Thursday, March 5, 2015

Noah, information equilibrium is the theory you're looking for

Noah Smith is back to criticizing macroeconomics [1], telling us that empirical accuracy isn't very highly regarded. As evidenced by Nick Rowe's comments on this post of mine, we can see how empirical accuracy works as a line of argument. Noah also says that novelty is prized by macroeconomists, but I haven't seen any of that in my attempts to get any to look at the information transfer model.

One thing I want to discuss, however, is this:
Economists will respond that seismologists can’t forecast earthquakes, and meteorologists can’t forecast hurricanes ...
I am so tired of this analogy. The reason? These two pictures (from wikimedia commons):



Robert Waldmann put it nicely:
In any case, there is an overwhelming consensus -- when there is an earthquake seismologists publicly agree about things that happened under the earth.
The ring of fire -- where earthquakes and volcanos are much more likely -- is part of a general theory that is consistent with geology (plate tectonics) paleontology (the fossil record). Seismologists can predict where earthquakes are likely to occur and meterologists can predict where various weather events are likely to occur. Tornado watches and warnings are useful.

Back to Waldmann:
I think the problem here is that the analogy is much too kind to macroeconomists. It is true that macroeconomists can't predict recessions. It is also true that macroecomists almost all admit this. However, macroeconomists don't agree on the explanation of what happened. Also macroeconomic models have lots of implications which can be confronted with the data. However they don't fit the data as the implications of models of plate tectonics do.
There is no analogous macroeconomic theory of plate tectonics that shows where earthquakes (i.e. recessions) are likely.

Or is there? I've started to see some possibilities for predicting when recessions may be likely including a model of recessions as avalanches that could be the 'plate tectonics' of macroeconomics.

Noah Smith also discusses two major themes to major scientific theories -- unification and broad applicability:
Other times, a theory will predict things we have seen before, but will describe them in terms of other things that we thought were totally separate, unrelated phenomena.


... [and] can predict more than just the phenomena that inspired the theory.
The information transfer model has both of these qualities ... it creates a unified framework for describing several economic models and unifies the quantity theory of money and the liquidity trap. And the theory wasn't even designed to work for economics at all.

Footnotes:

[1] Noah Smith is also back to mis-representing physics:
... quantum mechanics has gained a lot of support from predicting the strange new things like quantum tunneling or quantum teleportation.
Quantum mechanics gained a lot of support for being able to explain atoms in the 1920s and 1930s, and a few years later (1947) for being freakishly accurate. Tunneling was important in the description of the fusion reactions in the sun.

It was the fact that quantum mechanics was accepted for being empirically accurate and making qualitative sense of many phenomenon that people took it seriously enough to try teleportation. The idea for teleportation comes in the 1990s -- a time when quantum mechanics was well established.

If you want something new that was predicted by a theory and only later discovered the prime example would be antimatter, in particular the positron, predicted by Dirac's quantum theory of the electron. But again, Dirac's treatment of spin-1/2 particles was accepted because it was accurate.

The latest PCE inflation numbers are out ...

... and the information transfer (IT) model is still looking good (data from FRED):


I took off the error ranges as the different measures aren't really comparable and the highest resolution we have from the Fed predictions is quarterly data so it'll be when the March numbers come out at the end of April or beginning of May that we'll have a new true data point. Here's the previous time where we had both an end-of-year result for the FOMC prediction (purple) and an end-of-quarter result for the NY Fed DSGE model. 

However, since we now have the January number, we have 1/3 of the next quarterly prediction for the NY Fed DSGE model. In order to meet their quarterly prediction, core PCE inflation will have to average 1.7% in February and March. In order to meet the IT model prediction, February and March only have to average 1.3% core PCE inflation. The former scenario has happened roughly 28% of the time since January 2009, while the latter has happened 54% of the time.

This doesn't mean the models should be discarded. It's just that the FOMC and NY Fed DSGE model are predicting inflation should start heading back to "normal" (2%), like, now.

Wednesday, March 4, 2015

The hot potato effect is an entropic force

According to an oft-repeated but probably embellished account, when Laplace gave a copy of Mécanique Céleste to his physics-literate friend Napoleon Bonaparte, Napoleon asked him what role God played in the construction and regulation of the heavens. "Sire," Laplace replied, "I have no need of that hypothesis."
Neil deGrasse-Tyson 
The "hot potato effect" describes how injections of so-called high powered money, like cash or monetary base reserves, are like hot potatoes. Individuals are already holding as much money as they want in equilibrium at a given price level, so the additional cash is 'sold' to re-balance their portfolios. That ends up in a cascade of exchanges until everyone is back in equilibrium at a new higher price level.

