and λ jointly by assuming the whole distributionof the error term.Then, we could in principle estimate this model using GLS since we knowthe exact form of the Heterogeneity.

Preliminaries

Recall that the SEM model is given by:

yn = Xnβ0 + un,un = λ0Mnun + εn.

(24)

Homokedastic Errors (Kelejian and Prucha, 1999)

The innovations εi,n, 1 ≤ i ≤ n, n ≥ 1 are independently and identicallydistributed for all n with zero mean and variance σ2, where 0 < σ2 < b, withb <∞. Additionally, the innovations are assumed to possess finite fourthmoments.

Weight Matrix Mn (Kelejian and Prucha, 1999)

Assume the following:1 All diagonal elements of the spatial weighting matrix Mn are zero.2 The matrix (In − λ0Mn) is nonsingular with |λ0| < 1.

Preliminaries

Given Equation (24), and Assumption (Weight Matrix Mn), we can writeun = (In − λMn)−1εn. Therefore, the expectation and variance of un areE(un) = 0 and E(unu>n ) = Ωn(λ0), respectively, where:

Ωn(λ0) = σ2ε (In − λ0Mn)−1(In − λ0M>

n )−1.

Note that a row-standardized spatial weight matrix is typically not symmetric,such that Mn 6= M>

n and thus (In − λ0Mn)−1 6= (In − λ0M>n )−1.




SWLSThe spatially weighted least squares (SWLS) boils down to:

βSWLS =(X>s Xs

)−1 X>s ys, (25)

were Xs = X− λWX and ys = y− λMy, using a consistent estimate λ forthe autoregressive parameter.

Note that this model is basically and OLS applied to spatially filteredvariables.Furthermore, it should be noted that the SWLS are nothing but a specialcase of Feasible Generalized Least Squares (FGLS).To note this consider the homoskedastic case, with E

[ε>ε

]= σ2I.

Consequently:

E[uu>

]= Ω = σ2

[(In − λM)> (In − λM)

]−1, (26)

and the corresponding Generalized Least Squares (GLS) estimator for βis:

βGLS =[X>Ω−1X

]−1 X>Ω−1y.

GLS & FGLS

The expression for the GLS estimator simplifies to:

βGLS =[X> 1

σ2 (In − λM)> (In − λM) X]−1

X> 1σ2 (In − λM)> (In − λM) y,

=[X> (In − λM)> (In − λM) X

]−1X> (In − λM)> (In − λM) y.

The FGLS estimator substitutes a consistent estimate for λ into thisexpression, as:

βFGLS =[X>

(In − λM

)> (In − λM

)X]−1

X>(

In − λM)> (

In − λM)

y,

which is the same as Equation (25).

Remarks

So, I we had a consistent estimate of λ0, we could use the FGLS.How to estimate λ0 consistently? Kelejian and Prucha (1999) propose toestimate it by Method of Moments (MM).However, they do not provide the asymptotic distribution for λ: λ0 isviewed purely as a nuisance parameter, whose only function is to aid inobtaining consistent estimates for β0.One advantage of the MM estimator (and of QML) is that they do notrely on the assumption of normality of the disturbances ε. Nonetheless,both estimators assume that εi are independently and identicallydistributed for all i with zero mean and variance σ2.




Moment Conditions

The basic idea behind a method of moments estimator is to find a set ofpopulation moments equations that provide a relationship between populationmoments and parameters.

Moment Conditions

Given the DGP in Equation (24), we can write:

ε = u− λMu,

where ε is the idiosyncratic error and u is the regression error. The MMestimation approach employs the following simple quadratic momentconditions:

E[n−1ε>ε

]= σ2,

E[n−1ε>MMε

]= σ2

nE[tr(M>Mεε>)

],

E[n−1ε>Mε

]= 0.

The Kelijian and Prucha (1999)’s MM estimator of λ is based on these threemoments. The final value of E

[ε>MMε

]will depend on the assumption

about the variance of ε.

Moment Conditions

Definition (Moment Conditions)

Under homoskedasticity (Kelijian and Prucha, 1999) the moment conditionsare:

E[n−1ε>ε

]= σ2,

E[n−1ε>MMε

]= σ2

ntr(M>M

),

E[n−1ε>Mε

]= 0.

