Brief Introduction to Spatial Econometrics

Sung Y. Park

Chung-Ang Univ.

References

Anselin, L. and Bera, A.K.“Spatial Dependence in Linear RegressionModels with an Introduction to Spatial Econometrics”, 1998, In:Handbook of Applied Economic Statistics, A. Ullah and D.E.A.Giles, Eds., Marcel Dekker, NY.

Cressie N.A.C., Statistics for Spatial Data, 1993, Wiley, NY.

Arbia, G., Spatial Econometrics, 2007.

Motivation

• Growth regression model:

Consider well-known simple growth regression to analyze economicconvergence of per capita income.

ln

[

yT ,i

y0,i

]

= α + β ln y0,i + ǫi ,

where yt,i represents the per-capita income at time t in region i . Theparameter are estimated by OLS method (Mankiw, Romer and Weil(1992), Barro and Sala-i-Martin(1995)).

Remarks:

β < 0 and significantly different from 0 ⇒ absolute convergence.

One can perform a test for economic convergence: H0 : β = 0,H1 : β < 0.

When OLSE is BLUE?

What if ǫi is not independent?

Motivation(Con’t)

• Some sayings...

...“it is difficult to believe that Belgium and Dutch economic growthwould ever significantly diverge, or that substantial productivitygaps would appear in Scandinavia. The omitted variables that arecaptured in the regression residuals seem ex ante likely to take onsimilar values in neighbouring countries. This suggests that residualsin nearby nations will be correlated” (De Long and Summers (1991))

... “for the reported standard errors to be correct, the residuals forCanada must be uncorrelated with the residuals for the US. Ifcountry residuals are in fact correlated, as it is plausible, then datamost likely contain less information than the reprted standard errorindicate” (Mankiw (1995))

...“without more evidence that the disturbances are independent, thestandard errors in most growth regression should be treated with acertain degree of mistrust” (Temple (1999))

Motivation(Con’t)

Distribution of the per-capita GDP (expressed in natural log) in the 129European NUTS-2 regions in 1996.

Motivation(Con’t)

Distribution of the per-capita GDP growth rates (expressed in natural log) inthe 129 European NUTS-2 regions in the period 1980-1996.

Motivation(Con’t)

Map of the empirical standardized residuals on the 129 regions at a NUTS-2level over the period 1980-95. Residuals are classified in the 4 interquartile

classes.

Spatial Model

• “Dependence” (violation of independence in our class)

Time-series data: serial dependence

Cross-sectional data: spatial autocorrelation

Anselin (1988, p.7) “the collection of techniques that deal with thepeculiarities caused by space in the statistical analysis of regional sciencemodels”.

recent empirical works: urban and regional economics; real estateeconomics; transportation economics; economic geography;empirical growth analysis...

spatial dependence: a renewed focus on Marshallian externalities,spatial spillovers, new economic geography og Krugman (1991),endogenous growth theory of Romer (1986), analysis of localpolitical economy of Besley and Case (1995)...

Spatial Model(Con’t)

In econometrics, an attention to serial correlation has been the maininterest of time series analysis; typical focus for cross-sectional datais heteroskedasticity.

Until recently, in many studies spatial autocorrelation was largelyignored. In other discipline, primarily in physical science: geology,ecology, geography and socialogy and psychology, the spatialdependence is of central interests.

Spatial Autocorrelation

the coincidence of value similarity with location similarity

high or low values for a random variable tends to cluster in space orlocations tend to be surrounded by neighbors with very similarvalues

“location similarity” can be treated by “neighbor”

the existence of spatial autocorrelation maybe represented by

Cov(yi , yj) = E (yiyj) − E (yi)E (yj) = 0, for i 6= j,

where yi and yj are obs on a random variable at location i and j inspace and i , j can be points.

Spatial Autocorrelation

• ProblemFor a set N obs. on cross-sectional data, it is not possible to estimateN × N covariance terms directly from the data ⇒ imposing constraintson the covariance matrix.

• Two approaches

[Direct approach] variogram (semi-variogram) : all pairs of locationsare sorted according to the distance and the strength of covarianceis expressed as a continuous function of this distance. [not useful inempirical analysis]

Lattice perspective : A “neighborhood set” should be defined. Foreach obs i , j ∈ Si which can be exploited to specified a spatialstochastic process

Spatial Autocorrelation

• Spatial weights.

