'The Role of Money in New-Keynesian Models'

Should New Keynesian models include a specific role for money (over and above specifying the interest rate as the policy variable)? This is a highly wonkish, but mostly accessible explanation from Bennett McCallum:

The Role of Money in New-Keynesian Models, by Bennett T. McCallum, Carnegie Mellon University, National Bureau of Economic Research, N° 2012-019 Serie de Documentos de Trabajo Working Paper series Octubre 2012

Here's the bottom line:

...we drew several conclusions supportive of the idea that a central bank that ignores money and banking will seriously misjudge the proper interest rate policy action to stabilize inflation in response to a productivity shock in the production function for output. Unfortunately, some readers discovered an error; we made a mistake in linearization that, when corrected, greatly diminished the magnitude of some of the effects of including the banking sector. There seems now to be some interest in developing improved models of this type. Marvin Goodfriend (MG) is working with a PhD student in this topic. At this point I have not been able to give a convincing argument that one needs to include M. ...

There is one respect in which it is nevertheless the case that a rule for the monetary base is superior to a rule for the interbank interest rate. In this context we are clearly discussing the choice of a controllable instrument variable—not one of the "target rules" favored by Svensson and Woodford, which are more correctly called "targets." Suppose that the central bank desires for its rule to be verifiable by the public. Then it will arguably need to be a non-activist rule, one that normally keeps the instrument setting unchanged over long spans of time. In that case we know that in the context of a standard NK model, an interest rate instrument will not be viable. That is, the rule will not satisfy the Taylor Principle, which is necessary for "determinacy." The latter condition is not, I argue, what is crucial for well-designed monetary policy, but LS learnability is, and it is not present when the TP is not satisfied. This is well known from, e.g., Evans and Honkapohja (2001), Bullard and Mitra (2002), McCallum (2003, 2009). ...