I am looking for a few good books pitched at different levels to help analyse discrete data.
Specifically: Specification, Estimation and the application of econometric methods to model discrete choices by individuals, households or firms.
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Personally, for choice analysis, I like
(1) is a fairly short, readable classic. (2) is much more introductory, with lots of intuition, though it devotes a lot of space to the nlogit software. (3) has the same features as (2), but for a more focused set of topics. (4) is also pretty introductory and focuses on using Stata. It is perhaps the least "economisty" of the four, but still quite good.
The books you have listed are good, particularly Wooldridge, but they cover much more ground than just discrete data at the expense of depth.
I would suggest to look also into Alan Agresti's "Categorical Data Analysis".
The book is not focused on econometric applications, and this is exactly why I am suggesting it. It will give you the more general point of view on the matter, which sometimes brings fresh ideas in.
My experience is with the 2nd edition. The book does not target other scholars but students and practitioners. From the extras, I particularly like ch. 16 "Historical Tour of Categorical Data Analysis", since it always benefits my understanding of the material to have a sense of how a scientific field evolved.
Very extensive. Worth having as a reference at least because it covers an enourmous set of models and estimation techniques. It is a little bit light on the intuition though. Also, a kindle edition is available which is a lot cheaper than the very expensive hard cover
I haven't looked at this one so can't speak from experience. It seems popular.
In my experience, this one is a bit easier to read and does a better job of attempting to explain itself. It isn't as extensive as Greene. I would go to this book first because I find it more accessible. If I don't get the answer I need, it will probably be in Greene.