Scott Sumner wrote a post about this effect awhile ago, calling it the sine qua non of monetary economics. He said something remarkable in that post:
"[people] want explanations they can understand at the individual behavior level.  But that just won’t work in this case."
Something that doesn't exist at the individual behavior level? Sounds like an entropic force! Here's Sumner from the same post explaining the hot potato effect using a parable with gold:
Because before the discovery [of additional gold] people were already in equilibrium, they held as much gold as they wanted to hold at existing prices.  The extra gold is a sort of “hot potato” that people try to get rid of.  But obviously not by throwing it away!  They get rid of it by selling it.  But notice that while that works at the individual level, it doesn’t work in aggregate.  Now someone else has the extra gold. (That’s why attempts to understand money at the level of the representative consumer fail.)  The only way for society as a whole to get rid of the extra gold is by driving down the price of gold [i.e. driving up the price level] until people want to hold the new and larger quantity.
The thing is that this explanation doesn't really explain why the price level goes up. I may have more gold than I want to hold, but I don't want to give it away or get a bad deal. It seems like Sumner is implying that the price level just happens to rise over the course of several 'mistakes' (the price of bacon is too high at this store, but I have the money, i.e. extra gold, and I'm too lazy to go to the other store so I'll buy it anyway even though its not optimal). This is remarkably close to the entropic force view.

The entropic force view would say that the additional 'gold' (high powered money) randomly moves around (and prices randomly fluctuate) until the new most likely configuration (of prices and gold held) with the new larger amount of gold is happened upon by chance. In the simplest case that new most likely configuration is a uniform distribution across the agents. The new most likely price level is also higher since people will randomly accept both high and low prices, but more high prices (bad deals) will be accepted than were accepted before the additional gold was added because those 'mistakes' were made possible (the state space was opened up) by the additional gold in the market.

In the language of thermodynamics, if you add energy to a system, that opens up new parts of phase space with higher momentum states, raising the temperature of the system.

I made a short animation showing how a large injection of high powered money into a segment of the economy eventually finds its way across the entire economy through random exchanges. You can imagine each vertical light blue (well, purplish in the compressed youtube version) bar is an agent, firm or market sector (e.g. imagine the injection happens in the banking sector). The horizontal blue line is the average and median before the injection, the horizontal red dashed line is the average after the injection.



The solid red line is the median level; 50% of the agents are holding high powered money above this level and 50% below. When the median meets the average, that means the typical agent won't make a biased 'mistake', i.e. randomly accept a price that is too high more often than one that is too low. If the median is below the average -- as it is when the high powered money is injected -- then more than half of the agents will tend to increase their holdings since they have below the average level [1].

Effectively, the injection of high powered money is "thermalized" and heads toward a new equilibrium (uniform) distribution at the new average. We see the median (solid red line) rise from the previous average (blue line) to the new average (dashed red line).

The difference here is that there is no real requirement for human behavior in the explanation -- even at the macro level. Agents aren't thrown off of their original "desired" equilibrium and don't "want" to get rid of excess holdings of high powered money. The new equilibrium is just the new most likely state and the agents, if they have more than the average, just tend to reduce their holdings by random exchanges.

Utility maximizing agents? I have no need of that hypothesis.

Footnotes:

[1] This assumes non-preferential attachment -- i.e. all agents are equal. This illustration selects which agents trade money from a uniform distribution. A different mechanism leads to different distributions. See e.g. Bouchaud and Mezard http://arxiv.org/abs/cond-mat/0002374.



Tuesday, March 3, 2015

Theories of identities are nonsensical; information equilibrium conditions are better

In reading David Glasner's two posts (so far) on accounting identities:
I suddenly realized I was looking at an argument that these so-called accounting identities represent information equilibrium (~) conditions. Let's posit something called aggregate demand N (for NGDP) that is an information source for national aggregate variables. Additionally, let's say aggregate demand is in information equilibrium with income (Y) and expenditure (E):

N ~ Y
N ~ E

Now if we have ideal information transfer ('economic equilibrium'), information equilibrium is an equivalence relation, so that we can immediately say:

E ~ Y

log E = k log Y + m

In order to allow E = Y at some point, we must have k = 1  and m = 0, which means that E = Y in equilibrium (ideal information transfer). Now away from economic equilibrium (non-ideal information transfer), we have

a log Y + b < log N
a log E + b < log N

so in general E ≠ Y. (The coefficients of the logs, i.e. a, and the intercepts, i.e. b, must be equal in order to allow the possibility that E = Y.) The information transfer model doesn't tell you how far Y or E fall; it just says there should be a trend where Y ~ E if markets are typically in equilibrium.

