Wednesday, January 27, 2016

Models and frameworks

Given that I recently put forward the idea that inflation and growth are all about labor force growth, I thought I'd clarify some things. Some of you might have asked yourself (or did ask me in comments) about how this "new model" [1] relates to the "old model" [2] that's all about money. I know I did.

The key thing to understand is that the information transfer framework (described in more detail in my arXiv paper) is just that: a framework. It isn't a model itself, just a tool to build models. Those models don't have to be consistent with each other. So there really is no "new model" or "old model", just different models that may be different approximations (or one or both might become empirically invalid as more data comes in).

And as a tool, it's basically an information-theoretic realization of Marshallian supply and demand diagrams. What you do is posit an information equilibrium relationship between A and B, which I write A ⇄ B, or an information equilibrium relationship with an abstract price p = dA/dB, which I write p : A ⇄ B, and here's what's included (act now!) ...

  • A general equilibrium relationship between A and B (with price p) where A and B vary together (that always applies). Generally, more A or more B leads to more B or more A, respectively.
  • A partial equilibrium supply and demand relationship between A and B (with price p) with B being supply and A being demand -- it applies when either A or B is considered to move slowly with respect to the other (it's an approximation to the former where A or B is held constant).
  • The possibility of "market failure" where we have non-ideal information transfer that I write A  → B (all of the information from A doesn't make it to B). This leads to a non-ideal price p* < p as well as a non-ideal supply B* < B.
  • A maximum entropy principle that describes what (information) equilibrium between A and B actually means, including a causality that can go in both directions along with potentially emergent entropic forces that have no formulation in terms of agents.

So in the information transfer framework there are information equilibrium relationships A ⇄ B and more general information transfer relationships A  → B. I tend to refer to these individual relationships as "markets". Given these basic "units of model", you can construct all kinds of relationships. Traditionally crossing-diagrams are easiest. Things like the AD-AS model or the IS-LM model can be concisely written as the market

P : AD ⇄ AS

where AD is aggregate demand and AS is aggregate supply, or the markets

(r ⇄ p) : I ⇄ M

PY ⇄ I

PY ⇄ AS

for the IS-LM model where PY is nominal output (i.e. P × Y = NGDP, I also tend to write it N on this blog and in the paper), I is investment, M is the "money supply", p is the "price of money" and r is the interest rate.

Another aspect of the model is that information equilibrium is an equivalence relation, so that AD ⇄ M and M ⇄ AS implies AD ⇄ AS (this makes an interesting definition of money). This means that if you find a relationship (as I did in [2])

CPI : NGDP ⇄ CLF

there could be some other factor(s) X (, Y, Z, ...) such that

NGDP ⇄ X ⇄ Y ⇄ Z ⇄ CLF

Relationships like this can be inferred from a price that doesn't follow CPI* < CPI, but can be above or below the ideal price CPI (CPI* < CPI or CPI* > CPI) that follows from being careful about the direction of information flow and the intermediate abstract prices p₁ and p₂ in the markets

p₁ : NGDP ⇄ X
p₂ : X ⇄ CLF

These would probably find their best analogy in "supply shocks" (price spikes due to non-ideal information transfer) as opposed to "demand shocks" (price falls due to non-ideal information transfer). Note that in the model CPI : NGDP ⇄ CLF with intermediate X,  CPI = p₁ × p₂ because CPI = dNGDP/dCLF = (dNGDP/dX) (dX/dCLF) via the chain rule.

In the end, however, the only way to distinguish among different information equilibrium models (or information transfer models) is empirically. This framework works much like how quantum field theory works as a framework (as a physicist, I like to have a framework ... anything else is just philosophy). You observe something in an experiment and want to describe it. One group of researchers models it as a real scalar field and writes down a Lagrangian

ℒ = ϕ (∂² – m) ϕ

Another group models it as a spin-1/2 field

ℒ = ψ (i ∂ – m) ψ

(ok, that one's missing a slash and a bar). Both "theories" are perfectly acceptable ex ante, but ex post one or both may be incompatible with empirical data.

