Estimating Multipliers: Sub-National Cross-Sectional Studies Are a Perfectly Good Way to Produce a Lower Bound to the Constant Financial Conditions Multiplier
I think sub-national cross-sectional studies are a perfectly good way of producing a lower bound to the constant financial conditions multiplier. It is a lower bound because it does not capture demand spillover effects--and at the sub-national level those are bound to be large. It is a constant financial conditions multiplier because all regions of a country share the same financial conditions--thus it estimates what the multiplier would be if the central bank's reaction function was to keep financial conditions constant as the government spending and debt issuance too hold.
You may not want a lower bound to the constant financial conditions multiplier. But I think it is a perfectly fine thing to want to know. And sub-national cross-sectional studies produce perfectly fine estimates of it.
Facts & other stubborn things: Krugman is wrong to say that sub-national cross-sectional studies are good for estimating multipliers…. (1) Demand spillover means that any impact on demand in one unit is going to raise demand in other units. It's not just interstate commerce either, particularly since Krugman is looking at county-level data here…. (2) One of the most important impacts of fiscal stimulus is what it does in the loanable funds market. This market is national (international, actually), so it's going to be a wash in county-level comparisons….
Nakamura and Steinsson ask… "what is the impact of fiscal policy for geographic sub-units within a monetary union" (presumably with Europe in mind). If that's what you're interested in, then county-level comparisons are a great idea…. [T]his is a very appropriate use of county-level comparisons. That does not mean that Krugman is right to praise the method in general. It's certainly no good for measuring the effects of ARRA - as some have tried to do….
This [other] county-level analysis by Sufi and Mian… can verify the importance of balance sheets as one link in the chain…. Unless Krugman claims "tight money is definitely not the problem here", he doesn't seem to have said anything out of line with respect to the Sufi and Mian study. Krugman is quite sympathetic to the idea that tight money causes recessions, I think.
What readers need to be careful of, though, is assuming that geographical sub-unit analysis is a panacea for multiplier estimation - it's not.