This general argument would apply to any national income identity, such as savings and investment (S ≡ I), that isn't based on a definition (e.g. per Glasner purchases equal sales). More interestingly, it applies to another definition: the equation of exchange.

I got in an argument with Scott Sumner on his post that says MV = PY just means V ≡ PY/M for saying that, sure, in the economics profession it's just a definition, but I think the equation of exchange can be usefully restated as an information equilibrium condition.

If we look at the market P:N→M [1] where the price level P is a detector, N = PY is aggregate demand (NGDP) is an information source and M is the money supply (we'll say base money minus reserves), we can write down the equations (in economic equilibrium, i.e. ideal information transfer):

N ~ M

log N = k log M + c1

log P = (k - 1) log M + c2

And we can show:

PY/M = N/M = (1/k) P ≡ V

How does this compare with empirical base velocity? Well, if we take the expected value <P/k> in 1000 random markets with random k values between 0 and 2, we get a pretty good fit (for such a simple model):


This is the Monte Carlo result with 10 different random sets of 1000 markets, hence the 10 gray lines. The blue points are the data (from FRED). Again, it's the trend we're capturing here, and the fluctuations represent non-ideal information transfer and/or shocks.

Footnotes:

[1] The notation A:X→Y means that X is an information source, Y is an information destination and A is a detector per the definitions in the original information transfer model paper.

Sunday, March 1, 2015

Information equilibrium is an equivalence relation


Something for the math nerds. I've said it a couple times, but haven't actually shown the proof. However, it is true that information equilibrium is an equivalence relation. If we define the statement $A$ to be in information equilibrium with $B$ (which we'll denote $A \cong B$) by the relationship (i.e. ideal information transfer from $A$ to $B$):

$$
\text{(1) }\;\; \frac{dA}{dB} = k \frac{A}{B}
$$

for some value of $k$, then, first we can show that $A \cong A$ because

$$
\frac{dA}{dA} = k \frac{A}{A}
$$

$$
1 = k \cdot 1
$$

and we can take $k = 1$. Second we can show that $A \cong B$ implies $B \cong A$ by re-deriving the relationship (1), except moving the variables to the opposite side:

$$
\frac{dB}{dA} = \frac{1}{k}\;\; \frac{B}{A} = k' \; \frac{B}{A}
$$

for some $k'$ (i.e. $k' = 1/k$). Lastly we can show that $A \cong B$ and $B \cong C$ implies $A \cong C$ via the chain rule:

$$
\text{(2a) }\;\; \frac{dA}{dB} = a \frac{A}{B}
$$

$$
\text{(2b) }\;\; \frac{dB}{dC} = b \frac{B}{C}
$$

such that

$$
\frac{dA}{dC} = \frac{dA}{dB}\; \frac{dB}{dC} = a b \; \frac{A}{B} \frac{B}{C}
$$

$$
\frac{dA}{dC} = k \; \frac{A}{C}
$$

with information transfer index $k = a b$. That gives us the three properties of an equivalence relation: reflexivity, symmetry and transitivity.


Saturday, February 28, 2015

Sumner to quantitative analysis: drop dead

Tony Yates calls out market monetarists to get quantitative:
I’m sure these mix-ups would get ironed out if [market monetarists] stopped blogging and chucking words about, and got down to building and simulating quantitative models.
Scott Sumner decides instead to set Western civilization back a thousand years:
In my view economists should forget about “building and simulating quantitative models” of the macroeconomy, which are then used for policy determination. Instead we need to encourage the government to create and subsidize trading in NGDP futures markets (more precisely prediction markets) and then use 12-month forward NGDP futures prices as the indicator of the stance of policy, and even better the intermediate target of policy.
Sumner apparently doesn't even want to simulate an economy with an NGDP futures market before using one to guide the economy. And it's not like you can't just set one up and say it's not going to influence policy decisions. If you set up a liquid NGDP market, but said it was only an experimental measure with regard to policy, markets could still crash on news of a crash in the NGDP market.

"We think this NGDP market represents the wisdom of crowds, but please ignore it if it does anything weird."

Note that Sumner is trying to set up an NGDP prediction market. The thing is that inasmuch as people don't believe the prediction market actually works is it not dangerous to the economy. If people believed it was working (was liquid enough, had enough volume, enough diversity of participants, or whatever), then its movements could strongly affect economic sentiment and spark a panic. This is where market monetarism's reliance on expectations comes back to haunt it.

Now let's crowd-source a launch vehicle and send people to Mars without testing it!