Actually one of the goals of this blog (and the information transfer model) was to introduce exactly this kind of model rejection to economics:
I was inspired to do this because of Noah Smith's recent post on why macroeconomics doesn't seem to work very well. Put simply: there is limited empirical information to choose between alternatives. My plan is to produce an economic framework that captures at least a rich subset of the phenomena in a sufficiently rigorous way that it could be used to eliminate alternatives.
I've come up with several different information equilibrium relationships -- or models built from collections of relationships (see below) -- and I am testing their abilities with forecasts. Some might fail. Some have failed already. For example, the IS-LM model does not work if inflation is high (but represents an implicit assumption that inflation is low, so it is best to think of it as an approximation in the case of low inflation). A few of Scott Sumner's versions of his "market monetarist" model can be written as information equilibrium relationships (see below) ... and they mostly fail.

In a sense, I wanted to try to get away from the typical econoblogosphere (and sometimes even academic economics) BS where someone says "X depends on Y" and someone else (such as myself) would say "that doesn't work empirically in magnitude and/or direction over time" and that someone would come back with other factors A, B and C that are involved at different times. I wanted a world where someone asks: is X ⇄ Y? And then looks at the data and says yes or no. DSGE almost passes this test -- these models are at least specific enough to compare to data. However they don't ever seem to look at the data and say no ... it's always "add a financial sector" or "add behavioral economics". There isn't enough data to support that kind of elaboration.

A good example is the quantity theory of money. It says PY = MV. Now this was great in a world where people thought V was constant (i.e. the old Cambridge k). But that turns out not to be the case and now V could depend on E[PY] or E[P] or E[M] or something else. What are these specific expectation models? Is E[P] = TIPS? Or is V ≡ PY/M is now a definition? And what is M? M2? MB?

Essentially various versions of the quantity theory of money have been falsified empirically (or at best a loose approximation when inflation is high) ... but it keeps trucking along because it doesn't exist in a framework where either its scope or validity can be challenged.

It's probably a naive hope, but it's the kind of naive hope that distinguishes "science" from "mathematical philosophy".

...

Addendum: information equilibrium models

Note that just because these models can be formulated does not mean they are correct.

I. The "quantity theory of labor" [1]

P : PY ⇄ CLF

See this post for this one.

II. "The" IT model [2]

P : PY ⇄ M0
(r¹⁰ʸ ⇄ pᴹ⁰) : PY ⇄ M0
(r³ᵐ ⇄ pᴹᴮ) : PY ⇄ MB
P : PY ⇄ L

where the r's represent the long and short term interest rates (3 month and 10 year), M0 is base minus reserves, MB is the monetary base (including reserves) and L is the labor supply (the last relationship is essentially Okun's law). I usually measure the price level P with core PCE, but empirically it is hard to tell the difference between core PCE and core CPI (or the deflator). This model also allows the information transfer index in the first market to slowly vary. This represents a kind of analytic continuation from a "quantity theory of money" to an "IS-LM model with liquidity trap".


Both this model and the next one are in my paper.

III. Solow model (plus IS-LM)

PY ⇄ L
PY ⇄ K ⇄ I
K ⇄ D
1/s : PY ⇄ I
(r³ᵐ ⇄ p) : I ⇄ MB

where the last market is the IS-LM piece, K is capital and D is depreciation. This is a bit different from the traditional Solow model in that it is written in terms of nominal quantities. This may sound problematic, but it throws out total factor productivity as unnecessary and is remarkably empirically accurate in describing output as well as the short term interest rate.

IV. Scott Sumner's various models (1), (2) and (3)

1) u : NGDP ⇄ W/H

... this is just empirically wrong over more than a few years. H is total hours worked and W is total nominal wages.

2) (W/H)/(PY/L) ⇄ u

... but  H/L ⇄ u has almost no content (higher unemployment means fewer worked hours per person) and the relationship c : PY ⇄ W has a constant abstract price meaning  PY = c W with c constant. The model reduces to (1/c) (H/L) ⇄ u or just the content-less H/L ⇄ u.

The correct version of both of these is P : PY ⇄ L or P : PY ⇄ H, which are just Okun's law (above).

3) (1/P) : M/P ⇄ M

This may look a bit weird, but it could potentially work if Sumner didn't insist on an information transfer index  k = 1 (if k is not 1, that opens the door to a liquidity trap, however). As it is, it predicts that the price level is constant in general equilibrium and unexpected growth shocks are deflationary in the short run.

20 comments:

  1. I'm very happy to see this post corral your current library of models and explain how they relate to your framework (including the failed ones). Great discussion of using data to validate or reject a hypothesized information equilibrium / transfer relationship.

