In response to my criticism, Arnold writes:

So my challenge for Bryan (and for the rest of the profession, because
I am the one who is out on a limb on this) is to come up with a
definition of money that satisfies two criteria. First, your monetary
aggregate is correlated with nominal GDP in a reliable way (with
“reliable” including that if you were to target it, the relationship
would not fall apart). Second, the Fed controls that monetary aggregate
reasonably closely.

I think Arnold’s Criteria #1 conflates two distinct positions:

Variant A: Increasing/decreasing some measure of the money supply reliably increases/decreases Nominal GDP.

Variant B: Nominal GDP only increases/decreases when some measure of the money supply increases/decreases.

Arnold’s implicitly assuming Variant B.  On this assumption, I can’t meet Arnold’s challenge.

However, my position (and I think the position of most economists, at least pre-2008) is Variant A.  And under Variant A, my answer to Arnold’s challenge is simply the monetary base.

Think of my position as “noisy monetarism.”  NGDP fluctuates for lots of reasons, but changing the monetary base still reliably changes NGDP relative to what it would have been.  Arnold’s correct to observe that the Fed massively increased the base in 2008 as NGDP fell.  But if the Fed hadn’t massively increased the base in 2008, I say that Nominal GDP would have fallen vastly more than it did.  There was a huge increase in money demand in 2008, and the Fed only partly accomodated it.

I can see why Arnold might find my position dogmatic, but I think it’s the most reasonable position.  As I I explained in my original post, the quantity theory isn’t just intuitive; it also passes the clear-cut tests – historical episodes of high inflations and high deflations.  The rest of the time, the world is too noisy to convincingly confirm or falsify the quantity theory.  And if intuition and the most probative tests support a position, so should we.