A N × N positive and symmetric matrix W . wij = 1 when i and j

are neighbors and wij = 0 otherwise.

By convention, wii = 0.

Row-standardized weights matrix w sij = wij/

∑

j wij ⇒ all wij arebetween 0 and 1 ⇒ an averaging of neighboring values.

[Note] The spatial weights should be formed such that the spa tialprocess must satisfy the necessary regularity conditions such thatasymptotics maybe invoked to obtained the properties of estimatesand test statistics.

the weight matrix is taken to be exogenous, unless endogeneity isconsidered explicitly in the model specification

Spatial Autocorrelation

Which elements are non-zero?

[traditional] when they have a border in common (first ordercontiguity), wij = 1; wij = 1 for dij ≤ δ where dij is the distancebetween units i and j, and δ is a distance cutoff value (distancebased contiguity).

the above has been generalized to Cliff-Ord weights,

wij =b

βij

dαij

,

where bij is the share of the common border between units i and j

in the perimeter of i (note that bij is not necessarily equal to bji).

sociometrics: the weights reflect whether or not two individualsbelong to the same social network.

in some economic application: wij = 1/|xi − xj |.

Spatial Lag Operator

time-series: yt−k or yt+k : one-dimensional time axes.

spatial data: very ambiguous. Only on a regular grid structure thereis a potential solution.Rook criterion: vertex on a regular lattice, (i , j) has four neighbors(i + 1, j), (i − 1, j), (i , j + 1) and (i , j − 1).Queen criterion: rook criterion plus yi−1,j+1, yi−1,j−1, yi+1,j+1 andyi+1,j−1.

In the irregular spatial structure (common in real world) the abovenotion of spatial shift is impractical.

• Spatial lag operator: a weighted average of the values at neighboringlocation, Wy or

∑

j wijyj .

Spatially lagged dependent variable: Wy

Spatially lagged dependent variable: W ǫ

Spatial Lag Dependence

A mixed regressive spatial autoregressive model

y = ρWy + Xβ + ǫ. (1)

Note: (Wy)i is always correlated with ǫi (time-series case?) Moreover,(Wy)i is correlated with ǫj ⇒ OLSE is not consistent!

WHY?

y = (I − ρW )−1Xβ + (I − ρW )−1ǫ.

((I − ρW )−1 is a full matrix and not triangular.)

⇒ (I − ρW )−1 = (I + ρW + ρ2W 2 + · · · )ǫ

⇒ E [(Wy)i ǫi ] = E [W (I − ρW )−1ǫiǫi ] 6= 0.

• Note that (1) can be rewritten by

(I − ρW )y = Xβ + ǫ,

where (I − ρW )y is a spatially filtered dependent variable.

Spatial Error Dependence

y = Xβ + ǫ,

ǫ = λW ǫ + ξ.

(OLSE will be inefficient)

⇒ y = Xβ + (I − λW )−1ξ (2)

⇒ E [ǫǫ′] = σ2[(I − λW )′(I − λW )]−1 (3)

Note: the complex structure in the inverse matrices in (3) yieldsnon-constant diagonal elements in the error covariance matrix ⇒heteroskedasticity in ǫ irrespective of the heteroskedasticity of ξ.

• From (2) (multiplying (I − λW ) to both sides)

y = λWy + Xβ − λWXβ + ξ.

Spatial Durbin model.

Higher-Order Spatial Process

• SARMA(p,q) (Huang (1984))

y = ρ1W1y + ρ2W2y + · · · + ρpWpy + ǫ

ǫ = γ1W1ξ + γ2W2ξ + · · · + γqWqξ + ξ.

A regressive component Xβ can be also added. The above generalformulation is highly complex.

• A first-order spatial autoregressive lag with a first order spatialautoregressive error model [this model has been used in many application]

y = ρW1y + Xβ + ǫ (4)

ǫ = λW2ǫ + ξ (5)

(4) and (5) yield following reduced form

y = ρW1y + λW2y − λρW2W1y + Xβ − λW2Xβ + ξ. (6)

Higher-Order Spatial Process (Con’t)

W2W1 = 0, i.e., W1 and W2 do not overlap, (6): a biparametricspatial lag formulation with additional constraints on theparameters.