(27)

Under heterokedasticity (Kelijian and Prucha, 2010) the moment conditionsare:

E[n−1ε>ε

]= σ2,

E[n−1ε>MMε

]= n−1 tr

[W diag

[E(ε2i )

]W>] ,

E[n−1ε>Mε

]= 0.

(28)

Moment Conditions

In order to operationalize the moment conditions, we need to convertconditions on ε into conditions on u (since ε is not observed). Sinceu = λMu + ε if follows that ε = u− λMu, i.e., the spatially filtered regressionerror terms.

ε>ε = (u− λMu)>(u− λMu)= u>u− 2λu>Mu + λ2u>M>Mu (29)

ε>M>Mε = (u− λMu)>M>M(u− λMu)= u>M>Mu− 2λu>M>MMu + λ2u>M>MM>Mu (30)

ε>Mε = (u− λMu)>M(u− λMu)= u>Mu− 2λu>MMu + λ2u>M>MMu (31)

Let uL = Mu , uLL = MMu .

Moment Conditions

Taking the expectation over (29) and assuming Homokedasticity, we get:

E[ε>ε

]= E

[u>u

]− 2λE

[u>Mu

]+ λ2E

[u>M>Mu

]σ2 = 1

nE[u>u

]− λ 2

nE[u>uL

]+ λ2 1

nE[u>LuL

]since E

[n−1ε>ε

]= σ2

0 = σ2 − 1nE[u>u

]+ λ

2nE[u>uL

]− λ2 1

nE[u>LuL

]0 = λ

2nE[u>uL

]− λ2 1

nE[u>LuL

]+ 1nσ2 − 1

nE[u>u

]0 =

( 2nE[u>uL

]− 1nE[u>LuL

]1)( λ

λ2

σ2

)− 1nE[u>u

](32)

Moment Conditions

In similar fashion,

0 =( 2nE[u>LLuL

]− 1nE[u>LLuLL

] 1n tr(M>M)

)( λλ2

σ2

)− 1nE[u>LuL

](33)

0 =( 1nE[u>uLL + u>LuL

]− 1nE[u>LuLL

]0)( λ

λ2

σ2

)− 1nE[u>uL

](34)

At this point it is important to realized that we have have three equationsan three unknowns!, λ, λ2 and σ2.

Moment Conditions

Consider the following three-equations system implied by Equations (32), (33)and (34)

Γnα = γn (35)

where Γn is given in Equation (36), and α = (λ, λ2, σ2). If Γn where known,Assumption (Identification) implies that Equation (35) determines α asα = Γ−1

n γn where:

Γn =

2nE[u>uL

]− 1nE[u>LuL

]1

2nE[u>LLuL

]− 1nE[u>LLuLL

] 1n tr(M>M)

1nE[u>uLL + u>LuL

]− 1nE[u>LuLL

]0

(36)

and

γn =

1nE[u>u

]1nE[u>LuL

]1nE[u>uL

] (37)

Moment Conditions

Now we express the moment conditions as sample averages in observablesspatial lags of OLS residuals:

gn = Gnα+ υn(λ, σ2) (38)

which can be thought as a regression. Note also that

Gn =

2n u>uL − 1

n u>L uL 12n u>LLuL − 1

n u>LLuLL 1n tr(M>M)

1n

[u>uLL + u>L uL

]− 1n u>L uLL 0

(39)

and

gn =

1n u>u

1n u>L uL1n u>uL

(40)

where Gn is a 3× 3 matrix, and where υn(λ, σ2) can be viewed as a vector ofresiduals.

Moment Conditions

We can define the GM estimator for λ and σ2 as the Nonlinear Least Square(NLS) estimator corresponding to Equation (38):

(λNLS,n, σ2NLS,N ) = argmin

υn(λ, σ2)>υn(λ, σ2) : ρ ∈ [−a, a], σ2 ∈ [0, b]

(41)

Note that (λNLS,n, σ2NLS,N ) are defined as the minimizers of[gn −Gn

(λλ2

σ2

)]> [gn −Gn

(λλ2

σ2

)]

Assumptions

Bounded Matrices (Kelejian and Prucha, 1999)The row and column sums of the matrices Mn and (I− λMn) are boundeduniformly in absolute value.

