Economics Letters Volume 28 Issue 4 1988 [Doi 10.1016_0165-1765(88)90009-2] Luc Anselin -- A Test for Spatial Autocorrelation in Seemingly Unrelated Regressions

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Economics Letters 28 (1988) 335-341

North-Holland

335

A TEST FOR SPATIAL AUTOCORRELATION IN SEEMINGLY UNRELATED

REGRESSIONS *

Luc ANSELIN

University of Calif ornia Santa Barbara CA 93106 USA

Received 19 May 1988

Accepted 6 July 1988

A Lagrange multiplier test is proposed for spatial autocorrelation in the error term of the equations in a seemingly unrelated

regression (SUR) model. This test extends approaches developed for single equation models to the SUR context.

1 Introduction

Spatial econometrics is a subfield of applied econometrics concerned with complications in

estimation and testing that may result from the presence of spatial dependence and spatial

heterogeneity [e.g., Paelinck and Klaassen 1979) Ancot et al. 1986) and Anselin 1988b)]. These

complications are typically encountered in empirical analyses of cross-sectional data in regional

science and urban economics.

A common problem is the lack of independence of the regression disturbance term as a

consequence of spatial spill-over effects spatial externalities), and the arbitrariness of the boundaries

of aggregate spatial units of observation e.g., states, counties). This special case of a non-spherical

disturbance is commonly referrred to as spatial autocorrelation, and may lead to misleading

inference.

In the context of a single equation specification, spatial autocorrelation is well understood. For

example, a well-known test for its presence in a regression error term is the Moran coefficient [Cliff

and Ord 1981) Ring 1987)]. Although the same effect would also tend to be present in a panel data

context, when observations are pooled across space and over time, it is typically ignored. Extensions

of the Moran coefficient to a space-time situation have been suggested, but lack rigorous distribu-

tional properties and therefore cannot be used in a formal model specification.

In regional econometric modeling the data often consist of cross-sections for a small number of

time periods e.g., a few decennial censuses), or a cross-section of cross-sections e.g., employment by

county for different sectors). In this situation a seemingly unrelated regression SUR) is typically the

specification of choice. It is sometimes called a spatial SUR, since the equations pertain to

cross-sections.

In this note, I outline an asymptotic test for spatial autocorrelation in the error of a spatial SUR

based on the Lagrange multiplier principle. The test is an extension of the procedures introduced in

Anselin 1988a) in a single equation context. As shown in that paper, standard simplifying results for

LM tests with time series data [e.g., Breusch and Pagan 1980) Davidson and MacKinnon 1984)] do

* The research on which this paper is based was supported by Grant SES-8600465 from the National Science Foundation.

0165-1765/88/ 3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)



not hold for spatial models. due to the multidirectional nature of the dependence in space, and the

resulting complex structure of the Jacobian in the likelihood function. Therefore, a derivation is

necessary that takes into account the special problems encountered in the spatial domain.

2. Spatial SUR with spatial error autocorrelation

The formal specification of a spatial SUR model with spatial error autocorrelation consists of T

equations, one for each time period or sector, product, etc.):

where the index r is used to refer to each equation. Alternatively, in stacked form, all equations can

be summarized as

where Y is an NT by 1 vector of dependent variables, X is a block diagonal matrix of dimensions

NT by K, j3 is the overall coefficient vector of dimension K by 1. and 6 is an NT by 1 error vector.

The presence of spatial dependence in the error term for each eq. can be expressed as a spatial

autoregression,

where X, is the associated spatial autoregressive parameter, w is a weight matrix that reflects the

spatial pattern of dependence with zero diagonal terms), and p, is a spherical error term. In this

general specification the spatial parameter and the spatial weight matrix are allowed to differ foreach equation. The dependence between equations is in the usual SUR form:

where uts is the error covariance between equation t and s, combined in a matrix 2‘ for all t, s.

The spatially dependent error vector c, can be expressed as a transformation of the independent

CL, as

Consequently, it follows that

where, for notational simplicity,

B = (1 - h,W,).

The inverse error covariance for the full system, aPi, takes the form

0-l = B'(Z-' Q I)B, (8)



L. Anselin / Spatial autocorrelation in seemingly unrelated regressions 331

where B is a blockdiagonal NT by NT matrix with the B, as diagonal elements, and @ is the

Kronecker product.

