Aymptotics in spatial stochastic processes
As in time series analysis, the properties of estimators and tests for spatial series are derived from the asymptotics for stochastic processes. However, these properties are not simply extensions to two dimensions of the time series results. A number of complicating factors are present and to date some formal results for the spatial dependence case are still lacking. While an extensive treatment of this topic is beyond the scope of the current chapter, three general comments are in order. First, the intuition behind the asymptotics is fairly straightforward in that regularity conditions are needed to limit the extent of spatial dependence (memory) and heterogeneity of the spatial series in order to obtain the proper (uniform) laws of large numbers and central limit theorems to establish consistency and asymptotic normality. In this context, it is important to keep in mind that both SAR and SMA processes yield heteroskedastic variances, so that the application of results for dependent stationary series are not applicable.13 In addition to the usual moment conditions that are similar in spirit to those for heterogeneous dependent processes in time (e. g. Potscher and Prucha, 1997), specific spatial conditions will translate into constraints on the spatial weights and on the parameter space for the spatial coefficients (for some specific examples, see, e. g. Anselin and Kelejian, 1997; Kelejian and Prucha, 1999b; Pinkse and Slade, 1998; Pinkse, 2000). In practice, these conditions are likely to be satisfied by most spatial weights that are based on simple contiguity, but this is not necessarily the case for general weights, such as those based on economic distance.
A second distinguishing characteristic of asymptotics in space is that the limit may be approached in two different ways, referred to as increasing domain asymptotics and infill asymptotics.14 The former consists of a sampling structure where new "observations" are added at the edges (boundary points), similar to the underlying asymptotics in time series analysis. Infill asymptotics are appropriate when the spatial domain is bounded, and new observations are added in between existing ones, generating a increasingly denser surface. Many results for increasing domain asymptotics are not directly applicable to infill asymptotics (Lahiri, 1996). In most applications of spatial econometrics, the implied structure is that of an increasing domain.
Finally, for spatial processes that contain spatial weights, the asymptotics require the use of CLT and LLN for triangular arrays (Davidson, 1994, chs. 19, 24). This is caused by the fact that for the boundary elements the "sample" weights matrix changes as new data points are added (i. e. the new data points change the connectedness structure for existing data points).15 Again, this is an additional degree of complexity, which is not found in time series models.