Residuals (Kelejian and Prucha, 1999)Let ui,n denote the i-th element of un. We then assume that

ui,n − ui,n = di,n∆n

where di,n and ∆n are 1× p and p× 1 dimensional random vectors. Let dij,n be thejth element of di,n. Then, we assume that for some δ > 0, E |dij,n|2+δ ≤ cd <∞,where cd does not depend on n, and that

√n ‖∆n‖ = Op(1). (42)

This assumption should be satisfied for most cases in which u is based on√n-consistent estimators of the regression coefficients (non-linear OLS, linear OLS).

Assumption Residuals (Kelejian and Prucha, 1999) comes from (Kelejian andPrucha, 2010) and is a bit stronger than the same assumption in (Kelejian andPrucha, 1999).

Assumptions

Identification (Kelejian and Prucha, 1999)

The smallest eigenvalues of Γ>n Γn is bounded away from zero, that is,ωmin(Γ>n Γn) ≥ ω∗ > 0, where ω∗ may depend on λ and σ2

Theorem (Consistency)

Let (λNLS,n, σ2NLS,N ) given by:

(λNLS,n, σ2NLS,N ) = argmin

υn(λ, σ2)>υn(λ, σ2) : ρ ∈ [−a, a], σ2 ∈ [0, b]

Then, given Assumptions (Heterokedastic errors), (Weight Matrix Mn),(Bounded Matrices), (Residuals), and (Identification),

(λNLS,n, σ2NLS,N ) p−→ (λ, σ2) as n→∞ (43)

An important remark is that this Theorem states only that the NLS estimatesare consistent, but it does not tell us about the asymptotic distribution ofλNLS,n.




Asymptotics

Limiting Behavior

The elements of X are non-stochastic and bounded in absolute value bycX , 0 < cX <∞. Also, X has full rank, and the matrixQX = limn→∞ n−1X>X is finite and nonsingular. Furthermore, the matricesQX(λ) = limn→∞ n−1X>Ω(λ)−1X is finite and nonsingular for all |ρ| < 1

Asymptotics

Theorem (Asymptotic Properties of FGLS Estimator)

If assumptions (Homokedastic errors), (Weight Matrix Mn), (BoundedMatrices), and (Limiting Behavior) hold:

1 The true GLS estimator βGLS is a consistent estimator for β, and√n(βGLS − β

)d−→ N

(0, σ2QX(λ)−1) (44)

2 Let λn be a consistent estimator for λ. Then the true GLS estimatorβGLS and the Feasible GLS estimator βFGLS have the same asymptoticdistribution.

3 Suppose further than σ2n is a consistent estimator for σ2. Then

σ2n

[n−1X>Ω(λn)−1X

]is a consistent estimator for σ2QX(λ)−1.

Asymptotics

The Theorem assumes the existence of a consistent estimator of λ and σ2. Itcan be shown that the OLS estimator:

βn =(X>X

)−1 X>y

is√n-consistent. Thus, the OLS residuals ui = yi − x>i βn satisfy Assumption

47 with di,n = |xi| and ∆n = βn − β. Thus, OLS residuals can be used toobtain consistent estimators of ρ and σ2.Then, the FGLS estimator is given by

βFGLS =[X>(λ)X(λ)

]−1X>(λ)y(λ) (45)

where:

X(λ) = (I− λM)Xy(λ) = (I− λM)y

(46)

Asymptotics

The variance covariance matrix of βFGLS is estimated as:

Var(βFGLS

)= σ2

[X>(λ)X(λ)

]−1, (47)

where:

σ2 = ε>(λ)ε(λ)

ε(λ) = y(λ)−X(λ)βFGLS = (I− λM)uu = y−XβFGLS

(48)

Asymptotics

Theorem (CLT for triangular arrays with homokedastic errors, Kelejianand Prucha (1999))

Let vi,n, 1 ≤ i ≤ n, n ≥ 1 be a triangular array of identically distributedrandom variables. Assume that the random variables vi,n, 1 ≤ i ≤ n arejointly independently distributed for each n with E(vi,n) = 0 andE(v2

i,n) = σ2 <∞. Let aij,n, 1 ≤ i ≤ n, n ≥ 1 , j = 1, ..., k be triangulararrays of real numbers that are bounded in absolute value. Further let

vn =

v1,n...

vn,n

, An =

a11,n . . . a1k,n...