For W1 = W2 = W ,

y = (ρ + λ)Wy − λρW 2y + Xβ − λWXβ + ξ

coefficients of Wy and W 2y alone do not identified. the non-linear constraint between the β and −λβ yielded

estimate of λ. at least one exogenous variable should be included in X to

identify ρ and λ.

CAR and SAR

Conditionally and Simultaneously specified Spatial Gaussian Model:

Statistics: Conditional Autoregression (CAR)Econometrics: Simultaneous Autoregression (SAR)

Any difference?

• Easy and natural departure from the independence is a Markov chain,Z (t) : t = 0, 1, · · · .

Definition by its joint probability

Pr(z(1), · · · , z(i)|z(0)) =i∏

t=1

Qt(z(t); z(t − 1)), ∀i ≥ 1, (7)

where Qt is some function of z(t) and z(t − 1).

Definition by conditional probability

Pr(z(i)|z(0), · · · , z(i − 1)) = Pr(z(i)|z(i − 1)), ∀i ≥ 1 (8)

CAR and SAR (Con’t)(8) ⇒ (7):

Pr(z(1), · · · , z(i)|z(0)) =

i∏

t=1

Pr(z(t)|z(0), · · · , z(t − 1))

=i∏

t=1

Pr(z(t)|z(t − 1))

(7) ⇒ (8):

Pr(z(i)|z(0), · · · , z(i − 1)) =Pr(z(1), · · · , z(i)|z(0))

Pr(z(1), · · · , z(i − 1)|z(0))

=

∏i

t=1 Qt(z(t); z(t − 1))∑

z(i)

∏i

t=1 Qt(z(t); z(t − 1))

=Qi(z(i); z(i − 1))

∑

z(i) Qi(z(i); z(i − 1))

⇒ This equivalence is due to the unidirectional flow of time. The abovetwo models are difference in general in the spatial domain.

CAR and SAR (Con’t)

Consider

D = s = (u, v)′ : u = · · ·−2, −1, 0, 1, 2, · · · ; v = · · ·−2, −1, 0, 1, 2, · · ·

• Joint approach:

Pr(z) =∏

(u,v)∈D

Quv (z(u, v); z(u−1, v), z(u+1, v), z(u, v −1), z(u, v +1))

• Conditional approach:

Pr(z(u, v)|z(k , l) : (k , l) 6= (u, v))

= Pr(z(u, v)|z(u − 1, v), z(u + 1, v), z(u, v − 1), z(u, v + 1)),

for all (u, v)′ ∈ D.

SARSimultaneously Specified Spatial Gaussian Model:

[Whittle (1954)] supposeǫ(u, v) : u = · · · − 1, 0, 1, · · · ; v = · · · − 1, 0, 1, · · · is a process ofindependent and identically distributed random variables. defineZ (u, v) by

φ(T1, T2)Z (u, v) = ǫ(u, v),

where T1 and T2 are translation operator,

T1Z (u, v) = Z (u + 1, v), T −11 Z (u, v) = Z (u − 1, v)

T2Z (u, v) = Z (u, v + 1), T −12 Z (u, v) = Z (u, v − 1)

andφ(T1, T2) =

∑

i

∑

j

aijTi1T

j2

Example: the nearest-neighbor dependence

φ(T1, T2) = 1 − ξ1(T1 + T −11 ) − ξ2(T2 + T −1

2 )

SAR (Con’t)

[Gaussian model] Assume thatZ (s) : s ∈ D = Z (si) : i = 1, 2, · · · , n is defined on a finitesubset of the integer lattice in the plane. Let ǫ ∼ N(0, Λ) be a n-dimnormal distribution with mean 0 and diagonal variance matrix Λ.

One can define

(I − W )(Z − µ) = ǫ (9)

Clearly, E (Z ) = µ andVar(Z ) = E(Z − µ)(Z − µ)′ = (I − W )−1Λ(I − W ′)−1 since(Z − µ) is a linear combination of ǫ

Z ∼ N(µ, (I − W )−1Λ(I − W ′)−1) (10)

(9) can be rewritten as

Z (si) = µi +n∑

j=1

wij(Z (sj) − µj) + ǫi , i = 1, 2, · · · , n.