Under the assumption of normality, the log-likelihood function ignoring constants) for the model

in stacked form is

L= - 1/2)lnIQl- 1/2) Y-Xp)‘Q-‘ Y-Xp), 9)

which becomes, after some straightforward matrix manipulations:

L= - N/2)lnIZl+Z, lnIB,I- 1/2) Y-XP)‘B’[Z-‘@I]B Y-XP). 10)

Estimation necessitates a non-linear optimization of this likelihood and is not further considered

here.

3. A Lagrange multiplier test for spatial error autocorrelation

A Lagrange multiplier LM) or score test for the presnece of spatial error autocorrelation is

equivalent to a test for the null hypothesis H,: X = 0, where h is a T by 1 vector which contains the

h, coefficients for each equation. Following the standard LM approach, the coefficient vector is

partitioned as

8= [~ldl (11)

where u contains the upper triangular elements of 2.

The test statistic is constructed in the usual fashion, as

LM= d’I”d, 12)

where d is the score vector and I1t is the partitioned inverse of the information matrix that

corresponds to the coefficients in X, both evaluated under the null hypothesis.

Based on the log-likelihood lo), the score for each h, is

a.c/ax,= -tr Y I-x,w,)-’ +E’B’ E-1 caI,,,) z?~ rq)]c, 13)

where E” is a T by T matrix of zeros, except for a one in position t, t.

Under the null hypothesis, h, = 0 and thus B, = I. Also, since y has zero diagonal elements by

convention, tr K = 0, and thus 13) becomes

aL/ax,=c[ P. Et’) @ w+, 14)

which can be expressed succintly, for all T A,, as in row form):

l’ P*u’ur), 15)

where L is a T by 1 vector of ones, U is an N by T matrix with the error vector for each equation

corresponding to the columns, U,_ is a similar matrix of spatially lagged errors w . ct), and * stands

for the Hadamard product.



338 L. Anselin / Spatial autocorrelation in seemingly unrelated regressions

In contrast to the situation with serial error autocorrelation in the time domain [e.g., Magnus

1978, p. 311)], the information matrix in the spatial SUR model is not block diagonal between the

parameters X and u. This is due to the multidirectional nature of the dependene in space and the

structure of the Jacobian.

The relevant elements of the information matrix can be found in the usual way from the second

partial derivatives of 10) as:

ICAt uhk) = tr E”Z-‘Ehk) . tr D,, 16)

where E” is as before, and E h k is a T by T matrix of zeros, except for elements h, k and k , h,

which equal one. The matrices E” and E h k are used to select the relevant elements from the inverse

Y’. The matrix D , is introduced for notational simplicity, and equals

D,= W,(-A,W,)-‘. 17)

In general, the expression for this information submatrix will be non-zero. However, under the null

hypothesis of h, = 0, D , becomes equal to W,. Therefore the trace of D , becomes zero, and so does

16). As a result, the elements for the partitioned information matrix needed in 12) can be found

from the corresponding elements for X,, h,< only. These are

Z h,, X,)=tr D,)2+of’-a,,-trD,‘D,, 18)

I A,, A ,) = u r s. u ,,~. r D, ‘D, r , 19)

where CT” and a,, are the t, s elements of 2-l and 2, respectively. Under the null, these expressions

become

Io( A,, A, 1 = tr Wt 2 + ~7”. a ,, tr W’W), 20)

1, X,, X,) =u”.u,~. tr W’W,). 21)

Or, in matrix form, for all T parameters,

&(A, A) = T,, + T,:(Ii-‘*x), (22)

where T,, is a diagonal T by T matrix with as elements tr W,*, and T,, is a symmetric T by T matrix

with as elements tr Y’W,.

After substituting 15) and 22) in 12) the full expression for the LM test statistic for spatial error

autocorrelation in the spatial SUR model becomes

LM = I --I*U’L )[~, + T,,;(E-‘*z)] -‘(z-‘*u’u, , 23)

which is distributed asymptotically as x2 with T degrees of freedom. The special cases where the A,,

the W, or both are the same in each equation can be found as a straightforward extension.



L . Anselin / Spatial autocorr elation in seemingly unr elaied regressions 339

4. Concluding remarks

The test presented here has well-known asymptotic properties, in contrast to the various ad hoc

procedures that are sometimes suggested. Also, since the LM approach is based on estimation under

the null hypothesis only, no special non-linear optimization is needed. The statistic can therefore be

constructed fairly easily from the output of a SUR estimation in a regression package, by means of

standard matrix operations that are increasingly available in commercial econometric software.