...an1,n . . . ank,n

Assume that limn→∞ n−1A>nAn = QAA is finite and nonsingular matrix.Then

1√n

A>nvnd−→ N(0, σ2QAA)

Asymptotics

Sketch of Proof of Theorem.We first prove part (a). Recall that the GLS and FGSL estimator are given by:

βGLS =[X>Ω(λ)−1X

]−1 X>Ω(λ)−1y

βFGLS =[X>Ω(λ)−1X

]−1X>Ω(λ)−1y

Since y = Xβ + u, the sampling error of βGLS is,

β = β +[X>Ω(λ)−1X

]−1 X>Ω(λ)−1u

β − β =[X>Ω(λ)−1X

]−1 X> (In − λM)> (In − λM) (In − λM)−1ε

β − β =[X>Ω(λ)−1X

]−1 X> (In − λM)> ε√n(β − β) =

[1n

X>Ω(λ)−1X]−1 1√

nA>ε

where A = (In − λM) X.

Asymptotics

Sketch of Proof of Theorem.By Assumption (Limiting Behavior):

1n

X>Ω(λ)−1X→ QX(λ)

Since QX is not singular:[1n

X>Ω(λ)−1X]−1→ Q−1

X (λ)

Since A is bounded in abolute value, by Theorem it follows that:

1√n

A>ε d−→ N(0, limn→∞

n−1σ2A>A)

(49)

where limn→∞ n−1σ2A>A = σ2 limn→∞ n−1X> (In − λM)> (In − λM) X =σ2QX(λ).

Asymptotics

Sketch of Proof of Theorem.Consequently:

√n(β − β) =

[ 1n

X>Ω(λ)−1X]−1

︸︷︷︸→Q−1

X(λ)

1√n

A>ε︸︷︷︸d−→N(0,σ2QX (λ))

a∼ N[0,Q−1

X (λ)σ2QX(λ)Q−1X (λ)>)

]a∼ N

[0, σ2Q−1

X (λ)]

This also implies that βGLS is consistent. To show part (b), we can show that:√n(βGLS − βFGLS) p−→ 0

Following Kelijian and Prucha (1999), if suffices to show that

1n

X>[Ω(λn)−1 −Ω(λ)−1

]X p−→ 0 (50)

and

1n

X>[Ω(λn)−1 −Ω(λ)−1

]u p−→ 0

Asymptotics

Sketch of Proof of Theorem.Since:

Ω(λn)−1 −Ω(λ)−1 = (λ− λn)(M + M>) + (λ2 − λ2n)M>M

Then using the fact the we have summable matrices,

1n

X>[Ω(λn)−1 −Ω(λ)−1

]X = (λ− λn)︸︷︷︸

p−→0

n−1X>(M + M>)X︸︷︷︸

O(1)

+ (λ2 − λ2n)︸︷︷︸

p−→0

n−1X>M>MX︸︷︷︸

O(1)

where (λ− λn) p−→ 0 since λn is a consistent estimate of λ, and :

1n

X>[Ω(λn)−1 −Ω(λ)−1

]u = (λ− λn)︸︷︷︸

p−→0

n−1/2X>(M + M>)u︸︷︷︸

Op(1)

+ (λ2 − λ2n)︸︷︷︸

p−→0

n−1/2X>M>Mu︸︷︷︸

Op(1)

= op(1) ∗Op(1) + op(1) ∗Op(1)= op(1) + op(1)= op(1)p−→ 0

(51)

Asymptotics

Sketch of Proof of Theorem.To see that n−1/2X>(M + M>)u = Op(1) note

E[n−1/2X>(M + M>)u

]= 0

Var[n−1/2X>(M + M>)u] = n−1 X> (M + M>)Ω(M> + M)︸︷︷︸absolutely summable

X

︸︷︷︸O(n)

= O(1)

A similar result holds for n−1/2X>M>Mu.Part 3 of the theorem follows from (50) and the fact that σ2 is a consistentestimator for σ2.

Asymptotics

A Feasible GLS (FGLS) can be obtained along with the following steps:

GLS (FGLS) Algorithm of SEMThe steps are the following:

1 First of all obtain a consistent estimate of β, say β using either OLS orNLS.

2 Use this estimate to obtain an estimate of u, say u,3 Use u, to estimate λ, say λ, using MM estimator (NLS estimator),4 Estimate β using the FGLS estimator.




