y= Xb + e

Minimize e’e, e = (y-Xb)

b = (X’X)

^{-1}X’y

Generalized Least Squares: A Review

y = Xb + u

u = a vector of random errors with mean 0 and var-cov matrix C

b = (X’ C

^{-1}X)

^{-1}X’C

^{-1}y

Geographically Weighted Regression

y = X b(t) + e

b(t) = (X’ W X)

^{-1}X’Wy or Dy

W = spatial weight matrix , which may reflect contiguity between observations, or distance between observations

Spatial Lag Model

y = p Wy + e p = spatial autoregression parameter estimated from the data

Spatial lag Model (Mixed)

y = Xb + p Wy + e

Spatial Error Model

y = Xb + e where e = λ W e + u and u= y – Xb

y = Xb + λ W (y – Xb) + u

= Xb +λ Wy - λ WXb + u

Moran's I: test for spatial autocorrelation

I =(N S)(e'We/e'e)

e = OLS residuals and S = ΣΣ w

References:

Anselin, Luc. ‘Spatial Econometrics.’ Bruton Center. School of Social Sciences. University of Texas. Richardson, TX 75083-0688. http://www.csiss.org/learning_resources/content/papers/baltchap.pdf

Geographically Weighted Regression: The Analysis of Spatially Varying Relationships. A. Stewart Fotheringham et al. Wiley 2002.

George Casella and Edward I. George ‘Explaining the Gibbs Sampler’ The American Statistician, Vol. 46, No. 3, (Aug., 1992), pp. 167-174 American Statistical Association

Geospatial Analysis: A Comprehensive Guide. 2nd edition © 2006-2008 de Smith, Goodchild, Longley

http://www.spatialanalysisonline.com/output/

Rao, C. R. (1973). Linear Statistical Inference and its Applications (2nd Ed.). Wiley, New York.

Tiefelsdorf, Michael. ‘Modeling Spatial Processes: The Identification and Analysis of Spatial Relationships in Regression Residuals by Means of Moran’s I.’ Springer –Verlag Berlin Heidelber 2000. Germany

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