Spatial weight matrix and spillover effects in Spatial Durbin Model

[Lom Newton]

Dear All,

I have a little confusion about the use of spatial weight matrix that I want to get clarity on. I created two inverse distance weight matrices ( 0.5 mile and 1mile radius) using spmat as follows:

The 0.5 mile radius matrix is:

spmat idistance M05 lon lat , id(ID_) dfunction(dhaversine, miles) norm(row) vtruncate(1/0.5)

The 1 mile radius matrix is:

spmat idistance M1 lon lat , id(ID_) dfunction(dhaversine, miles) norm(row) vtruncate(1)

Let say I use M05 to run a spatial regression, specifically, Spatial Durbin Model (SDM) as:

xtset ID year
   xsmle y x1 x2 x3 x4,  wmat(M05) dmatrix(M05) ///
  model(sdm) durbin(x1 x2 x3 x4) re vce(cluster ID) effect nolog

And let say I use M1 to run a spatial regression, specifically, Spatial Durbin Model (SDM) as:

xtset ID year
   xsmle y x1 x2 x3 x4,  wmat(M1) dmatrix(M1) ///
  model(sdm) durbin(x1 x2 x3 x4) re vce(cluster ID) effect nolog

My question is, would it be correct to have the indirect (spillover) marginal effects of any of the covariates in the M05 model to be less than its counterpart from the M1 model? The basis of my question is that I thought the M05 model which defines a closer relation should produce bigger estimates than the M1 model. At least that's what Tobler's first law of geography implies: “everything is related to everything else, but near things are more related than distant things” (Tobler 1970).

Any help to remove my confusion is highly appreciated.

[Dario Maimone Ansaldo Patti]

I think it might be. Suppose you have unit i. When you use M05 you assume that only units that are in the 0.5 mile radius may have an effect on i. But when you use M1, you assume that more areas unfold their effects on unit i. Hence the indirect effect could be larger, provided that it is correct to consider a larger radius. Maybe if you increase the radius more and more you might observe that the indirect effect becomes smaller or even insignificant. Think of this: is it reasonable to believe that what happens 100M far from unit i may have any effect on the latter? Probably the answer is not.

Hence using the M05 matrix could be too restrictive, since you exclude any other area that in fact may have effects on unit i.

When you use M05 you make the assumption that anything outside that radius have a zero effect. Within the radius you assume that units that are closer to i, say at 0.1M weight more that those at 0.3M. Hence Tobler first law is satisfied.

[Lom Newton]

Thank you Dario Maimone Ansaldo Patti for your response. It makes sense to me. So far, my quarter mile radius has the least of the indirect effects. I will run 5 miles and 10 miles(and may be more) to see if the effects become smaller and perhaps insignificant.

[Dario Maimone Ansaldo Patti]

Lom Newton defining the appropriate spatial matrix is not trivial, as you can observe by yourself. If you want to take into consideration what happens if you increase the radius, you might even try to use an inverse distance matrix without cut-off. This is the largest radius you may consider as it indicates that everything is affecting everything else (even though the closer a unit the larger its impact).

[Lom Newton]

Exactly, Dario Maimone Ansaldo Patti. Even before I put up the question here I started running my model without the truncation of the inverse distance matrix. It is still running and I guess it will take a couple of days more given how long 2 miles radius model took to run. And again, my initial confusion was also to the effect that since the SDM considers not just local spillover effects, but also global spillover effects, it could be that the larger radius models would have larger estimates after all and may be I do not need to run the model with any cut-offs.

[Lom Newton]

Just an update: My models finally run and the spillovers start to decay after 10 miles radius. Thanks.