Influences

Ok, this is entirely self-indulgent. I mentioned in a sentence at the bottom of the previous post where I got the favicon for the blog, but the ultimate source is my old TI-99 4A computer and the CALL CHAR instruction in TI BASIC. When I was a kid (about 8 or 9), this was the first time I connected data, numbers, and distributions of objects (e.g. black squares) (from here):

This entered into one of my early drawings (using penultimate) when I was thinking about information transfer (in this case, aggregate demand for money) to convey information flowing; the favicon is inside that red dashed box:

which I later used here. This was then used in a description of supply and demand as an allocation problem here and put into this figure of supply and demand "events" meeting as transaction "events":

Anyway, this gives a sense of the process I described in the previous post about one's vague intuitions and mental visualizations becoming more concrete for other people by using mathematics.

PS The people in the drawing above are a combination of XKCD and the people shaking hands on the cover of Radiohead's Meeting People is Easy:

Should I use math or just assume convergence of a vague algorithm?

Arnold Kling apparently doesn't think math is necessary (H/T Noah Smith):

Think of [recent Nobel winners Hart and Holmstrom's] work as consisting of three steps.

1. Identifying some real-world complexities that affect how businesses operate. ...

2. Construct a mathematical optimization model that incorporates such complexities.

3. Offer insights into designing appropriate compensation systems, including when to outsource an activity altogether.

...

In my view, step 2 is unnecessary.

Tell us what you really think:

But I do not think in terms of mathematical optimization. Instead, I think in terms of a dynamic process of trial and error. A manager tries an approach to compensation. As long as it seems to work, it persists. 

So exploring the compensation strategy space by trial and error, a manager determines if a given strategy meets his or her objective. Or another way, by randomly sampling the compensation domain and evaluating some objective function, the manager arrives at an optimal solution. Of course. That is not thinking in terms of mathematical optimization at all. It is completely different! Wait (wikipedia):

In evolutionary computation, differential evolution (DE) is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Such methods are commonly known as metaheuristics as they make few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. However, metaheuristics such as DE do not guarantee an optimal solution is ever found.

Joking aside, it's the last sentence that the math is good for. Like many optimization methods, there is no guarantee that an optimal solution can be found. Does Kling's "dynamic process of trial and error" converge to any solution? You can't just 'reckon' the location of the optimum and the path to it that can be discovered by agents. Sure, you can give the intuition of that process, but an explicit example using mathematical optimization gives me confidence in your eyeballed solution and assumed convergence.

[update + 30 min] The real function of math isn't to arrive at your intuition for a problem. It can for people who's minds work that way, but I generally visualize things in a way not entirely unlike Kling -- usually in pictures that only eventually become more rigorous like here or here or here (that last one is behind the sketch I turned into the favicon for this blog). The real function of math is to convince others who can't see your mental process (instead of Kling's manager, I saw a person exploring a diagram like the one at the top of this post), and to make sure your imagination dots the i's and crosses the t's (e.g. convergence or existence).

On volume and information

Random agents exploring the state space. 

But are we really satisfied to state that the existence of exchanges, and the fact that information percolates into prices via a series of trades, are facts only “explainable" by human folly, that would be absent in a more perfect (or perfectly-run) world?

John Cochrane

I think an excellent summary of the information equilibrium framework's take on prices and exchange is to answer "yes" to that question.

John Cochrane wrote an nice essay about financial market volume and information (what should really be called knowledge here, because it differs from information theory "information" in the sense that the knowledge people trade on is typically meaningful, and not just measured in bits). That's where I got that quote. The key issue is that, as Cochrane puts it:

Volume is The Great Unsolved Problem of Financial Economics. In our canonical models ... trading volume is essentially zero.

Cochrane then makes a really good analogy:

I gather quantum mechanics is off by 10 to the 120th power in the mass of empty space, which determines the fate of the universe. Volume is a puzzle of the same order, and importance, at least within our little universe.

Correct this to energy density because vacuum energy acts as an outward pressure (accelerating the expansion of the universe, contributing to the cosmological constant) while matter acts gravitationally, but overall this is a correct assessment. However, this is more of a back of the envelope calculation than hard core theory. Because we don't know the fundamental theory of gravity (strings, loops, triangles), we can only guess that it has something to do with the Planck length.

