Tyler Cowen uses the term "New Old Keynesian" to describe "Paul Krugman, Brad DeLong, Justin Wolfers and others." I don't know if I am part of the "and others" or not, but in any case I resist a being assigned a particular label.
Why? Because I believe the model we use depends upon the questions we ask (this is a point emphasized by Peter Diamond at the recent Nobel Meetings in Lindau, Germany, and echoed by other speakers who followed him). If I want to know how monetary authorities should respond to relatively mild shocks in the presence of price rigidities, the standard New Keynesian model is a good choice. But if I want to understand the implications of a breakdown in financial intermediation and the possible policy responses to it, those models aren't very informative. They weren't built to answer this question (some variations do get at this, but not in a fully satisfactory way).
Here's a discussion of this point from a post written two years ago:
There is no grand, unifying theoretical structure in economics. We do not have one model that rules them all. Instead, what we have are models that are good at answering some questions - the ones they were built to answer - and not so good at answering others.
If I want to think about inflation in the very long run, the classical model and the quantity theory is a very good guide. But the model is not very good at looking at the short-run. For questions about how output and other variables move over the business cycle and for advice on what to do about it, I find the Keynesian model in its modern form (i.e. the New Keynesian model) to be much more informative than other models that are presently available.
But the New Keynesian model has its limits. It was built to capture "ordinary" business cycles driven by pricesluggishness of the sort that can be captured by the Calvo model model of price rigidity. The standard versions of this model do not explain how financial collapse of the type we just witnessed come about, hence they have little to say about what to do about them (which makes me suspicious of the results touted by people using multipliers derived from DSGE models based upon ordinary price rigidities). For these types of disturbances, we need some other type of model, but it is not clear what model is needed. There is no generally accepted model of financial catastrophe that captures the variety of financial market failures we have seen in the past.
But what model do we use? Do we go back to old Keynes, to the 1978 model that Robert Gordon likes, do we take some of the variations of the New Keynesian model that include effects such as financial accelerators and try to enhance those, is that the right direction to proceed? Are the Austrians right? Do we focus on Minsky? Or do we need a model that we haven't discovered yet?
We don't know, and until we do, I will continue to use the model I think gives the best answer to the question being asked. The reason that many of us looked backward for a model to help us understand the present crisis is that none of the current models were capable of explaining what we were going through. The models were largely constructed to analyze policy is the context of a Great Moderation, i.e. within a fairly stable environment. They had little to say about financial meltdown. My first reaction was to ask if the New Keynesian model had any derivative forms that would allow us to gain insight into the crisis and what to do about it and, while there were some attempts in that direction, the work was somewhat isolated and had not gone through the type of thorough analysis needed to develop robust policy prescriptions. There was something to learn from these models, but they really weren't up to the task of delivering specific answers. That may come, but we aren't there yet.
So, if nothing in the present is adequate, you begin to look to the past. The Keynesian model was constructed to look at exactly the kinds of questions we needed to answer, and as long as you are aware of the limitations of this framework - the ones that modern theory has discovered - it does provide you with a means of thinking about how economies operate when they are running at less than full employment. This model had already worried about fiscal policy at the zero interest rate bound, it had already thought about Says law, the paradox of thrift, monetary versus fiscal policy, changing interest and investment elasticities in a crisis, etc., etc., etc. We were in the middle of a crisis and didn't have time to wait for new theory to be developed, we needed answers, answers that the elegant models that had been constructed over the last few decades simply could not provide. The Keyneisan model did provide answers. We knew the answers had limitations - we were aware of the theoretical developments in modern macro and what they implied about the old Keynesian model - but it also provided guidance at a time when guidance was needed, and it did so within a theoretical structure that was built to be useful at times like we were facing. I wish we had better answers, but we didn't, so we did the best we could. And the best we could involved at least asking what the Keynesian model would tell us, and then asking if that advice has any relevance today. Sometimes if didn't, but that was no reason to ignore the answers when it did.
[So, depending on the question being asked, I am a New Keynesian, an Old Keynesian, a Classicist, etc.]