SAR (Con’t)

The likelihood

(2π)−n/2|Λ|−1/2|I−W | exp−(1/2)(z−µ)′(I−W ′)Λ−1(I−W )(z−µ)(11)

Cov(ǫ, Z ) = E (ǫZ ′) = Λ(I − W ′)−1 ⇒ Not diagonal, i.e., the erroris not independent of the autoregressive variable.

CAR

Conditionally Specified Spatial Gaussian Model:

f (z(si )|z(sj) : j 6= i) ∼ N(θi , τ2i )

= (2πτ2i )−1/2 exp[−z(si) − θi(z(sj) : j 6= i)2/2τ2

i ], i = 1, 2, · · · , n.

Under a regularity condition of “pairwise-only dependence”

θi(z(sj) : j 6= i) = µi +

n∑

j=1

cij(z(sj) − µj), i = 1, 2, · · · , n,

where cijτ2j = cjiτ

2i and cii = 0.

By factorization theorem (Besag(1974) but, originally, Hammerseyand Clifford Theorem (1971))

Z ∼ N(µ, (I − C)−1M),

where C ≡ (cij) is an n × n matrix and M ≡ diag(τ21 , · · · , τ2

n ).

CAR

The likelihood

(2π)−n/2|M|−1/2|I − C |1/2 exp −(1/2)(z − µ)′M−1(I − C)(z − µ)

If v are defined by v ≡ (I − C)(z − µ),

z(si ) − µi ≡n∑

j=1

cij(z(sj) − µj) + vi , i = 1, 2, · · · , n.

Even though Var(v) = M(I − C ′) is not diagonal, E [vZ ′] = M

(diagonal).

Estimation• MLE:

Spatial Lag Model

In the likelihood (11) one major concern is the Jacobian of thetransformation: |I − ρW | and |I − λW | in the spatial lag and spatialautoregressive error models, respectively. Time-series model: thedeterminant of a triangular matrix. Spatial model: the determinantof a full matrix.

Ord (1975) showed

|I − ρW | =

N∏

i=1

(1 − ρωi),

where ωi is the eigenvalue of W .

Then,

L =∑

i

ln(1−ρωi)−N

2ln(2π)−

N

2ln(σ2)−

(y − ρWy − Xβ)′(y − ρWy − Xβ)

2σ2

⇒ clearly show that why OLS is not MLE.

Estimation

From FOC (conditional on ρ)

βML = (X ′X)−1X ′(I − ρW )y

σ2ML =

(y − ρWy − Xβ)′(y − ρWy − Xβ)

N

a concentrated log-likelihood (why?)

Lc = −N

2ln

[

(e0 − ρeL)′(e0 − ρeL)

N

]

+∑

i

ln(1 − ρωi), (12)

where e0 and eL are residuals in a regression of y on X and Wy ,respectively.

ρML is obtained from maximizing (12) with respect to ρ.

Estimation (Con’t)

AsyVar [ρ, β, σ2] =

Tr[WA]2 + Tr[W ′AWA] + [WAXβ]′[WAXβ]

σ2

(X ′WAXβ)′

σ2

Tr[WA]σ2

X ′WAXβσ2

X ′Xσ2 0

Tr[WA ]σ2 0 N

2σ4

−1

,

where WA = W (I − ρW )−1 (lack of block diagonality)

Estimation (Con’t)

Spatial error autocorrelation Model

Ω(λ) = [(I − λW )′(I − λW )]−1

The log-likelihood

L = −1

2ln |Ω(λ)|−N

2ln(2π)−N

2ln(σ2)− (y − Xβ)′Ω(λ)−1(y − Xβ)

2σ2

From FOC (conditional on λ)

βML = [X ′Ω(λ)−1X ]−1X ′Ω(λ)−1y .

⇒ familiar GLS form. But estimation of λ is not as easy as in thetime series case.

Estimation (Con’t)

Magnus (1978, p.283) the iterative solution of the first ordercondition

Tr

[(

∂Ω−1

∂λ

)

Ω

]

= e′

(

∂Ω−1

∂λ

)

e,

where e = y − Xβ are GLS residuals,∂Ω−1/∂λ = −W − W ′ + λW ′W .