Therefore the scale that we should use to cut off our integrals is the only scale we know, i.e. the Planck length ℓ. And 1/ℓ⁴ [the 4 is for 4-D spacetime] is off by 120 orders of magnitude from the real value of basically zero. The takeaway is not that quantum mechanics is wrong, but that obviously the Planck length does not set the scale of the vacuum energy density. The fundamental theory of quantum gravity should tell us what the real scale we should use is — our naïve guess is wrong.

Getting back to the volume problem in finance, the assumption in finance is that the information (knowledge) content of trades should set the scale of volume in financial markets. Unfortunately, this tells us that volume should be zero if agents are rational.

The obvious conclusion is that the information (knowledge) content of trades does not set the scale of volume in financial markets. Stiglitz and Grossman (1980) set up a new scale — that of noise traders. In that case, the number of noise traders will set the scale of market volume.

If we look at random agents fully exploring the economic state space (maximum entropy), then simply the number of agents participating in the market will set the scale of market volume. Gary Becker wrote a paper about random agents, calling them "irrational" (see here for some slides I put together). I prefer to say they are so complex they appear random. Much like how a sufficiently advanced technology is indistinguishable from magic, sufficiently complex behavior should be indistinguishable from randomness. (Note that this is actually a good definition of algorithmic randomness.)

This is not terribly different from the "noise trader" concept, except it makes no assumptions about individual behavior. I may be irrationally selling Apple stock if you just look at it as a number (observed in the market), but maybe I need to pay medical bills (not observed in the market). But it does make sense of a few of the stylized facts that Cochrane mentions. For example, rising markets are associated with rising volume. This would make sense because more people are drawn into the market when the indices are rising (it doesn't matter the details of the source of the boom). It also makes sense of that great contraindicator: when random people in your neighborhood seem to be talking about markets, it's time to get out.

The thing is that uncorrelated agents fully exploring the state space (maximum entropy, or information equilibrium) does seem to be consistent with informationally efficient markets — at least until there is some kind of correlation (e.g. panic and non-ideal information transfer).

I want to comment on a couple of things Cochrane says towards the end:

Behind the no-trade theorem lies a classic view of information — there are 52 cards in the deck, you have three up and two down, I infer probabilities, and so forth.

Well, this is a classic game theory view of information mixed with information theory information. The three cards face up have zero information entropy, but would contribute to perfect information in game theory (probably a good example to see the difference).

But I did like this:

For a puzzle this big, and this intractable, I think we will end up needing new models of information itself.

Maybe information equilibrium?

...

PS There is also this post on rate distortion and Christopher Sims I wrote last month that is relevant. The limited number of bits of information getting through tell us the vast majority of trades are "entropy trades" (a better term than noise trades) by random agents.

PPS + 3 hrs As a side note, the basic asset pricing can be expressed as a maximum entropy condition.

I am not sure Steve Keen understands the arguments against his framework

So the simple argument against Steve Keen's Forbes piece is that there is insufficient data to tell the difference between a nonlinear model and a linear model with stochastic shocks. Keen claims that this paper from Kocherlakota is somehow a response to Roger Farmer's simple argument:

Thank you again Narayana for saving me effort of tedious rejoinder Remarkable refusal by others to consider fundamental failings of paradigm https://t.co/arN6luCE3K

— Steve Keen (@ProfSteveKeen) October 7, 2016

Ironically, I think it is Keen who is showing remarkable refusal to consider fundamental failings of his paradigm.

Keen's Forbes piece is also not about the failings of the mainstream DSGE paradigm (I have a discussion of the piece here). He simply asserts it has failings in the first couple paragraphs and then continues on to the Lorenz attractor. Actually, his main argument seems to be that general equilibrium is "no longer necessary" and that nonlinear dynamics has arisen to act as a new starting point in the sense that the use of slide rules is "no longer necessary" because we have computers now.

If this is his argument, then he should accept that machine learning and big data have arisen in the 21st century and can act as a new starting point. In fact, Keen's argument for his paradigm would be like if I started using the fact that information theory was developed in the 20th century and has many uses across a variety of scientific disciplines as the only motivation for using the information transfer framework. [Actually, information theory is is useful in understanding chaotic systems (which can also be connected to neuroscience, see here or here), and is useful [pdf] in empirical application of Takens theorem as I discuss below.]

