Brad DeLong has a nice post that starts off with a concise point about "taking models seriously". In physics, taking the mathematics of the model seriously leads to things like discovering relativity, developing quantum mechanics, or predicting antimatter. Promoting a symmetry of Maxwell's equations to a real symmetry of the universe -- taking it seriously -- gives us special relativity. Taking the convergence of the discrete sum over the integral seriously gives us quanta of electromagnetic radiation -- photons. And when you add relativity to the Schrodinger equation, you end up with a pair of spinors in a 4-component vector instead of just a 2-component spinor. Taking the other 2-component spinor seriously leads to predicting antimatter. DeLong mentions the first two; the third is my favorite.
DeLong notes that the New Keynesian model probably doesn't have the "hooks" into the underlying process to take the model seriously and discover something new. He then illustrates this with the example of the central bank:
In the basic New Keynesian model, you see, the central bank "sets the nominal interest rate" and that, combined with the inflation rate, produces the real interest rate that people face when they use their Euler equation to decide how much less (or more) than their income they should spend. When the interest rate high, saving to spend later is expensive and so people do less of it and spend more now. When the interest rate is low, saving to spend later is cheap and so people do more of it and spend less now.
But how does the central bank "set the nominal interest rate" in practice? What does it physically (or, rather, financially) do?
I set the Taylor rule up as an information equilibrium model, and I think it helps illustrate this point. The basic model has the nominal interest rate R in information equilibrium with the price level П and real output Y.
R ⇄ П
R ⇄ Y
Taking this model seriously would mean that the central bank is just inflation, output, and a nominal interest rate. This doesn't give the central bank anything physical (financial) to "hook" into. You can really make this sound silly if you think about it in terms of supply and demand: the nominal interest rate is the demand for inflation and output. This is not to say this is incorrect, just that it seems a bit too superficial (maybe abstract is a better word) to have any connection to the underlying process.
In contrast, the information equilibrium model of the short term interest rate looks something like this
r ⇄ p
p : N ⇄ MB
p : N ⇄ MB
where N is nominal output and MB is the monetary base. Here, the short term nominal interest rate is effectively a price of money (p), where N is aggregate demand -- including demand for base money (MB). Taking this model seriously, the central bank controls the supply of base money.
Now this may not be what happens in real life (in fact, it is almost certainly an effective description where the central bank acts "as if" it controls the supply of base money). But it's the kind of thing you need if you want to "take a model seriously".
PS I actually discussed "taking the model seriously" a year ago in a post script to this post. I called it "economic reality"
[elements of the model] must represent an economic reality. An equilibrium condition must be measurable economic construct that has dynamics just like a Dirichlet boundary condition is a D-brane with an energy density flying through an 11-dimensional universe. A good example (in my head) is that your Taylor rule in your New-Keynesian DSGE model has a physical reality as the central bank. The Taylor rule is the central bank. At least, in that specific model.