IntuitionConsider the following SAC model:

y = Xβ + ρWy + u = Zδ + uu = λMu + ε

(52)

Applying the spatial Cochrane-Orcutt transformation to the first equation:

y = Zδ + (I− λM)−1ε

(I− λM) y = (I− λM) Zδ + εys(λ) = Zs(λ)δ + ε

(53)

where the spatially filtered variables are given by:

ys(λ) = y− λMy= y− λyL= (I− λM) y

Zs(λ) = Z− λMZ= Z− λZL= (I− λM) Z

IntuitionIf we knew λ, we would be able to apply an IV approach on thetransformed model. For the discussion below, assume that we know λ.Note that the ideal instruments in this case will be:

E (Z) = E [X,WE (y)]E (MZ) = E [MX,MWE (y)]

Given that all the columns of E(Z) and E(MZ) are linear in

X,WX,W2X, ...,MX,MWX,MW2X, ... (54)

Let the matrix of instruments, H, be a subset of the columns in (54), forexample

H =[X,WX, ...,WlX,MX,MWX, ...,MWlX

],

where typically, l ≤ 2. Then:

PZ = (X,PWy)PMZ = (MX,PMWy) .

IntuitionSince we have the instruments H, and we have assumed that we have λ suchthat λ p−→ λ0 we might apply a GMM-type procedure using the followingmoment conditions:

m(λ0, δ0) = E[

1n

H>u]

= 0

Obviously, the corresponding GMM estimator is just the 2SLS estimator.Note that for the transformed model (53), the moment conditions would be

m(λ0, δ0) = E[

1√n

H>ε]

= 0

Now let λ some consistent estimator for λ0 which can be obtained in aprevious step, then the sample moment vector is:

mδ(λ, δ) = 1√n

H>[ys(λ)− Zs(λ)δ

]︸︷︷︸

ε

,

where we explicitly state that the moments depends on δ—which will beestimated—and a consistent estimate of λ.

Intuition

Under homoskedasticity the variance-covariance matrix of the momentvector g(λ0, δ0) is given by:

Var(m(λ0, δ0)) = E(m(λ0, δ0)m(λ0, δ0)>) = σ2n−1H>H,

which motivates the following two-step GMM estimator for δ0:

δ = argminδ

gδn(λ, δ)>Υ δδn gδn(λ, δ)

with

Υ δδn =[

1n

H>H]−1

.

IntuitionNote that:

Jn =[

1√n


]]> [ 1n

H>H]−1 [ 1√

nH>

[ys(λ)− Zs(λ)δ

]]= 1n

[ys(λ)− Zs(λ)δ

]>H[

1n

H>H]−1


]=[ys(λ)− Zs(λ)δ

]>H[H>H

]−1 H>[ys(λ)− Zs(λ)δ

]=[ys(λ)− Zs(λ)δ

]>PH

[ys(λ)− Zs(λ)δ

]Then, the estimator of δ will be:

δ =[Zs>

Zs]−1

Zs>

ys

where Zs = H(H>H

)−1 HZs. This estimator has been called the FeasibleGeneralized Spatial Two-stage Least Squares (FGS2SLS) estimator (Kelijianand Prucha, 1998). However, this estimator is not fully efficient.

Intuation

The question is: How to obtain a consistent estimator of λ? As probably youcan guess, this consistent estimator is obtained in a previous step by GM.




Moment conditions

Consider the homokedastic model and the following three momentconditions:

E[ε>ε

]= σ2

E[ε>MMε

]= σ2 tr

(M>M

)E[ε>Mε

]= 0

Substituting out σ2 into the second moment equation yields:

E[ε>MMε

]− E

[ε>ε

]tr(M>M

)= 0

E[ε>MMε− ε>ε tr

(M>M

)]= 0

E[ε>MMε− ε> tr

(M>M

)ε]

= 0E[ε>(MM− tr

(M>M

)I)ε]

= 0E[ε>A1ε

]= 0.

Moment conditions

Generalizing this expression for the third moment we end up with two insteadof three quadratic moment conditions:

1nE[ε>A1ε

]= 0

1nE[ε>A2ε

]= 0

(55)

with

A1 = MM− n−1 tr(M>M

)I

A2 = M.