Note that project Vanguard actually did test the components of that rocket before that happened. Because reason.

Thursday, February 26, 2015

Market monetarism and the Keynesian beauty contest

Attention: conservation notice. Over 1500 word essay about universality classes and kittens.
In my recent post on expectations, I wrote this:
Overall, I'd sum up my problem with the centrality of expectations with a question to Scott Sumner: what happens to your theory if markets don't believe market monetarism? In a sense, this question cannot be addressed by market monetarism. The "market" piece presumes markets believe the theory (i.e. market expectations are consistent with market monetarism, i.e. assuming rational expectations in market monetarism ... I called this circular reasoning before, but charitably, this could be taken that market monetarism is the only self-consistent theory of expectations as I mention in a comment at that link).
One thing I forgot about until I was doing some searching on Sumner's blog was that Sumner had basically assumed it explicitly:

Markets are market monetarists (23 Mar 2012)
It’s not surprising that the markets are market monetarist, as my views of macro were largely developed by watching how markets responded to the massive and hence easily identifiable monetary shocks of the interwar period.  That’s why I never lose any sleep at night worrying about whether market monetarism will ever be discredited; I know the stock market agrees with me.
I'd like to expand on what I meant by market monetarism claiming to be the only self-consistent theory of expectations. I borrowed the phrase self-consistent from physics; let me elaborate on what I mean by that [1]. I'd also like to better explain why I think market monetarism is more an ideological movement than an economic theory.

Let's call market monetarism M, which is a functional of expectations E, i.e. M = M[E]. But additionally, the ouptut of "market monetarism" as an economic theory defines the expectations one should have given a set of economic variables n, m, ... (say, NGDP, base money, ...). That is to say, M[E] gives us E as one of the outputs. This is rational expectations, aka model consistent expectations. So what we have is this:

(1) E(n, m, ...) =  M[E = E(n, m, ...)]

There are many paths of variables (n, m, ...) that can lead to the same expectations E (it's called indeterminacy), but that's not important right now [2]. Basically, expectations held by the market represent a fixed point of M ... like a Nash equilibrium of some expectations game. This is all well and good, and is really just a straightforward application of rational expectations. You could say the same of a New Keynesian theory ... E = NK[E]. In fact, a wide class of theories can have fixed points like this (any RBC or DSGE model, for example).

The thing is that market monetarism doesn't think there is that kind of freedom, and the reason is that market monetarism is almost entirely expectations. This is an uncontroversial categorization of market monetarism. For example, Scott Sumner wrote a post to that effect:

Money and Inflation, part 5: It’s (almost) all about expectations (1 Apr 2013)

And here is a quote from a Nick Rowe comment at Tony Yates' blog that's even more concise:
Monetary policy is 99% expectations, so how monetary policy is communicated is 99% of how it works.
Why does market monetarism's insistence that the theory is 99% expectations lead inexorably to the conclusion that market monetarism makes the (erroneous) claim to be the only theory of expectations and therefore indisputable? Let me tell you.

In general, the other theories have explicit dependencies on observable quantities x like the monetary base (i.e. not only do expectations have something to do with whether x is well above or below trend but that expectations can be anchored by real observable quantities):

E(x) = NK[E(x), x]

but in the market monetarist model we have, expanding around x = 0:

E(x) = M[E(x), x]

E(x) ≈ M[E(x)] + α x  ... with α << 1

Concrete steppes (like QE) have little to do with expectations and as Nick Rowe said above, the theory is 99% expectations (i.e. α ~ 0.01). Scott Sumner would have to lump what I call α x into a error term or "systematic error" term SE in his post here. It represents the difference between pure expectations (an NGDP futures market or credible central bank target) and reality. I'd call it the influence of the actual value of NGDP on the expected value of NGDP.

So what's wrong with that, you say? Well, the explicit dependence on x above is what makes these theories with expectations different from each other since they're all based on people making decisions. It's what forms the basis of the model dependence of the expectations. In Keynesian models, x includes things like interest rates and unemployment. In RBC models, x includes "technology". The lack of an explicit dependence on x in M is what I mean by model independent expectations in this post. It also couples the theory to the empirical data. Without it the only dependence on x is via E(x) -- that is to say the value of x doesn't matter, it's what people think x is (like what Allan Meltzer thinks inflation is, or what Republicans think Obamacare is).

Here's the kicker, though. Since all of these theories are based on economic agents (people) with human expectations, market monetarism is making the claim to be the only expectation-based theory. If we expand around x in a generic theory T ...

E(x) = T[E(x), x] ≈ T[E(x)] + τ x + ...

