Friday, July 4, 2014

What does "model" mean?

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).


  1. Jason, your final sentence: it's really NGDP ~ k log M0, and not log NGDP ~ k log M0?

    I'm not as familiar with your NGDP model (but I'm used to seeing log P in the P model).

    Also, what if people were to just naturally migrate away from using cash and do more purchases with bank cards or credit cards. Do you suppose that M0 would still continue to be a good indicator (perhaps only a scale factor changes slowly over time or something)?

    I especially liked this part:

    "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."

    1. You are correct; that is a typo. It's been updated.

      The basic idea behind the price level model is that

      dN/dM = k N/M

      so that

      log N ~ k log M

      which we substitute into

      P = dN/dM = k N/M ... i.e. the RHS of the first equation so that

      P = k (M^k)/M = k M^(k - 1)

      Regarding the cash and bank cards, I am under the impression that physical currency fixes the value of (anchors) the information content of a dollar unit. It may not matter how much people use electronic money vs physical currency as long as the latter is still around.

      My impression is that if electronic currency became the only thing there is, the full monetary base (including reserves, which are mostly electronic these days) would become the aggregate you'd use in the price level.

      I don't really know what would happen if we switched. That's just my impression from the model.

  2. Correlation of the rates of change or your referenced graph:
    But the levels are lost.

    And the logs of levels are correlated much more than their % rates of change,
    derivative/level (GDP´/GDP, M´/M) aproximated above.

    1. Completely anti correlated rates of change could lead to correlated levels (imagine two time series where one grows 5% when the other shrinks 1% and vice versa) both levels would be almost perfectly correlated, but the rates of change would be perfectly anti correlated.

      Any two exponentially growing functions are going to be correlated in log space, so I'm not sure the correlation between the levels of M2 and NGDP is telling us much.

    2. It shows what many economist attemt to divert the public from and what every one knows. It shows the long run rip off of the rising price to purchas every thing we produce in a year, GDP, due to dillution of money. Those conomists are not will not be sucessful because almost every one knows the problem first hand.

      Have you lived through a great money dillution event? It was terrible! It produces great continual losses for the majority of people.

      And a mild dillution rate over a long time is also a significant loss but with less chaos per time period.

  3. "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."

    And, I think this is the proper approch to describe economics.

    1. We're going to have to agree to disagree on that one. If one made up M2 in order to correlate with NGDP, telling me that NGDP correlates with M2 isn't adding any new information.

      If you build a house to keep the rain out, I won't be surprised when the house keeps the rain out. But I would be interested in the theory where building that house made a city appear.

  4. So is it really that cooked up?

    1. I claimed that velosity V was a computed variable and pointed you to where I gave evidence to that at where a more broad theory seems to work better.*
    In short they seem to be computing v by v1:=GDP/M1, v2:=GDP/M2, etc.

    2. You claim that M2 is made up. That could be. Making up M2 would be fraud.

    If an eqation of three variables has two variables that are cooked up that is a bummer. Then it is totally unscientific. If that is so I’m wasting my time on this.

    Repeated from linked in a previous comment
    June 27,2014
    *Just bouncing ideas arround:

    Graph of GDP=v1*M1=v2*M2:

    “Gross Domestic Product
    M2 Money Stock*Velocity of M2 Money Stock
    Veloc¬ity of M1 Money Stock*M1 Money Stock”

    This seems to show by the data v of the money type is derived by GDP/(Money type). GDP-v*M=0 seems to be with in round off error. This may be a circular logic that does not prove the equation but if the equa¬tion is cor¬rect then it could work.

    So they seem to be computing v by v1:=GDP/M1, v2:=GDP/M2, etc.

    Velocity seems to be computed from data of the other measured variables in the equation. And there are different velocities for each type of money. So this assumes that GDP is only paid for in that kind of money.

    *** At the same time there are dif¬fer¬ent money types assumed to pay for all GDP.

    (This is an inconsistancy. It is like saying we pay for all of GDP with only M2 money and we pay for all of GDP with only M1 money.)


    RE: v mea­sured “indi­rectly”. I would say com­puted not mea­sured. In short search have not foud direct mea­sure­ment. Fig­ure I would not find it in a short time.

    Accord­ing to :
    “In prac­tice, attempts to mea­sure the veloc­ity of money are usu­ally indirect:”
    So I googled for “direct measurement”-“velocity of money” but did not find one in a short time.

    Accord­ing to:

    “This is in con­trast to the mod­ern way used by the Fed, where their veloc­ity num­ber is derived from the for­mula V=GNP/M. This sim­ple model for­mula acts as the mea­sur­ing device for veloc­ity. There are no inde­pen­dent or sep­a­rate num­bers which con­sti­tute direct mea­sure­ments (or even guesses) of mon­e­tary cir­cu­la­tion or velocity.”

    - See more at:

    1. I don't think M2 is completely "cooked up", however different things have been added to monetary aggregates over time because economists believed monetary aggregates were theoretically relevant. Originally we had the "money supply" referring to just notes and coins. That was expanded to "M1" (although it wasn't called that at the time) and then to M2, M3, MZM, Divisia M4 ... etc.

      If you care about money being a medium of exchange, then the correct definition would be something like MZM. If your theory doesn't work with MZM, then the proper approach is to say your theory is wrong -- not come up with a new definition of "money".

      Likewise if you think money is a unit of account, the the monetary base should be your measure of choice since that is what defines money.

      And yes, velocity is computed from the other variables in the equation. If you have NGDP, then velocity of Mx is V = NGDP/Mx where Mx is your monetary aggregate (M1, M2, MZM etc)


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