Conceptually the spatial lag models and spatial error models are substantially different, yet the results from their use on a given dataset may be similar. Furthermore, both are related to a process known as spatial differencing which is a form of spatial filtering. This is similar to simple differencing methods applied in time series analysis, but instead of analyzing values at each time slot, yt, one examines the differences between sequential time slots (i.e. first differences), yt‑yt-1. In the spatial context the expression:

can be written with a first-order spatially autocorrelated difference and spatial autocorrelation coefficient ρ as:

Thus a model with a spatially autoregressive error term is a form of standard regression equation on spatially filtered data. The term:

is the spatial filter in this instance. Similarly the mixed spatial lag model, mrsa, described earlier:

can be written with a first-order spatially autocorrelated difference as:

Hence, once again the model can be seen as a form of spatial filter where in this case only the dependent variable is filtered.

A somewhat different approach has been pioneered by Getis and Griffith (2002, [GET1]). They apply spatial filtering to potential independent variables in a regression equation using one of two procedures: (i) using their Gi statistic to transform the source data (Getis method); or (ii) decomposition of the Moran I statistic into orthogonal and uncorrelated map pattern components (Griffith). The approach of Getis is the most straightforward to describe and apply, since it involves a simple transformation of the original data values. Let Gi(d) be the local Getis and Ord spatial autocorrelation statistic applying to observation (point) i:

where d is the estimated range of observed spatial autocorrelation. This expression is the observed value of the statistic for location i. The expected value of the statistic for location i is of the form Wi/(n‑1). The filtering transformation of the observed data values xi applied is then:

where Wi is the sum of the binary weights, wij, within a distance d of i, and n is the number of observations. The original dataset can then be viewed as being comprised of a spatial component, xsp={xi‑xi*}, and a non-spatial (filtered) component, xf={xi*}. These two components are then used in standard linear OLS as dependent variables and analysis proceeds as described earlier. Hence for each filtered dependent variable there will be two additional terms in the regression equation.

Getis and Griffith (2002, [GET1]) applied unfiltered OLS, standard spatial autocorrelation, Getis filtered OLS and Griffith filtered OLS to sample US state expenditure data (observation taken as applying to state centroids). For OLS, standard spatial autocorrelation and the Getis filtering the coefficient of determination was found to be 75%, whilst with the Griffith method a slightly higher value of 80% was obtained. The authors conclude that both filtering methods provide a more appropriate specification of the regression model, facilitating meaningful interpretation of all of the regression coefficients, with the advantage of not requiring formal specification of an appropriate spatial autocorrelation model. Overall the Griffith approach appears somewhat more powerful, but the Getis approach is simpler to apply, although it does require that the filtered variables are positive and have a natural origin (i.e. are positive ratio-scale variables).

References

[GET1] Getis A D, Griffith D A (2002) Comparative spatial filtering in regression analysis. Geog. Anal., 34, 130-40