Kocherlakota's paper

Kocherlakota's paper constructs a case where a worse fit to existing data can potentially give better policy advice:

This paper uses an example to show that a model that fits the available data perfectly may provide worse answers to policy questions than an alternative, imperfectly fitting model. ... [the author] urges the use of priors that are obtained from explicit auxiliary information, not from the desire to obtain identification.

...

In this paper, I demonstrate that the principle of fit does not always work. I construct a simple example economy that I treat as if it were the true world. In this economy, I consider an investigator who wants to answer a policy question of interest and estimates two models to do so. I show that model 1, which has a perfect fit to the available data, may actually provide worse answers than model 2, which has an imperfect fit.

But it is important to remember where the motivating information for this worse fitting model is coming from -- microeconomics: 

There is an important lesson for the analysis of monetary policy. Simply adding shocks to models in order to make them fit the data better should not improve our confidence in those models’ predictions for the impact of policy changes. Instead, we need to find ways to improve our information about the models’ key parameters (for example, the costs and the frequency of price adjustments). It is possible that this improved information may come from estimation of model parameters using macroeconomic data. However, as we have seen, this kind of estimation is only useful if we have reliable a priori evidence about the shock processes. My own belief is that this kind of a priori evidence is unlikely to be available. Then, auxiliary data sources, such as the microeconometric evidence set forth by Bils and Klenow (2004), will serve as our best source of reliable information about the key parameters in monetary models.

Emphasis mine. Kocherlakota wants the evidence to come from microeconomics. As I said in my piece:

Basically, while there isn't enough data to assert the macroeconomy is a complex nonlinear system, if Keen were to develop a convincing microfoundation for his nonlinear models that get some aspects of human behavior correct [ed. i.e. "microeconometric evidence"], that would go a long way towards making a convincing case.

Kocherlakota is saying the exact opposite thing that Keen says in his Forbes piece:

The obvious implication for economists is that macroeconomics is not applied microeconomics. So what is it, and how can economists do it, if they can’t start from microfoundations? It’s macroeconomics, and it can be built right from the core definitions of macroeconomics itself.

While I have no problem with this idea, I do have a problem with Keen claiming Kocherlakota's paper is any kind of rejoinder defending Keen's approach from Farmer's blog post -- that you can't tell the difference between nonlinear dynamics and stochastic linear dynamics.

But the elephant in the room here isn't that we're comparing between Kocherlakota's model that kind of fits the existing data and his other model that fits the existing data better. In fact, DSGE models are actually quite good at describing existing data [e.g. pdf]  -- they are just poor forecasters. However Keen's models do not remotely look like the data at all. Take this paper [pdf] for instance. It says real growth looks like this:

As far as I can tell, those are years, so that over twenty years the data should look like that. But real growth data looks like this:

And Keen's paper is called "Solving the Paradox of Monetary Profits"; however, if your theoretical model looks nothing like the data, then I'm not sure how you can assert you are solving anything. And if your model looks nothing like the data, I'm not sure how a paper that has two models where one perfectly fits the data and the second fits a little worse is relevant to your argument.

This is the part I am most troubled by. The main problem with economics is that it doesn't reject models, or it creates models that are too complex to be rejected. In both cases, it is theory ignoring the empirical data. Keen's attempted break with mainstream economics makes exactly the same mistakes. People call it a heterodox approach, but when it comes to the data Keen is just another orthodox economist.

Takens theorem

Ian Wright made a great contribution to this conversation by citing Takens theorem:

@ProfSteveKeen Takens' theorem might be relevant to your spat with Farmer; e.g.https://t.co/HckCQj4sLW

— Ian Wright (@ianpaulwright) October 7, 2016

Here is a good description on YouTube

Taken's theorem can be used to capture important properties of a nonlinear (chaotic) system by letting us use lagged data to visualize the low dimensional subspace. You can e.g. create a "shadow manifold" of the original low-dimensional subspace by looking at data at different times. For example, if unemployment was part of a nonlinear dynamical system then unemployment at times t - n τ (for some τ)can be used to construct a shadow S of the original manifold:

S = {U(t), U(t - τ), U(t - 2 τ), U(t - 3 τ), ...}

One interesting application is that you can use this to de-noise data coming in from nonlinear dynamical systems because assuming you are going to be close to the the low-dimensional (shadow) subspace lets you filter out high dimensional noise so that if the observed U*(t + τ) is off of the manifold, you can take U*(t + τ) = U'(t + τ) + n where U'(t + τ) is the closest point on the manifold (that you've found from earlier observations). It's kind of like measuring the distance to the moon and using knowledge of the moon's orbit to remove the noise in your measurement.

