st: Clarification requested about the at() option of -margins-

[Trevor Zink <tzink@bren.ucsb.edu>]

Long-time lurker, first-time post. I couldn't find a good explanation in 
the archives.

I'm confused about what, specifically, -margins- is doing with the at() 
option, such that it can calculate margins for values of variable that 
don't exist in the data. To articulate with an example:

sysuse auto
summarize weight //maximum weight is 4840
logit foreign weight  //nonsensical, but ok for the example
margins, dydx(weight) at(weight=(0(1000)10000 100000))

Here I ask for the slope of the function at a variety of weights from 0 
to 10,000 and also 100,000. The maximum weight observed in the data is 4840.

My understanding of -margins- with at() was that it calculates the slope 
of the function holding the specified variables constant at the 
specified levels. But if the specified level doesn't appear in the data, 
how can Stata determine what the slope is at this value? Ok, it's 
clearly extrapolating, but based on what information? The only other 
information included in the above model is a constant. When I try the 
above but specifying the nocons option to -logit- Stata returns an 
error, so it must be forecasting based on the constant; but specifically 
how?

What's even more strange to me is that the standard errors *shrink* as 
the estimates extend beyond the observed data. If Stata is forecasting 
based on only the constant this seems counter-intuitive to me.

Thanks, and sorry if this is silly.

Trevor Zink

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