[R-sig-Geo] Heteroskedasticity and different Spatial Weigth Matrices

[Roger Bivand]

On Fri, 24 Sep 2010, Angela Parenti wrote:

> Hey everyone,
>
> I'm puzzled about a recent result, and I'm wondering if anyone can help 
> explain it.
>
> I am currently estimating a growth model with many controls by OLS. 
> Looking at the residual tests I find that the Breusch-Pagan test points 
> the presence of heteroskedasticity. Moreover, looking for spatial 
> dependence in the residuals using the Moran's I test I find that with 3 
> definitions of spatial weight matrix I cannot reject the null hp of no 
> spatial dependence while with 3 different spatial weight matrices I can.
>
> Here the results in details:
>
> Breush-Pagan Test:
> BP=66.3478, p-value=0
>
> Moran'I test on regression residuals:
>
> 1) W1 is binary matrix with a cut-off=660.8 km (row-standardized):
>     Observed Moran's I=-0.02938, p-value=/0.57547/
>
> 2) W2 is the first-order contiguity matrix(row-standardized):
>     Observed Moran's I=0.1389, p-value=_0.00021_
>
> 3)  W3 is the second-order contiguity matrix(row-standardized):
>     Observed Moran's I=0.0694 , p-value=_0.00724_
>
> 4)  W4 is the matrix s.t each region has at least one neighbour- max distance 1124.710 km -(row-standardized):
>     Observed Moran's I=-0.01286, p-value=/0.91883/
>
> 5)  W5 is the matrix where weights are 1/d_ij^2  with no cut-off(row-standardized):
>     Observed Moran's I=0.03239, p-value=_0.00350_
>
> 6)  W6 is the matrix where weights are 1/exp(2*d_ij)  with no cut-off(row-standardized):
>     Observed Moran's I=0.06847, p-value=/0.15462/
>
>
> Therefore, my questions are:

> - since I find evidence of heteroskedasticity shouldn't I look for 
> aheteroskedasticity-robust version of the Moran's I? If yes, is there 
> the possibility to implement it with the function "lm.morantest"?

No, it would make no sense at all. So-called heteroskedasticity-robustness 
is used to manipulate coefficient standard errors, so has no effect on 
residuals. Believe the three tests showing that spatial autocorrelation is 
present. Also understand that BP and Moran's I may be detecting the same 
misspecification in your model.

> - what should I conclude from such different results using different 
> spatial weight matrices? It seems that the lower the number of 
> neighbours is the higher is the probability of finding spatial effects. 
> In this latter case how can I decide the "right" matrix? By looking at 
> the AIC in the maximum likelihood?
>

More neighbours will smooth more (like a larger bandwidth), so may include 
more varied residuals. In general the more parsimonious weights (fewer 
neighbours, but not fewer than sensible) will be preferable, but the 
scheme should have some motivation in your scientific field. Most models 
of (economic) growth are badly affected by the range of relative sizes of 
the observations - often leading to observed heteroskedasticity and 
residual spatial autocorrelation. In other fields than economics, it is 
usual that weighted regression is used, or more advanced methods to 
acknowledge the greater uncertainties associated with rates estimates (the 
dependent variable) for small observations.

Roger

> Thank you very much!
>
> Angela Parenti
>
>
>
>
>
>
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>
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-- 
Roger Bivand
Economic Geography Section, Department of Economics, Norwegian School of
Economics and Business Administration, Helleveien 30, N-5045 Bergen,
Norway. voice: +47 55 95 93 55; fax +47 55 95 95 43
e-mail: Roger.Bivand at nhh.no