Heteroskedasticity and different Spatial Weigth Matrices
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 [[alternative HTML version deleted]]
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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