Introduction

For this discussion, assume there arensample observations of the dependent variableyat unique locations. In spatial samples, often each observation is uniquely associated with a particular location or region, so that observations and regions are equivalent. Spatial dependence arises when an observation at one location, sayyiis dependent on “neighboring” observationsyj,yj∈ϒi. We use ϒito denote the set of observations that are “neighboring” to observationi, where some metric is used to define the set of observations that are spatially connected to observationi. For general definitions of the sets ϒi,i=1,…,n, typically at least one observation exhibits simultaneous dependence, so that an observationyj, also depends onyi. That is, the set ϒjcontains the observationyi, creating simultaneous dependence among observations. This situation constitutes a difference between time series analysis and spatial analysis. In time series, temporal dependence relations could be such that a “one-period-behind relation” exists, ruling out simultaneous dependence among observations. The time series one-observation-behind relation could arise if spatial observations were located along a line and the dependence of each observation were strictly on the observation located to the left. However, this is not in general true of spatial samples, requiring construction of estimation and inference methods that accommodate the more plausible case of simultaneous dependence among observations.