Professor Kerr: Everything you said is exactly right. Except, technically that was the methodology of my QJE paper with Milyo, not necessarily the methodology of my book.
My book uses three (and if you count Ch. 14, four) methodologies. Only one of those is the QJE-article methodology.
Here is a copy of the QJE paper. In case anyone has a copy of my book, I describe that methodology in Ch. 13.
For readers who want a less technical description of the methodology of the QJE article, here is a critique by Geoffrey Nunberg. And here is a response by Milyo and me.