Abstract
The analysis of the effects of spatial dependence in the error terms of the linear regression model was the first specifically spatial econometric issue to be addressed in the regional science literature. Initial problem descriptions and the suggestion of some solutions were formulated in the early 1970’s, e.g., by Fisher (1971), Berry (1971), Cliff and Ord (1972), McCamley (1973), Hordijk (1974), Martin (1974), Bodson and Peeters (1975), and Hordijk and Paelinck (1976). This was followed by many further assessments of the properties of various estimators and test statistics, which continue to be formulated to date. Spatial error autocorrelation is also the only spatial aspect of inference that has been recognized in the standard econometric literature, though only very recently and to a limited extent, e.g., in Johnston (1984) and King (1981, 1987).
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Notes on Chapter 8
This approach to spatial autocorrelation leads more directly to the use of spatial correlograms, in analogy to time series analysis. See also Granger (1989), for a similar viewpoint.
In Cliff and Ord (1972), a finite sample approach is presented as well, based on an analogy with the Durbin Watson bounds in time series analysis. However, in contrast to the time series context, the generality of the bounds is limited. In essence, each different spatial weight matrix would necessitate the computation of a new set of bounds. Therefore, in most applications, the asymptotic approach is taken.
For a detailed and a more rigorous derivation, the reader is referred to the texts by Cliff and Ord (1973, 1981) and Upton and Fingleton (1985). The essential difference between the approach for regression residuals and that for other variates is the existence of dependence in residuals. Indeed, it is well known that the OLS residuals are related to the error terms as e = Me, where M is an idempotent projection matrix I-X(X’X)-1X’. Consequently, the e will not be uncorrelated, even when the a are. An alternative approach is to ignore the special nature of residuals and to use a randomisation framework. The moments of the Moran statistic for that case are given in Cliff and Ord ( 1981, p. 21 ).
For a more detailed derivation, see Anselin (1980, pp. 130–2).
This is a special case of the general structure in Magnus (1978), and the invariance result presented in Breusch (1980).
For example, see the comments in Cliff and Ord (1981), and Upton and Fingleton (1985).
Overviews are given in Magnus (1978), and Raj and Ullah (1981). Estimation in these models is further considered in Section 9. 4.
For a detailed derivation, see Anselin (1988a).
Since the variance matrix is asymptotic, the estimate for Q2 in (8.19) can be obtained by dividing the sum of squared residuals by either N or N—K (for very large N, the difference between the two will become negligible). In finite samples, the latter is less likely to lead to overly optimistic (small) estimates of variance.
Specifically, this is assumption A.3. (Andrews 1988, p. 892). The other conditions relate to the estimator for 13 and are clearly satisfied by the ML approach.
See the detailed derivations in Ord (1975), Hepple (1976), Anselin (1980, 1981 ), Cliff and Ord (1981), and Upton and Fingleton (1985).
Davidson and MacKinnon (1985a) consider various forms of 11(u), but the one mentioned here shows superior power in finite samples.
For a detailed derivation, see Anselin (1988b).
This strict requirement may potentially be replaced by a parameterization in function of a distance metric, as in Granger (1969) and Cook and Pocock (1983). In either case, consistency will require a large number of observations in each distance—class.
See White (1980, 1984) for details. The application of this approach to other than the standard regression context is discussed more extensively in, e.g., Cragg (1983), Hsieh (1983), Nicholls and Pagan (1983), Chesher (1984), and Robinson (1987). However, most of those extensions are based on the assumption that OLS provides a consistent estimate for the model parameters. As shown in Section 6.1, this is not the case in the specifications with spatially lagged dependent variables. Adjustments that result in better finite sample performance are discussed in MacKinnon and White (1985), and Chesher and Jewitt (1987).
For a rigorous discussion, see White (1984), and White and Domowitz (1984). An extension and suggestions for operational implementation are given in Newey and West (1987).
For details, see Anselin (1988b).
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© 1988 Springer Science+Business Media Dordrecht
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Anselin, L. (1988). Spatial Dependence in Regression Error Terms. In: Spatial Econometrics: Methods and Models. Studies in Operational Regional Science, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7799-1_8
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DOI: https://doi.org/10.1007/978-94-015-7799-1_8
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