Note that A1 is symmetric with tr(A1) = 0 (you should be able to prove this),but its diagonal elements are non zero (In the heteroskedasticity case it is!). InDrukker et al. (2013), an additional scaling factor is included as:

ν = 1/[1 +

[(1/n) tr

(M>M

)]2].

Moment Conditions

Under this case the weighting matrices for quadratic moments are:

A1 = ν[MM− n−1 tr

(M>M

)I]

A2 = Mn.

If the errors are heterokedastic, then:

A1 = M>M− n−1 diag(M>M

)= M>M− n−1 diag

(m>i mi

)A2 = M,

where mi is the ith column of the weights matrix M. Note that diag(m>i mi

)consists of the sum of the squares of the weight in the ith column. Denote thismatrix as D.

Moment ConditionsSince u = λuL + ε, it follows that ε = u− λuL = us, the spatially filteredresiduals. Then:

1nE[u>s A1us

]= 0

1nE[u>s A2us

]= 0

(56)

or more general

E[u>(I− λM>)Aq

(I− λM>)u

]= 0 (57)

Now, we can express the sample moment conditions as :

m2×1

= g2×1− G

2×2

(λλ2

)= 0

The elements of g are the following (see Kelijian and Prucha, 2010, pag.56):

g1 = 1n

u>A1u = 1n

[u>LuL − u>Du

](58)

g2 = 1n

u>A2u = 1n

u>uL (59)

Moment Conditions

The G matrix is given by:

G11 = 2n−1u>M>A1u (60)G12 = −n−1u>M>A1Mu (61)G21 = −n−1u>M> (A2 + A>2

)u (62)

G22 = −n−1u>MA2Mu (63)

A more compact notation is:

G = 1n

u>(A1 + A>1

)us −u>s A1u>s

......

u>(Aq + A>q

)us −u>s Aqu>s

g = 1

n

u>A1u...

u>Aqu

(64)




Assumptions

Now we will state the assumption for the SAC model under heteroskedasticityfollowing Arraiz et al. (2010). The assumptions regarding the spatial weightmatrix are the following:

Spatial Weights Matrices (Arraiz et al., 2010)

Assume the following:(a) All diagonal elements Wn and Mn are zero.(b) λ ∈ (−1, 1), ρ ∈ (−1, 1).(c) The matrices In − ρWn and In − λMn are nonsingular for all λ ∈ (−1, 1)

and ρ ∈ (−1, 1).

Part (a) is a normalization rule: a region cannot be a neighbor of itself. Part(b) has to do with the parameter space. This assumption is discussed byKelejian and Prucha (2010, section 2.2). part (c) ensures that y and u areuniquely defined.

Assumptions

Thus, under these assumptions, we can write the model as:

yn = (In − ρWn)−1 [Xnβ + un]un = (In − ρMn)−1

εn.

The reduced form is:

y = (I− ρW)−1Xβ + (I− ρW)−1(I− λM)−1ε

The reduced form represents a system of n simultaneous equations. As in thestandard spatial lag model, we can include endogenous explanatory variableson the right hand side of model specification. In this case:

y = ρWy + Xβ + Yγ + (I− λW)−1ε.

Assumptions

Heteroskedastic Errors (Arraiz et al., 2010)

The error term εi,n : 1 ≤ i ≤ n, n ≥ 1 satisfy E(εi,n) = 0, E(ε2i,n) = σ2i,n, with

0 < aσ ≤ σ2i,n ≤ aσ <∞. Furthermore, for each n ≥ 1 the random variables

ε1,n, ...., εn,n are totally independent.

This assumption allows the innovations to be heteroskedastic with uniformlybounded variances. This assumption also allows for the innovations to dependon the sample size n, i.e., to form a triangular arrays.

Assumptions

Bounded Spatial Weight Matrices (Arraiz et al., 2010)The row and column sums of the matrices Wn and Mn are bounded uniformly inabsolute value, by , respectively, one and some finite constant, and the row andcolumn sums of the matrices (In − ρWn)−1 and (I− ρMn)−1 are bounded uniformlyin absolute value by some finite constant.