(2) E(x) ≈ T[E(x)]

T is completely unspecified right now. Now there are two possibilities here. First, is that equation (2) doesn't specify T. In that case, whether T = NK or T = M depends on what humans believe and there is no specific theory of pure expectations (or you just have to convince the market that T = X and it is entirely political ... X could be communism or mercantilism or the Flying Spaghetti Monster). 

Obviously market monetarists don't believe that. Therefore they must believe that equation (2) specifies T. In that case, market monetarists are claiming T = M. Basically, the first term in any expansion of any theory in x is M (i.e. the first term is unique) ...

T[E(x), x] ≈ M[E(x)] + τ x + ...
NK[E(x), x] ≈ M[E(x)] + k x + ...
M[E(x), x] ≈ M[E(x)] + α x + ...

But also, α = τ = k is small (according to market monetarism), so the first term is all that matters! If you expanded a theory and didn't end up with M[E(x)] as a first term, then whatever that expansion was would be a theory equally valid to market monetarism.

Physicists out there probably recognize this idea: market monetarism is making the claim (without proof or comparison to empirical data) to be the universality class of macroeconomic systems. Universality classes are why the same kinds of processes happen in totally different systems, or things like the normal distribution show up everywhere.

This is a lot different from e.g. Scott Sumner just saying "my theory is right". It is Scott Sumner saying "every theory reduces to my theory" [3]. He is not explicitly saying this; it is implicit in his argument that the theory is primarily expectations and somehow unique (or at least has a reason to be advocated besides pure opinion).

Now that I've gone off the deep end of abstraction, let me close with something concrete to show how preposterous this is.

Kittens.

Yes, kittens. NPR's Planet Money did an experiment to illustrate Keynes' "beauty contest" in markets, but it gives us an excellent illustration of expectations and theories of expectations. Planet Money put up three pictures of animals (a kitten, a slow loris -- my personal favorite, and a polar bear cub) and asked people not only which one they thought was the cutest, but which one they thought everyone else would think is the cutest ... i.e. the expected winner of the poll.

Here are the images and results:

Picture from NPR's Planet Money.



I have a couple of theories for the result. The first one is that the most commonly experienced critter will be expected to win the poll. Call this theory MC, and it depends on the actual data of which critter is most common. Call that c. The second theory is that the one with the biggest eyes (relative to body size) will be expected to be the winner. Call this BE and it depends on eye size e. Both of these theories will expect the kitten to win. In our notation above, we have the self-consistent (e.g. Nash) equilibria: 

E(c) = MC[E(c), c]

E(e) = BE[E(e), e]

(If you repeat the "game" with the same pictures, the result would rapidly converge to nearly 100% for the kitten for the expected result.)

Now if we make the market monetarist assumption that the empirical values (c and e) have little influence on the result we can say:

MC[E(c), c] ≈ MC[E(c)] + α c
BE[E(e), e] ≈ BE[E(e)] + α e

With α being small. The alpha terms measure how much the fact that kittens are actually the most common critter of the three influences what people think the most common critter is (or how big the critters' eyes are measured to be in the pictures influences how big people think they are). Now take α → 0 and look at our self-consistent theories above:

E(c) ≈ MC[E(c)]
E(e) ≈ BE[E(e)]

These are not the same theory! However, the market monetarist claim is effectively saying that the "Most Common" theory and the "Big Eyes" theory must be equivalent -- or else they're effectively advocating something that is pure opinion (taking α → 0 has decoupled our theory from the 'concrete steppes' of empirical data).

Update 27 Feb 2015:

The Keynesian beauty contest above also illustrates Noah Smith's contention of uninformative data being the reason we can't select between different theories in macro. We'd need to see a lot more critters to determine which of the "big eyes" or the "most common" theories were correct (or maybe neither of them).

Footnotes:

[1] What I write here is actually borrowed a lot from physics; E(x) is a quantum field, M[...] is essentially a path integral given a Lagrangian ('the theory') in an abuse of notation. So one would view E(x) = M[E(x)] as a matrix element/expectation value, in a self-consistent field approach.

[2] Sumner:
Unfortunately the role of expectations makes monetary economics much more complex, potentially introducing an “indeterminacy problem,” or what might better be called “solution multiplicity.”  A number of different future paths for the money supply can be associated with any given price level.  Alternatively, there are many different price levels (including infinity) that are consistent with any current money supply.
[3] I should add "as you decouple it from empirical data" to that quote. As α → 0 (or k or τ) the theory decouples from direct contact with empirical data. It is no longer about empirical data, but what you interpret markets (or important economic actors like the central bank) to think about empirical data.