By “breaks down” I should have said “gives strange results”. In particular, the code as written gives very strange posterior results, which are not similar to the results from the original book.
Ah, yes, there is an identification problem! It seems that the model in the book is the same as the one in An and Schorfheide (2007), and there has been discussion previously that this model is not identified. See for example, the thread Identification problem of Taylor rule coefficients! , and there is also a discussion about this by An and Schorfheide (https://cpb-us-w2.wpmucdn.com/web.sas.upenn.edu/dist/e/242/files/2017/04/er-rejoinder-final-15j9hdj.pdf), as well as Mutschler (2015) (https://www.sciencedirect.com/science/article/pii/S0165188915000731).
This is a little frustrating, as I would think a model used to introduced Bayesian methods in a textbook wouldn’t have this problem.
I think I will probably move on from this model for now.