In my challenge to macroeconomists to show a theoretical model of the price level or inflation compared to empirical data, I got a couple of responses that put forward quantity theory models where P = k MB (which I noted at the beginning of the post) or NGDP = k M2; also David Andolfatto asked what I meant by model (H/T Tom Brown).
To answer David's question, I was being pretty loose with what I meant by model. I intended model to mean a set of equations that are motivated by some theory, that then take some empirical inputs to define the parameters and subsequently output some other variable. On the surface, NGDP = k M2 fits this definition. You fit the parameter k so that k M2 fits NGDP and then the subsequent model NGDP = k M2 outputs NGDP given a value of M2. The theory motivating the equation is essentially the quantity theory of money including the suppositions that "velocity" is stable in the long run.
But what does it mean when we say NGDP = k M2? Are we saying that increased M2 causes NGDP to go up? That is the exogenous money assumption in the quantity theory. Are we saying that, starting from an initial condition M2(0) and NGDP(0), there is a complex relationship where M2 causes NGDP to grow which causes M2 to grow, which feeds back to NGDP, etc, much like the changing electric and magnetic field generate each other in a radiating electromagnetic wave? This is something closer to Wicksell's endogenous money. Is NGDP = k M2 a long run equilibrium, where market fluctuations occur around it? This is closer to Milton Friedman's view.
As an anonymous commenter noted, the relationship between NGDP and M2 can break down (e.g. in Japan), which implies that NGDP = k M2 is not the real model, but instead an approximation to some other model. However, the reason we have M2 as opposed to just "M" is because the relationships between macro variables and the aggregates broke down over time [pdf]:
Although financial innovation has been an important factor, the evolution of the Federal Reserve Board staff's definitions of monetary aggregates primarily been governed by economists changing empirical perceptions of the appropriate concept of money.
That is M2 is constructed to be a better indicator than M1 of some economic variable. That is to say, the monetary aggregates definitions over time has been changed in order to make them fit better to e.g. NGDP. Where does that leave us? We've defined M2 to match NGDP and then we turn around and say that NGDP = k M2 is a model of the economy? This is a completely circular argument: our model uses NGDP to define M2, which we use to show how our model compares to NGDP.
That's why I would put more trust in a theoretical monetary model that looks at the monetary base or MZM which have definitions independent of macro variables. The former is essentially printed currency, although it does contain reserves which means that the measure is "adjusted" for reserve requirements, which, if your model has some important place for e.g. excess reserves or interest on reserves, should bring it under suspicion (are you sure the effect caused by excess reserves isn't used to define the adjusted monetary base?). The latter aggregate (MZM) is a pretty nice definition of money. It ostensibly includes anything that has effectively zero maturity making its definition independent of macro variables (it seems to have a connection with the long term interest rates in the information transfer model).
Now I didn't call for this in the original challenge, but I would like to take what a model should be a step further. Regardless of the problems listed above with NGDP = k M2, there is the additional question of what does the model mean? I could easily see a story where causality goes the other way -- the size of the economy (NGDP) dictates how much money is created by banks via fractional reserve banking (part of M2), so that the growth rates of both are fairly correlated. Is the story really that banks create money through fractional reserve banking which causes the economy to grow? How does that work?
That's where the information transfer model comes in. First, empirically, it's just currency, not M2. What money does is allow people to move information around. Yes it's a medium of exchange, but it's also a unit of account -- the latter is better stated as a unit of information. When the Treasury prints new currency and the Fed releases it into circulation, more information coming from the aggregate demand (NGDP) can be "captured" by the new base money, causing the economy to grow. And the equation is really log NGDP ~ k log M0 where M0 is the currency in circulation (and k changes).