Spatial Lag 

 In my last blog entry  (here) ,  I wrote that associations like space can mess up the assumptions  underlying standard estimation techniques.  This entry is about the first problem I mentioned, spatial lag:  when neighboring observations affect one another.  Such dependencies can lead to inconsistent and biased estimates in an OLS model.  And even if you don't care about "space" in a geographic sense, you might be interested in related topics like technology diffusion among farmers, network effects, countries that share the same membership in international organizations (an idea picked up in Beck, Gleditsch and Beardsley; see below) etc.  The point is that spatial lag is pervasive in many contexts and though it might be called different names, the basic problem remains the same. 

 Spatial lag models are similar to lagged dependent variable autoregression models in time series analysis but the problem is that the correlation coefficient cannot be easily estimated.  That's a problem because to estimate the coefficient, a spatial weights matrix is needed but it is often not clear what that matrix should look like, i.e., what the actual spatial relation is. 

 So how much can it matter? James LeSage (in an excellent guide to spatial econometrics and his MATLAB functions, also below) provides an example of OLS and spatial lag estimations of the determinants of house values. The idea is that -- apart from the influence of the independent variables like county population density or unemployment rates -- areas with high house values might be adjacent to other high value areas, and therefore there is a spatial trend in the outcome variable.  The example shows that an interesting variable like population density can become statistically insignificant when spatial dependence is taken into account, and that coefficients of other variables can change in magnitude.  In addition, taking spatial lag into account also improves the model fit. 

 So one should really take space into account if it matters. How would you know if it does? There are a number of tests to check for spatial lag, but for most part just starting to think about it helps. 

 For some more information of spatial lag, take a look at the sources mentioned: 

 -- James LeSage's Econometrics Toolbox (www.spatial-econometrics.com), which has an excellent workbook discussing spatial econometrics and examples for the MATLAB functions provided on the same site; and 
-- Beck, Gleditsch and Beardley (draft of April 14, 2005) "Space is more than Geography: Using Spatial Econometrics in the Study of Political Economy" (http://www.nyu.edu/gsas/dept/politics/faculty/beck/becketal.pdf).