I have a question with regard to a possible bias as a result of how one constructs the spatial weights matrix. Consider a case where one works with areal data, for instance we're interested in whether the crime rate in one neighbourhood depends on the rate in other states. The spatial weights matrix is this case can be constructed using contiguity,e.g. queen contiguity based on 1st order neighbours, or using a distance cut off.

Working with areal data the polygons used as unit of analysis are rarely equally size. So it seems to me that if one uses the distance based spatial weights matrix the results might be biased due to this variability in size among the polygons.

I wondered whether anyone knows of any good references from the literature on this subject? I didn't have much success with my internet search so far. Thanks in advance.


1 Answer 1


Most of the results derived in spatial econometrics assume you know the spatial weights matrix beforehand (and it is fixed). There is likely some research that uses simulations to address bias in particular circumstances, but they are unlikely to be generalizable to your actual research design in which you don't know what form the weights matrix will take.

Fortunately, slight differences in the weights matrix tend to be highly correlated, and so results don't tend to change dramatically even with differing weights matrices. See The Biggest Myth in Spatial Econometrics by Lesage and Pace. Abstract below:

There is near universal agreement that estimates and inferences from spatial regression models are sensitive to particular specifications used for the spatial weight structure in these models. We find little theoretical basis for this commonly held belief, if estimates and inferences are based on the true partial derivatives for a well-specified spatial regression model. We conclude that this myth may have arisen from past applied work that incorrectly interpreted the model coefficients as if they were partial derivatives, or from use of mis-specified models.

  • I don't address the differing populations part of the question, as that is more about the model you estimate than it is about the spatial weights matrix specifically. Certain types of spatial models deal with differing populations pretty explicitly (e.g. Bayesian CAR models using the population as a denominator).
    – Andy W
    May 30, 2014 at 20:48

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