This assumption limits the extent of spatial autocorrelation between u and y. Itensures that the disturbance process and the process of the dependent variableexhibit a “fading” memory. Note that:

E [un] = E[(In − λMn)−1 εn

]= (In − λMn)−1 E [εn]= 0 by Heteroskedastic Errors

(65)

E[unu>n

]= E

[(In − λMn)−1 εnε

>n

(In − λM>

n

)−1]

= (In − λMn)−1 E[εnε

>n

] (I− λM>

n

)−1

= (In − λMn)−1Σ(In − λM>

n

)−1

(66)

where Σ = diag(σ2i,n).

Assumptions

Regressors (Arraiz et al., 2010)

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

This assumption rules out multicollinearity problems, as well as unboundedexogenous variables.

Instruments I (Arraiz et al., 2010)The instruments matrices Hn have full column rank L ≥ K + 1 (for all n largeenough). Furthermore, the elements of the matrices Hn are uniformlybounded in absolute value. Additionally, Hn is assumed to, at least, containthe linearly independent columns of (Xn,MnXn)

There are some papers that discuss the use of optimal instruments for thespatial (see for example Lee, 2003; Das et al., 2003; Kelejian et al., 2004, Lee,2007)

Assumptions

Instruments II (Identification) (Arraiz et al., 2010)The instruments Hn satisfy furthermore:(a) QHH = limn→∞ n−1H>nHn is finite and nonsingular.(b) QHZ = plimn→∞ n−1H>nZn and QHMZ = plimn→∞ n−1H>nMZn are

finite and have full column rank. FurthermoreQHZ,s(λ) = QHZ − λQHMZ has full column rank.

(c) QHΣH = limn→∞ n−1H>nΣnHn is finite and nonsingular, whereΣn = diag(σ2

i,n)

In treating Xn and Hn as non-stochastic our analysis should be viewed asconditional on Xn and Hn.




Procedure

Consider again the transformed model:

ys(λ0) = Zs(λ0)δ0 + ε

where ys(λ0) = y− λ0My and Zs(λ0) = Z− λ0MZ. If we would know λ0,then we could apply the S2SLS to the transformed model. However, λ0 isunknown and therefore we need to estimate it in a first place in order toestimate δ. The steps will be:

1 An initial IV estimator of δ leads to a set of consistent residuals.2 With these residuals, derive the moment conditions that provide a

consistent estimate of λ0 using GMM Estimation procedure.3 The estimate of λ0 is then used to define a weighting matrix for the

moment conditions in order to obtain a consistent and efficient estimator.4 An estimate of δ0 is obtained from the transformed model.5 Finally, a consistent and efficient estimate of λ is based on GS2SLS

residuals.Now we will consider each step in detail

Step 1a: 2SLS estimator

In the first step, δ is estimated by 2SLS applied to the untransformed modelmodel yn = Znδ + un using the instruments matrix H. Then:

δ2SLS =(

Z>Z)−1

Z>y (67)

where Z = H(H>H

)−1 H>Z = PHZ = (X,Wy). The estimates δ2SLS yieldan initial vector of residuals, u2SLS as:

u2SLS = y− Zδ2SLS (68)

Although δ2SLS is consistent, it does not utilize information relating to thespatial correlation error term. We therefore turn to the second step of theprocedure.

Step 1b: Initial GMM estimator of λ based on 2SLSresidualsUsing the consistent estimate u in the previous step, now we create the samplemoments corresponding to Equation (57) for q = 1, 2 based on the estimatedresiduals, and us = Mu:

m(λ, δ2SLS) = 1n

(u>2SLS

(I− λM>)A1 (I− λM) u2SLS

u>2SLS(I− λM>)A2 (I− λM) u2SLS

)= G

(λλ2

)− g

(69)

where,

G = 1n

u>(A1 + A>1

)us −u>s A1u>s

......

u>(Aq + A>q

)us −u>s Aqu>s

g = 1

n

u>A1u...

u>Aqu

(70)

Step 1b: Initial GMM estimator of λ based on 2SLSresidualsThe initial GMM estimator for λ is then defined as

λgmm = argminλ

[G(λλ2

)− g

]> [G(λλ2

)− g

](71)

where Υ λλ = I. This estimator is consistent but not efficient. For efficiency weneed to replace Υ λλ by the variance-covariance matrix of the sample moments.Furthermore, the expression above can be interpreted as a NLS system ofequations. The initial estimate is thus obtained as a solution of the abovesystem.Now, we need to define the expression for the matrices As. Drukker et al.(2013) suggest, for the homokedastic case, the following expressions:

A1 = υ

[M>M− 1

ntr(M>M

)I]

A2 = M(72)

where υ is the scaling factor needed to obtain the same estimator of Kelejianand Prucha (1998, 1999).