A concentrated likelihood method can be also used.

AsyVar [β, σ2, λ] =

σ2[X ′LXL] 0 0

0 N2σ4

Tr(WB)σ2

0 Tr(WB)σ2 Tr(WB)2 + Tr(W ′

BWB)

−1

where WB = W (I − λW )−1 and XL denotes spatially filteredvariable, X − λWX . ⇒ knowledge of the precision of λ does notaffect the precision of the β estimates.

Estimation (Con’t)

IV/GM:

E [Wyǫ] 6= 0 ⇒ endogeneity problem

Let Q P × N matrix (P ≥ K + 1) of instruments (including“exogenous variables” from X).

IV or 2SLS estimate follows as

βIV = [Z ′Q(Q′Q)−1Q′Z ]−1Z ′Q(Q′Q)−1Q′y ,

where Z = [Wy , X ]. The AsyVar (βIV ) = σ2[Z ′Q(Q′Q)−1Q′Z ]−1

and σ2 = (y − Z βIV )′(y − Z βIV )/N

In Kelejian and Robinson (1993), the consistency of βIV is derivedformally.

Since E [(I − ρW )−1ǫ|X ] = 0 E [y |X ] = (I − ρW )−1Xβ.

⇒ E [Wy |X ] = W (I − ρW )−1Xβ

⇒ E [Wy |X ] = WXβ + ρW 2Xβ + ρ2W 3Xβ + · · ·

Estimation (Con’t)

Based on the expansion, Kelejian and Robinson (1993) suggest theuse of a subject of columns from X , WX , W 2X , W 3X , · · · as theinstruments.

Recent works has focused on the selection of “optimal” instruments.Lee (2003) [pure spatial lag model without error dependence]suggested

Q = [X , W (I − ρW )−1X β]

where the ρ and β are obtained in a first round estimation usingWX as the instrument [optimal instrument is the inverse of n × n

matrix].

Kelejian, Prucha and Yuzefovich (2004) introduce a series expansionto avoid the calculation of the inverse matrix

Q = [X ,r∑

s=0

ρsW s+1X β],

Estimation (Con’t)

Kelejian and Prucha (1999): a generalized moments (GM) approachto obtain a consistent estimate of the λ in ǫ = λW ǫ + u. u isassumed i.i.d with variance σ2. Three moment conditions(Tr(W ) = 0)

E [(1/n)u′u] = σ2

E [(1/n)u′W ′Wu] = (1/n)σ2tr(W ′W )

E [(1/n)u′W ′u] = 0

Using u = ǫ − λW ǫ and Wu = W ǫ − λWW ǫ the above conditionscan be expressed by error ǫ.

Estimation (Con’t)

Then, substituting ǫ by its sample counterpart yields the followingsystem with unknown λ, λ2 and σ2

[

(2/n)e′e (−1/n)e′e 1(2/n)e′e (−1/n)e′e (1/n)tr(W ′W )

(1/n)(e′e + e′e) (−1/n)e′e 0

][

λλ2

σ2

]

=

[

(1/n)e′e

(1/n)e′e

(1/n)e′e

]

where e = We and e = WWe. Using nonlinear least squares λ andσ2 can be solved. Plug λ into the error variance-covarianceestimates to get a consistent estimator for β.

Estimation (Con’t)• Some current studies... (GMM and HAC)

Lee (2001), Generalized Method of Moments Estimation of SpatialAutoregressive Processes

Lee, L. F. (2004). Asymptotic distributions of quasi-maximum likelihoodestimators for spatial econometric models, Econometrica 72: 1899.1926.

Lin, X. and Lee, L. F. (2006). Gmm estimation of spatial autoregressivemodels with unknown heteroskedasticity. Working paper, Department ofEconomics, Ohio State University.

Lee and Liu (2006), Efficient GMM Estimation of a SpatialAutoregressive Model with Autoregressive Disturbances, Working paper,Department of Economics, Ohio

Lee (2007), GMM and 2SLS estimation of mixed regressive, spatialautoregressive models, Journal of Econometrics.

Driscoll, J., Kraay, A., 1998. Consistent covariance matrix estimationwith spatially dependent panel data. The Review of Economics andStatistics 80, 549.560

Kelejian and Prucha (2008), HAC estimation in a spatial framework,Journal of Econometrics.