    1) Regarding the notation A ⇄ B, the ⇄ is what? An operator? A signifier? It seems A ⇄ B does not imply B ⇄ A. "Causation" (you say) flows both ways, but the order of A and B is always significant: A (on the left) is the information source and B (on the right) is the destination, true?

    2) Regarding your 2nd bullet point: you also (sometimes?) get two partial equilibrium relationships, right? I'm thinking of both the supply and demand curves. Also in your paper you bring up the case where both D and S are slow moving. dD/dS = D0/S0 I think it was.

    3) The (A ⇄ B) : C ⇄ D is a little confusing to me still. Is that just shorthand for something longer, such as:
    A ⇄ B
    and
    B : C ⇄ D
    ?
    Something else? If true, does that imply anything else, such as:
    A : B ⇄ D
    Perhaps?

    4) I think of PY = MV as the "equation of exchange," serving only to define V for a choice of M. (one of the possibilities you bring up: and one I've heard Sumner advocate). The quantity theory is what happens when you in addition assume V is approximately constant. But I learned that from blogs... so maybe it's not right.

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    1. Do F&B have any examples (in any of their paper's versions) that would be well described by the (A ⇄ B) : C ⇄ D relation?

      There are variations too I suppose, but do they make any sense?:

      (P : A ⇄ B) : C ⇄ D

      (A → B) : C ⇄ D

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    2. 1) It is notation. It is an equivalence relation in information equilibrium so that A ⇄ B means B ⇄ A, but not in the case of non-ideal information transfer.

      2) The supply and demand diagram represents what happens when you shift supply or demand curves, holding the other one constant. Calling it two relationships is not really a useful description, nor is calling it three relationships (shift supply curve, shift demand curve, don't shift either curve).

      3) Yes, it is a shorthand for

      A ⇄ B
      and
      B : C ⇄ D

      And no, it doesn't imply A : C ⇄ D. Two detectors being in information equilibrium with each other does not imply both detectors are detectors for another information equilibrium relationship.

      4) None of that theory is well-defined enough to warrant such precise definitions. What is the definition of quantity theory of money? Bennett McCallum says it is the EoE plus long run neutrality. The definitions only get more vague from there. The constant velocity version has the added bonus of being falsified empirically.

      Plus it is fairly obvious what I am talking about. It would be like a physicist calling

      ℒ = ϕ (∂² – m - g ϕ²) ϕ

      a ϕ^4 theory and some other physicist "correcting" him or her by saying it is a Lagrangian density for a ϕ^4 theory. At which point that second physicist would be bludgeoned with a handy copy of the "phone book".

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    3. Regarding your follow-ups:

      Nope.

      And sure both of those could make sense. The price P would be dA/dB, already implicit in the relationship A ⇄ B. You're just giving it a name. There is no difference between

      P : A ⇄ B
      and
      A ⇄ B

      It's just that "P" has a name in the first one.

      The second one would mean that A is a ceiling for B^k ... A > B^k for some k.

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  2. Also, I don't recall you covering an explicit connection between the (Gary Becker type diagram)-derived demand curves and the P:D⇄S-partial-equilibrium-derived demand curves in your draft Summer talk paper. Is there one? Did I overlook it? I realize the former (Becker) gets more interesting as goods (markets?) (i.e. dimensions) are added while the latter (information equilibrium) can apply to a single good / market.

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    1. Each point in the triangle is one of the n = S/dS bits (or bytes or whatever): the supply states. Demand is held constant (a demand bath). Expanding the triangle (or just expanding it in one direction as shown) increases the number of supply states leading to a fall in price (the centroid moves out) -- a demand curve.

      The other way is less intuitive in the same way that the supply curve analog of the PV isotherm "demand curve" with the opposite relationship is less intuitive.

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    2. Thanks. If that's not already there, you might consider adding a paragraph like that to your draft Summer talk paper to tie those two sections together.

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  3. Can you think of an example (in any field: physics, economics, whatever) for which the information transfer/equilibrium framework might seem perfectly appropriate at first, but after further contemplation, a researcher might rightly conclude to reject it (without ever getting as far as checking it empirically)? If so, what would be the reason to reject the ITM in that example?

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    1. I'm not sure how I'm supposed to answer this question. How do you come up with something that seems "perfectly appropriate at first" but would be rejected after thinking about it more?

