[R-sig-Geo] weighted spatial autoregression

[Sam Field]

Roger,  

The reason seems 
> to be that W+G did the same as spautolm() (in SAS?) - find the spatial 
> autoregressive coefficient first (optimise in one dimension), then use GLS 
> to find the regression coefficients. 

Don't the weights play a role in the optimization to find lambda? Certainly the 
location of lambda is influenced by the weights employed by W+G. Wouldn't they 
also influence the lcoation of rho in the spatial lag model?

A while ago I wrote some SAS code to fit a spatial lag model and calculate the 
variance covariance matrix of the parameters, so I am a somewhat familiar with 
the two step procedure in that case. I have never really messed with the 
spatial error model. 

I am actually pretty happy being confined to a weighted spatial error model for 
the moment, since it would seem to me that spill over effects from the X%*%beta 
can always be accomodated by including W%*%X%*%tau terms in a spatial error 
model (though one must assume that the influence of spatially lagged covariates 
stop at first order neighbors?).  I wondered if including W%*%X%*%tau into a 
spatial error model would lead to inconsistent paramter estimates since rho*W%*%
X is also in the model.  Some quick simulation suggested that the consistency 
of the parameter estimates was not affected.

I am not sure that a true spatial lag process is theoretically compelling in my 
case anyway, now that I think about it. In fact, the idea the spatial 
correlation among the residuals is due to a "pure" contagion process (as 
represented by the pWy term) would seem pretty rare in the case of most complex 
phenomona - which is why the type='mixed" option in lagsarlm() is so useful!  

I wonder if 

Y = Xbeta + WXtau + pWe + u

isn't a sensible alternative to the spatial lag model when a contagion process 
is not theoretically plausible but where spill over effects of neighboring 
covariates are.  


thanks again for your amazing support of the spdep package.  

cheers,

Sam







 


   


Quoting Roger Bivand <Roger.Bivand at nhh.no>:

> Sam,
> 
> On Wed, 29 Aug 2007, Sam Field wrote:
> 
> > Roger,
> >
> > One possibility in this limited case might be to replicate the aggregate
> > level cases based on their respective weights (since they are integers,
> > i.e. within unit sample sizes), then run a spatial lag model.  This
> > would be equivalent to recreating the individual level data from the
> > aggregate data (excluding measures that vary within the aggregate
> > units).  This would obviously inflate your sample size and one would
> > have to correct for this somehow in the variance covariance matrix of
> > the parameters estimates.
> >
> > You would have to do the same for your nb object as well of course.  I
> > have looked into this by creating a list of neighbor ids from the
> > original nb object, but nb2listw() requires an nb object not a list so I
> > am stuck.
> >
> 
> You could fake it with nb2blocknb, but that was not written for this case, 
> but for the case when the individual level variables were observed, but 
> that there was no address or coordinates, just a postal code. Here the LHS 
> and RHS would be replicated, which doesn't seem desirable.
> 
> > The other problem would be that you would end up with a potentially
> > large data set. In my case, 13,000 - maybe more then spautolm() could
> > handle?  Maybe this whole idea if flawed.
> >
> >
> > Thanks again for your input! The results change quite a bit with the
> > weighted SAR models.
> 
> One interesting conclusion that I've reached is that while the spdep code 
> in spautolm() replicates Waller and Gotway for unweighted and weighted SAR 
> and CAR, S-Plus SpatialStats fails on the weighted CAR. The reason seems 
> to be that W+G did the same as spautolm() (in SAS?) - find the spatial 
> autoregressive coefficient first (optimise in one dimension), then use GLS 
> to find the regression coefficients. But S+ seems to try to optimise all 
> the coefficients at once, and gets bitten by the fact that
> (I - \rho W) %*% diag(wts) in their case is not symmetric (W has to be 
> symmetric, and the wts have to "balance" - see Cressie etc. Now I'm not 
> sure that S+ is right here. If not, then the lag model can be given 
> weights too, by simply passing them to the auxilliary regressions used to 
> set up the framework for optimisation. The analytical covariance matrix of 
> the coefficients remains a problem, though. We'd need to use some other 
> mechanism to get there for the eigen method, though the LR tests used for 
> sparse methods would be, I think, OK. I've also been playing with sampling 
> from a fitted model, to generate synthetic "standard errors", like 
> mcmcsamp() in lme4, but I don't know if it is sensible, or how well it 
> would scale to many observations.
> 
> So I am thinking about how lagsarlm() could get weights, but it won't 
> happen too fast, maybe.
> 
> Best wishes,
> 
> Roger
> 
> >
> >
> > Sam
> >
> >
> >
> > Roger Bivand wrote:
> >> On Tue, 21 Aug 2007, Sam Field wrote:
> >>
> >>
> >>> Thanks Roger!
> >>>
> >>> Sorry about omitting the subject line.  I have been working with
> errorsarlm() -
> >>> did not know about spautolm().  Do you know if there is something
> analogous
> >>> possible in the case of the spatial lag model,
> >>>
> >>> Y = pWY + XB + e ?
> >>>
> >>
> >> I have not looked at it, but because it is a wierd animal, I don't think
> >> it will be too easy to provide a theoretical foundation for it. The
> >> heteroskedasticity is in the error term, but the autoregressive part
> >> isn't. I don't think there are any examples anywhere, either.
> >>
> >> It ought to be possible, though.
> >>
> >> Roger
> >>
> >>
> >>> I was going to start looking into it.
> >>>
> >>> thanks!
> >>>
> >>>
> >>> Sam
> >>>
> >>>
> >>>
> >>>
> >>> Quoting Roger Bivand <Roger.Bivand at nhh.no>:
> >>>
> >>>
> >>>> On Tue, 21 Aug 2007, Sam Field wrote:
> >>>>
> >>>>
> >>>>> List,
> >>>>>
> >>>>> I am looking for ways of estimating spatial autoregression models that
> >>>>>
> >>>> adjust
> >>>>
> >>>>> for a known source of heteroskedaticity and the Waller and Gotway
> (2004)
> >>>>>
> >>>> text
> >>>>
> >>>>> outline how this can be done in the case of the SAR model.  If I work
> at
> >>>>>
> >>>> it, I
> >>>>
> >>>>> think I can implement this myself in R, but I wanted to see if anybody
> else
> >>>>>
> >>>> had
> >>>>
> >>>>> done it. It seems like a pretty straightforward generalization of the
> SAR
> >>>>>
> >>>> model
> >>>>
> >>>>> and would make a very helpful addition to the spatial regression tools
> in
> >>>>> spdep - especially given the effects of heteroskedaticity on the
> >>>>>
> >>>> consistency of
> >>>>
> >>>>> the SAR parameters!
> >>>>>
> >>>> ?spautolm
> >>>>
> >>>> The examples reproduce the results in Waller & Gotway, perhaps apart
> from
> >>>> a flattish function to optimise in the weighted CAR case. spautolm()
> now
> >>>> provides weighted or unweighted SAR, CAR, and SMA. Sparse matrix
> methods
> >>>> are available for SAR and CAR, SAR when spatial weights are symmetric
> or
> >>>> similar to symmetric (CAR weights have to be symmetric).
> >>>>
> >>>> Roger
> >>>>
> >>>>
> >>>>> Sam
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>> --
> >>>> 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
> >>>>
> >>>>
> >>>>
> >>>
> >>>
> >>
> >>
> >
> >
> >
> 
> -- 
> 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
> 
> 


-- 
********Note the new contact information*******

Samuel H. Field, Ph.D. 
Senior Research Investigator
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