Step 1b: Initial GMM estimator of λ based on 2SLSresiduals

On the other hand, when heteroskedasticity is assumed, Kelejian and Prucha(2010) recommend the following expressions:

A1 = M>M− diag(M>M)A2 = M

(73)

Step 1c: Efficient GMM estimator of λ based on 2SLSresidualsThe efficient GMM estimator of λ is a weighted NLS estimator. Specifically, thisestimator is λ where:

λogmm = argminλ

[m(λ, δ)>Ψ−1m(λ, δ)

](74)

and where the weighting matrix is Ψ−1n . The matrix Ψ−1

n = Ψ−1n (λgmm) is defined as

follows. Let Ψ =[Ψrs

]r,s=1,2

with

Ψrs = (2n)−1 tr[(Ar + A>r )Σ(As + A>s )Σ

]+ n−1a>r Σas, (75)

where:

Σ = diagi=1,...,n(ε2i

)ε =

(I− λgmmM

)u

ar =(I− λgmmM

)HPαr

αr = −n−1 [Z> (I− λgmmM)

(Ar + A>r )(I− λgmmM

)u]

P =( 1n

H>H)−1 ( 1

nH>nZn

)[( 1n

H>Z)> ( 1

nH>H

)−1 ( 1n

H>Z)]−1

(76)

It is important to note that this step is not necessary since the previous estimator ofλ is already consistent.

Step 2a: G2SLS Estimator

Using λogmm from step 1c (or the consistent estimator from step 1b) in thetransformed model we have:

δn(λogmm) =[Z>s (λogmm)Z(λogmm)

]−1Z>s (λogmm)ys(λogmm) (77)

where

ys = y− λogmmMyZs = Z− λogmmMZZs = PHZs

PH = H(H>H

)−1 H>

(78)

Step 2b: Efficient GMM estimator of λ using GS2SLSresidual

An efficient GMM estimator of λ based on the GS2SLS residuals is obtainedby minimizing the following expression:

λ = argminλ

[G(λλ2

)− g

]>(Ψ λλ)−1

[G(λλ2

)− g

](79)

where Ψ λλ is an estimator for the variance-covariance matrix of the(normalized) sample moment vector based on the GS2SLS residuals. Thisestimator differs for the cases of homoskedastic and heteroskedastic errors.


For the homoskedastic case the r, s (with r, s = 1, 2) element of Ψ λλ is givenby:

Ψ λλrs =[σ2]2 (2n)−1 tr

[(Ar + A>r

) (As + A>s

)]+ σ2n−1a>r a>s+ n−1

(µ(4) − 3

[σ2]2) vecD (Ar)> vecD (As)

+ n−1µ(3) [a>r vecD (As) + a>s vecD (Ar)],

(80)

Step 2b: Efficient GMM estimator of λ using GS2SLSresidualwhere

ar = TαrT = HP,

P = Q−1HHQHZ

[Q>HZQ−1

HHQ>HZ]−1

Q−1HH =

(n−1H>H

),

QHZ =(n−1H>Z

),

Z =(

I− λM)

Z,

αr = −n−1 [Z> (Ar + A>r)ε]

σ2 = n−1εε,

µ(3) = n−1n∑i=1

ε3i ,

µ(4) = n−1n∑i=1

ε4i .

(81)


For the heteroskedastic case the r, s (with r, s = 1, 2) element of Ψ λλ isgiven by:

Ψ λλrs = (2n)−1 tr[(

Ar + A>r)Σ(As + A>s

)Σ]

+ n−1a>r Σa>s , (82)

where, Σ is a diagonal matrix whose ith diagonal element is ε2i .

Lecture 7: GMM Estimation of Spatial Models1 Estimation of SLM: Spatial Two Stage Estimation (S2SLS) Introduction Instruments Assumptions EstimatorandAsymptoticDistribution 2 Estimation

Documents