Test• Moran’s I test

Moran’s (1950) I test is a two dimensional analog of the test ofsignificance of the serial correlation coefficient in univariatetime-series

Cliff and Ord (72, 73) presented Moran’s I statistics„

I =N

S0

(

e′We

e′e

)

,

where e = y − X β, β = (X ′X)−1X ′y , N is the number of obs, andS0 is a standardization factor equal to the sum of the spatialweights,

∑

i

∑

j wij .

For a row-standardized weights matrix W , S0 = N .

I =e′We

e′e

Moran did not derive the statistic from any basic principle.

Test (Con’t)

• Similarity between Moran’s I and Durbin-Watson statistic

DW =e′Ae

e′e,

A =

1 −1 0 0 · · · 0 0 0−1 2 −1 0 · · · 0 0 00 −1 2 −1 · · · 0 0 0...

......

......

......

...0 0 0 0 · · · −1 2 −10 0 0 0 · · · 0 −1 1

The DW test is UMP test for one sided alternatives withǫt = λǫt−1, xit .

Test (Con’t)

Cliff and Ord (1972) established a link between the LR and I tests.If we take the alternative ǫ = λW ǫ + ξ. LR test (H0 : λ = 0;Ha : λ = λ1)

ǫ′W ǫ

ǫ′(I + λ21G)ǫ

,

where G is a function of W . As λ1 → 0 I → LR.

It is also possible to develop a finite sample bound test for I

following Durbin and Watson. Note that we need to the boundsindependent of not only X but also W [Imhof (1961), Koerts andAbrahamse (1968)]

Note that the computation of exact values of the Moran’s I statisticis very similar to the that of the DW statistic.

Test (Con’t)• Kelejian-Robinson Test

They consider the following assumption

Cov(ǫi , ǫj) = σij = Zijα,

where Zij is 1 × q vector which can be from X , α is q × 1 vector ofparameters, and i , j are contiguous. The null hypothesis of nospatial correlation is tested by H0 : α = 0.

We denote C by hN × 1 vector σij for i < j. A test for α = 0: run a

regression of C on Zhn×q ≡ (Zij). C can be replaced by C = ei ej ,where ei is OLS residual. The test is

KR =γ′Z ′Z γ

σ4,

where γ = (Z ′Z )−1Z ′C and σ4 denotes a consistent estimator of σ4

(eg. [e′e/N ]2 or (C − Z γ)′(C − Z γ)/hN)

Under H0, KR χ2q.

Test (Con’t)

Since γ = (Z ′Z )−1Z ′C

KR =C ′Z (Z ′Z )−1Z ′C

σ4

Under H0, C ′C/hN→pσ4. An asymptotically equivalent form of thetest

hN · C ′Z (Z ′Z )−1Z ′C

C ′C

⇒ the familiar NR2 form.

Test (Con’t)

• Similarity between KR and Moran’s I:

I2 =(e′We)2

(e′e)2=

1

N2σ4

(

N∑

i=1

N∑

j=1

Wijei ej

)2

=N∑

k=1

N∑

l=1

N∑

m=1

N∑

n=1

Wkl Wmn(ekel)(emen)

N2σ4

And

KR =

hN∑

i=1

hN∑

j=1

pij Ci Cj

σ4,

where pij are the elements of Z (Z ′Z )−1Z ′.

Test (Con’t)• Tests for Spatial Error Autocorrelation:

ǫ = λW ǫ + ξ

Wald Test:

WSλ =λ2

ˆAsyVar(λ)

where

AsyVar(λ) =

[

tr(W 2B) + tr(W ′

BWB) − tr(WB)2

N

]−1

LR Test:

LRλ = N [ln σ2 − ln σ2] + 2

N∑

i=1

ln(1 − λωi)

Test (Con’t)

Score Test:

dλ =∂L

∂λ

∣

∣

∣

∣

λ=0

=ǫ′W ǫ

σ2

RSλ =d2

λ

T=

[e′We/σ2]2

T,

where T = tr [(W ′ + W )W ].

