1. As in the time series case, this is due to the complex stochastic nature of the inverse term. This will typically contain elements which are a function of the y’s (and therefore the e) at every observation point. As a result, this term will not be uncorrelated with E. Moreover, while in time series E[yL’e]=0 if there is no serial residual correlation, this is not the case in space. Indeed, in the spatial model: E[yL’e]=E{[W.(I—pW)—le]’e) which is only zero for p=0.
2. This in contrast to some of the early suggestions of Hordijk (1974). A more rigorous demonstration of this point is given in Anselin (1981).
3. As pointed out in the introduction to this chapter, these approaches are discussed in detail in Cliff and Ord (1981), Ripley (1981), and Upton and Fingleton (1985). See also Anselin (1980) for a detailed description of issues involved in the derivation of maximum likelihood estimators for spatial autoregressive models.
4. Previous formal treatments of this issue can be found in, e.g., Silvey (1961), Bar—Shalom (1971), Bhat (1974), and Crowder (1976), based on a conditional probability framework. However, it is only in the more recent articles that the simultaneous case is considered. This situation is most relevant to econometric models with lagged dependent variables and general error variance structures. See also the various laws of large numbers and central limit theorems discussed in Chapter 5.
5. A more rigorous formulation, for the situation where a normal distribution is assumed, is given in Heijmans and Magnus (1986c). The conditions presented there can be taken as the formal structure within which ML estimation can be carried out for the spatial autoregressive models considered here.