Using GeoDa software, I have run a spatial error model and obtained coefficients for each of my independent variables, in addition to constant and lambda variables. I am now wondering how to write my spatial error equation. Any help would be appreciated.
I'm not exactly sure what you are requesting, so have patience with me. Spatial autocorrelation is the result of having a model that is incomplete. So if you work on the specification of your model, then the spatial autocorrelation might go away. In its place, you will have new variables and coefficients on those variables. Suppose, for example, that you ran a hedonic model for property sales data throughout a metropolitan model. Furthermore, suppose that there is no variable for distance to the urban center. Spatial autocorrelation would ordinarily be huge. This is like finding autocorrelation in time series data if you ignore seasonality.
Now consider other important spatial variables, proximity to amenities and disamenities. By including these variables, spatial autocorrelation should diminish. If you want to discuss how to structure the inclusion of these variables, that might be best done in another conversation. On the other hand, I may have misunderstood your question and so you might want to disregard this answer altogether.
Peter's answer is one way to look at it. In this approach, the 'neighbouring effect' is modeled through the proximity to some spatial 'benchmarks' (e.g. distance to city center) , assuming that all neighbour will have similar effect (because they are all located near the regress point.
However, there is another approach to account for the 'neighbouring effect, by using the "Geographical Weighted Regression" technique first derived by Stewart Fotheringham (read: "Geographically weighted regression : the analysis of spatially varying relationships" Fotheringham et.al 2002).
This is very gooad approach since basically it is an ordinary regression model but done simultaneously on each regressed points. So if you have 100 data locations, you will have 100 sets of coefficients (one for each). Each coefficients (for each regressed point) are estimated from the neighbouring data.
I hope it helps.
Ari
Thanks alot Peter and Ari for your great inputs.
@ Peter: I see your point. However, I am working on ecological dataset and spatial autocorrelation kinda should exist. Should it help, I am examining micro-climatic thermal regulation and theoretically, the temperature in each sample is affected by its neighboring's, and further confirmed by the significant spatial autocorrelation. Anyhow, as this is a violation of conventional statistical tests, I should use spatial statistics to kinda incorporate the space factor, spatial autocorrelation, into the modelling. In other words, I am looking at the autocorrelation as a source of meaningful data, even a variable for that matter, rather than a disturbing thing.
@ Ari: Thanks for the nice suggestion. I also performed GWR in ArcMap and gave me interesting insights into the relationships as well as some hints for further research. GWR is a brilliant tool. However, I also need to compare my variables in terms of their significance, and GWR does not provide such information. Specifically, I need to know that for example, how one unit change in one of my variables, affects temperature like I would do it if I used conventional methods like ordinary least squsres (OLS). I used the spatial error model for regression to incorporate autocorrelation as one additional variable to model the temperature and got the coefficients for each variable, however, it is not clear to me as to how to use these coefficients in an equation. It does not seem to me as straightforward as OLS. Any thoughts on this?
I do not want to criticize locally weighted regression, however I would ask if you really want 100 regression equations? Instead, you might be interested in a non-parametric approach. One regression equation is all you will need. See:
https://www.researchgate.net/publication/46552003_Estimating_a_Price_Surface_for_Vacant_Land_in_an_Urban_Area
https://www.researchgate.net/publication/5151500_A_Primer_on_Piecewise_Parabolic_Multiple_Regression_Analysis_via_Estimations_of_Chicago_CBD_Land_Prices?ev=prf_pub
Article Estimating a Price Surface for Vacant Land in an Urban Area
Article A Primer on Piecewise Parabolic Multiple Regression Analysis...
Hi Mahyar,
Have you tried using the actual GWR4 software?
free downlload from this:
http://gwr.maynoothuniversity.ie/other-gwr-software/
This software is developed by Fotheringham's own team. In ArcGIS the GWR output is mainly in graphical presentation (which is of course very helpful) but missing out on the actual model estimates. With the actual GWR estimates, I think (if i understand your problem correctly) you will not have any problem in doing those analysis, because basically it is basically just a regression (it just that it is done throughout the space continuum).
You might want to try it out.
I hope it helps
Ari
Hi Peter,
Thanks for the papers. Read the first one. Great work. Yup! You are totally right. I'd rather go for one single equation to model the dataset as a whole. However, I also do run GWR as a first step as the localized graphical representation of the relationships usually gives me some hints on the missing variables, etc. I was just reviewing a paper on the interpretation of coefficients derived using different types of spatial regression methods including the spatial error I had a couple of challenge with. I found it interesting and useful. I am attaching it to this reply.
http://link.springer.com/article/10.1007/s40980-015-0016-y
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