The test requires only OLSE

Under H0, RSλ χ21

Time-series: W = W ′, T = N − 1 ⇒ RSλ = (N − 1)λ2, whereλ =

∑

t etet−1/∑

t e2t−1.

Consider q-th order spatial autoregressive model

Test H0 : λ1 = λ2 = · · · = λq = 0

RSλ1···λq =

q∑

l=1

[e′Wle/σ2]2

Tl

where Tl = tr [W ′

l Wl + W 2l ], l = 1, 2, · · · , q.

Test (Con’t)

• Tests for Spatial lag dependence:

Score Test:

dρ =∂L

∂ρ

∣

∣

∣

∣

ρ=0

=ǫ′Wy

σ2

RSρ =d2

ρ

T1

=[e′Wy/σ2]2

T2

where T1 = [(WXβ)′M(WXβ) + Tσ2]/σ2. Note that I(θ) is not a

block-diagonal matrix.

Test (Con’t)

• Tests for the Possible Presence of both Spatial Error and LagAutocorrelation:

y = ρW1y + Xβ + ǫ

ǫ = λW2ǫ + ξ, ξ ∼ N(0, σ2I)

Test (Con’t)

Notes: Impact of local presence of ρ (λ) on the asymptotic nulldistribution of RSλ (RSρ), LRλ (LRρ), and WSλ (WSρ).

Let ρ = δ/√

N, δ < ∞. Under H0 : λ = 0 RSλ, LRλ, and WSλ

asymptotically converge to a noncentral χ21 with noncentrality

parameter

Rρ =δ2T 2

12

NT22

where Tij = tr [Wi Wj + W ′i Wj ], j = 1, 2.

Let λ = τ/√

N, τ < ∞. Under H0 : ρ = 0 RSρ, LRρ, and WSρ

asymptotically converge to a noncentral χ21 with noncentrality

parameter

Rλ =τ2T 2

12σ2

ND

where D = (W1Xβ)′M(W1Xβ) + T11σ2.

Test (Con’t)

• Joint Test: H0 : λ = ρ = 0

RSλρ = E −1

[

(dλ)2 D

σ2+ (dρ)2T22 − 2dλdρT12

]

where E = (D/σ2)T22 − (T12)2.

When W1 = W2 = W (in most applications),T11 = T21 = T22 = T = tr [(W ′ + W )W ]

RSλρ =d2

λ

T+

(dλ − dρ)2

σ−2(D − T σ2)

Under H0 RSλρ χ22 ⇒ loss of power compared to the proper

one-sided directional test when only one of the two forms ofmisspecification is present.

Test (Con’t)• Conditional Test: H0 : λ = 0 (ρ = 0) in the presence of ρ (λ).

H0 : λ = 0 conditional on ρ (based on the residuals of a MLE of thespatial lag model)

RSλ|ρ =d2

ρ

T22 − (T21A)2Var(ρ)

where T21A = tr [W2W1A−1 + W ′2W1A−1] with A = I − ρW1.

H0 : ρ = 0 conditional on λ (based on the residuals of a MLE of theerror autocorrelation model)

RSρ|λ =[ǫ′B′BW1y ]2

Hρ − HθρVar(θ)H ′θρ

where θ = (β′, λ, σ2)′, B = I − λW2 and

Hρ = trW 21 + tr(BW1B−1)′(BW1B−1) + (BW1Xβ)′(BW1Xβ)/σ2

Hθρ = [(BX)′BW1Xβ/σ2, tr(W2B−1)′BW1B−1 + trW2W1B−1, 0]

Test (Con’t)

• Robust Test (Anselin et al. (1996)): H0 : λ = 0 in the localmisspecification of ρ.

RS∗λ =

[dλ − T12σ2D−1dρ]2

T22 − (T12)2σ2D

When W1 = W2 = W ,

RS∗λ =

[dλ − T σ2D−1dρ]2

T (1 − T σ2D)

Under H0, RS∗λ χ2

1

Test (Con’t)

• Robust Test (Anselin et al. (1996)): H0 : ρ = 0 in the localmisspecification of λ.

RS∗ρ =

[dρ − T12T −122 dλ]2

σ−2D − (T12)2T −122

When W1 = W2 = W ,

RS∗ρ =

[dρ − dλ]2

σ−2D − T

Under H0, RS∗ρ χ2

1