      This involves defining a scale of thinking T0 where if T < T0, the model is good, but if T > T0 the model is bad. I have no idea what T0 means ... is it people who have had calculus? Is the number of thought experiments that have to be done? How many thought experiments do you have to do to reject a theory?

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    2. Ha... good points. How about this: have you ever had this experience personally, where you changed your mind about trying to apply the ITM before you got very far.

      I guess I was looking for a case that you would describe as follows:

      "Because of its properties A, B and C case X might at first seem like a good candidate to analyze using the IT framework but unfortunately, because of its properties D, E and F the IT framework is actually likely to be a poor choice."

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    3. Since it is so easy to test empirical data on FRED for plausibility with this framework, it's nearly always an empirical test.

      But the criterion is basically the existence of two process variables (A, B) that are "macro scale" aggregates of some underlying "micro scale" degrees of freedom that you don't necessarily know much about -- along with some possible path of information transfer from A to B.

      Example: biological activity effects on the atmospheres of two planets meets the first part, but fails in a path of information transfer. Fails the test.

      Example: neurons producing EEG signals in wake/sleep states. Passes the test (and Todd Zorick and I put together a paper on this).

      Example: humans playing chess. This probably fails because while the human brains are made up of neurons, there is the approximation of effective free will and so there effectively aren't any underlying micro-scale degrees of freedom. However an MMOG might pass.

      But there aren't that many criteria to check

      Two process variables
      Micro scale degrees of freedom
      Plausible information transfer channel

      So the deciding factor is usually just looking at empirical data.

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    4. Good examples. Thanks.

      Do tell more about this paper you and Todd put together. When did that happen and is it available to read?

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  4. Given

    p1 : A ⇄ B

    you say above that A ⇄ B implies B ⇄ A which must mean we have a separate p2 such that:

    p2 : B ⇄ A

    Under IE, then it appears p1 = 1/p2

    Now are there two separate non-ideal information flows we could consider?

    A → B, having a non-ideal price p1* < p1, and a non-ideal supply B* < B.

    but potentially simultaneously also (or assuredly also?)

    B → A with p2* < p2 and A* < A

    ?

    I guess what I'm asking is does A ⇄ B degenerating into A → B necessarily imply that B ⇄ A degenerates into B → A where all four of the above inequalities hold? I guess what's confusing about that to me is that as B* becomes less than B (as information transfer starts to become non-ideal for some reason) this in turn implies a new lower upper bound on A*, etc. It seems like a potential negative feedback loop that quickly approaches 0 for both A* and B*. I'm probably misinterpreting something here, but I'm reading B* (for the A → B relation) as the actual B achieved, as compared to the best case (ideal) B which results from information equilibrium (A ⇄ B). But if that's true, then this should give rise to a an actual A (A*) which is less than the ideal A, etc.

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    1. If you have ideal information transfer (information equilibrium), it doesn't matter the order of A and B. But for non-ideal, it does and the two systems B → A and A → B are different. Either A is the source or B is the source -- both can't be the source.

      You can use transitivity and reflexivity for information equilibrium, but not for non-ideal where you have to be careful about what mathematical operations do to your inequalities.

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    2. Ah, OK, so let me try to restate that. So given
      p1 : A ⇄ B
      it follows that we must also have
      p2 : B ⇄ A
      but should this degenerate into non-ideal transfer, then we will have (at most) one of:
      A → B
      or
      B → A
      but never both. So say A ⇄ B degenerates into B → A for which there's a p2* < p2 and an A* < A, then is it fair to say that A → B is just not happening at all at that point? I.e. no information flow from A to B?

      Is which way a system in IE degenerates entirely determined by the qualities of the system, or is it possible a system could be a priori indeterminate in regards to which way it degenerates into non-ideal information transfer? In other words, we won't know until it happens?

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    3. Sort of. You still don't know if there is an intermediate variable, so you won't necessarily know which model it has "degenerated" into. But assuming there isn't an intermediate variable, then yes.

      And yes, we wouldn't know until it happens. And we might not know it has already happened as the system might always have been non-ideal.

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  5. It would be great if you got together with Paul Cockshott and Allin Cottrill. I'm sure you'd have plenty to talk about and a fruitful collaboration could occur.

    http://www.dcs.gla.ac.uk/~wpc/

    pe

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    Replies
    1. Thanks for the link. I will check out their work.

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