In 2018 I was invited to give a talk at SOCHER in Chile, to give my opinions about using spatial methods for policy analysis. I like the idea of putting in spatial lags of policy variables to measure spillovers. Use fixed effects with panel data, compute fully robust ses. 
For the life of me, I couldn't figure out how putting in spatial lags of Y had any value. After preparing a course in July 2020, I was even more negative about this practice. It seems an unnecessary complication developed by theorists.
As far as I can tell, when spatial lags in Y are used, one always computes the effects of own policy changes and neighbor policy changes, anyway, by solving out. This is done much more robustly and much more easily modeling spillovers directly without spatial lags in Y.
Writing y(i) = a + b*w(i)Y + x(i)*c + u(i) makes no sense. It has no ceteris paribus interpretation because there's no interesting counterfactual. It's not at all like time series; that connection is superficial. Spatial lags in policy variables has a potential outcomes interp.
I made these points in my SOCHER talk and I don't think they went over well. But they were very nice to me and they didn't leave me in the Atacama Desert.
Imagine a model like
y(i) = a + b*w(i)Y + x(i)*c + w(i)X*d + u
What's the thought experiment for d? We hold y(j) fixed to see the effect that changing x(j) on y(i)? But x(j) directly affects y(j), so how can we hold y(j) fixed?
y(i) = a + b*w(i)Y + x(i)*c + w(i)X*d + u
What's the thought experiment for d? We hold y(j) fixed to see the effect that changing x(j) on y(i)? But x(j) directly affects y(j), so how can we hold y(j) fixed?
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