Appendix: Extended derivations

Al. Special matrix manipulations used in the score eq. 13)

For the SUR model with spatially dependent errors, the matrix ti can be expressed as

which is a special case of the model considered in Magnus 1978). More specifically, the special

structure considered by Magnus 1978) is of the form C5’ Q E @ A)Q’. The estimation equations for

the parameters p, 2 and h can be found by applying the conditions given in Magnus 1978) [see also

Anselin 1988b)].

Since B is a block diagonal matrix with elements Z - h w), its partial derivative with respect to a

particular h, consists of - W, in the t, t block on the diagonal, and zeros elsewhere, or

aB,/px, = - w, for h = t,

= 0, for h f t

which yields.

as/ax, = -E” Q W,,

with E” defined as in the paper.

Also, as a direct application of matrix calculus,

AZ. The elements of the information matrix eqs. 18-19)

As pointed out above, the information matrix for the ML estimator can be derived as a special

case of the results in Magnus 1978) for a parameterized non-spherical error variance. There, the

information matrix for the general model with a O), where B is a vector of parameters, is of the form

\k, = i/2) tr[ auf’/ae,)n afi-‘/ae,)s2]

for all combinations of 8, and 8, in 0.



340 L. Anselin / Spatial uutocorrelation in seemingly unrelated rrgressrons

In the spatial SUR model, the information matrix is block diagonal between the elements of fi

and those of [A, a]. The result for p is the usual X’Q2’X. The important elements of the

information matrix for the parameters in s2 A and a), say q, are

* ( A, A)=tr K, K, + tr K, ' ( Z‘ - ' c 3Z) K, ( 2‘ @Z) ,

* ( A, u) = tr K, ' ( 2 p1Eh @ ) ,

* a. u) = 1/2)N- tr XP’E”) ZP’Eh’),

with i, j, h, k , s the relevant elements of the coefficient vectors, E defined as before, and K as an

auxiliary matrix for each element A, of A. In the spatial SUR model, K, becomes

K, =( M/ X+B- '

= - E @ W ( Z- X, W ) - - '

The expressions needed to derive \k X, A) and Ik h, a) can be obtained by using the simplifying

notation D, W ( X, W ) - ' nd the following intermediate results:

= (Elf. E ) ~3 D, Q) ,

or,

K, KS=O, for tfs,

=E @ D, ) ' , for t=s,

tr K, K, = [ r Er r ] t r ( D, ) ' ] .

= tr D, ) 2,

K, ( X- ' Ehh @ I ) = ( E @ D, ' ) ( z - ' E' %Z) ,

tr K, ' ( F' Eh @ ) = [tr Er r Z‘ p ' Ehl ‘ ) ]tr D,].



L. Anselin / Spatial autocorrelation in seeming~v unrelated regressions 341

eferences

Ancot, J.-P., J.H.P. Paelinck and J. Prim, 1986, Some new estimators in spatial econometrics, Economics Letters 21, 245-249.

Anselin, L., 1988a, Lagrange multiplier test diagnostics for spatial dependence and spatial heterogeneity, Geographical

Analysis 20, 1-17.Anselin, L., 1988b, Spatial econometrics: Methods and models NiJhoff, Dordrecht).

Breusch. T. and A. Pagan, 1980, The Lagrange multiplier test and its applications to model specification in econometrics.

Review of Economic Studies 67, 239-253.

Cliff, A. and J.K. Ord. 1981, Spatial processes, models and applications Pion, London).

Davidson, R. and J. MacKinnon, 1984, Model specification tests based on artificial linear regression, International Economic

Review 25, 485-502.

King, M., 1987, Testing for autocorrelation in linear regression models: A survey, in: M. King and D. Giles, eds., Specification

analysis in the linear model Routledge and Kegan Paul. London) 19-73.

Magnus, J., 1978, Maximum likelihood estimation of the GLS model with unknown parameters in the disturbance covariance

matrix, Journal of Econometrics 7, 281-312.

Paelinck, J.H.P. and L. Klaassen, 1979, Spatial econometrics Saxon House. Farnborough).

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Economics Letters Volume 28 Issue 4 1988 [Doi 10.1016_0165-1765(88)90009-2] Luc Anselin -- A Test for Spatial Autocorrelation in Seemingly Unrelated Regressions