Economics Letters 28 (1988) 335-341 North-Holland 335 A TEST FOR SPATIAL AUTOCORRELATI ON IN SEEMI NGLY 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 t hat 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 encou ntered 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 autocorr elation, and may lead to misleading inference. In the context of a single equation specification, spatial autocorr elation 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 spac e 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 prop erties and therefore cannot be used in a formal model specificati on. 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)
7
Embed
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
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
8/12/2019 Economics Letters Volume 28 Issue 4 1988 [Doi 10.1016_0165-1765(88)90009-2] Luc Anselin -- A Test for Spatial A…
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)
8/12/2019 Economics Letters Volume 28 Issue 4 1988 [Doi 10.1016_0165-1765(88)90009-2] Luc Anselin -- A Test for Spatial A…