Spatial Regression Models
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Spatial Regression Models S UMEETA S RINIVASAN School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Synonyms Spatial autoregressive models; Dependence, spatial; Simultaneous autoregressive model (SAR); Moving average (MA) process model; Spatial lag model; Spatial error model; Geographic weighted regression (GWR) Definition Spatial dependence is measured by spatial autocorrelation, which is a property of data that arises whenever there is a spatial pattern in the values located on a map, as opposed to a random pattern that indicates no spatial autocorrelation. To measure the spatial pattern (spatial association and spatial autocorrelation), some standard global and local spatial statistics have been developed. These include Moran’s I, Geary’s C, Getis, LISA and GLISA statistics. Besides spatial dependence in the data, there can be spatial heterogeneity. This means that the underlying process being studied may vary systematically over space. This creates problems for regression and other econometric methods that do not accommodate spatial variation in the relationships being modeled. For an ordinary least squares (OLS) estimation of the regression model the assumption is that the values of the coefficients of the independent (explanatory) variables are constant across the spatial extent of the study. If there is spatial variation in the independent variables across space then it must be confined to the error term. Also, OLS estimates are found by minimizing the sum of squared prediction errors with an assumption that error has mean zero, the error terms are uncorrelated, have a constant variance (homoskedastic) and that they are normally distributed. If there is spatial dependence amongst the data it violates these assumptions about the error term. Therefore, the standard properties of OLS do not hold and more fundamentally, a model assuming that observations are independent is incorrect. Regression models that do account for this spatial autocorrelation are called spatial regression models.
Spatial Regression Models
Historical Background The theoretical work of Durbin and Watson in the 1950s on test statistics for serial autocorrelation and Imhof’s approach in the 1960s to calculate the exact distribution of centrall
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