Keen's models (e.g. from the paper linked above) have these limit cycles:

So does Takens theorem tell us anything from the data? Well, I looked at unemployment (the most likely candidate in my view to exhibit this behavior with regard to its lags). However I tried many different values of τ from a year to 10 years and did not see any sign of a low dimensional subspace of 3 dimensions:

And here's a smoothed version:

It's true that you might have to go to higher dimensions to see something, but that's exactly the issue Farmer points out -- in higher dimensions, you need more and more data to definitively say there is a low dimensional subspace. As it is, there is not a short enough limit cycle to show up in the post-war data, and won't be for a many more years (at this rate, it looks like hundreds of years of data would be needed).

Regardless, Keen should be able to show some kind of limit cycle behavior in the data -- you'd imagine that he'd definitely advertise it if he had a good graph. The fact that he doesn't means it probably doesn't exist.

Summary

We are still left with a major argument against Keen's framework that he hasn't addressed: there is insufficient data available to definitively select it over another one, and there won't be sufficient data for years to come.

That said, there are still things Keen could do to motivate acceptance of his framework:

1. Show that it fits empirical data (just basic curve fitting)
2. Show that empirical data is not inconsistent with nonlinear dynamical systems (via Takens theorem)
3. Show that microeconomics (theory or data) leads to his models

He is not doing any of these things. In fact:

1. He never shows theoretical curves going though data
2. He probably hadn't thought of using Takens theorem
3. He thinks microeconomics is unnecessary

It is not "others", but rather Keen who's demonstrating "remarkable refusal ... to consider failings of the paradigm" (I'll not that lots of economists have considered the failings of the DSGE paradigm). This lacks what Paul Romer called Feynman integrity, which I will leave up to Feynman himself to describe:

It’s a kind of scientific integrity, a principle of scientific thought that corresponds to a kind of utter honesty–a kind of leaning over backwards. For example, if you’re doing an experiment, you should report everything that you think might make it invalid–not only what you think is right about it: other causes that could possibly explain your results; and things you thought of that you’ve eliminated by some other experiment, and how they worked–to make sure the other fellow can tell they have been eliminated. 

Details that could throw doubt on your interpretation must be given, if you know them. You must do the best you can–if you know anything at all wrong, or possibly wrong–to explain it. If you make a theory, for example, and advertise it, or put it out, then you must also put down all the facts that disagree with it, as well as those that agree with it. There is also a more subtle problem. When you have put a lot of ideas together to make an elaborate theory, you want to make sure, when explaining what it fits, that those things it fits are not just the things that gave you the idea for the theory; but that the finished theory makes something else come out right, in addition.

Price growth (i.e. inflation) state distribution

I have started looking at some price data from Cavallo 2016 (related to MIT's billion price project) in order to make a better version of the graph at the bottom of this post: an animation of the distribution of CPI changes among the components of CPI. Effectively, I want to try and visualize the equilibrium price change distribution (the price state space) of individual prices. This is also some relevant background reading (hitting on zero price changes) -- I claim that although individual prices are not "sticky", there is an equilibrium distribution (a statistical equilibrium per Gregor Semieniuk).

Unfortunately, the files are really huge and my six year old desktop is choking on them; I'm in the process of finding a workaround. I managed to look at part of the data (it is price data from supermarkets from 05/2008 to 07/2010) and it largely shows a stable bimodal distribution (i.e. unchanging, not the specific distribution called a "stable distribution") of non-zero price changes.

In talking about "equilibrium" (e.g. here), we can think of the these price changes existing in an equilibrium distribution of price changes while individual prices are changing (similar to atoms being in an equilibrium velocity distribution even though each atom is changing its speed with every collision).

If you've looked at the animation, you might have noticed a big change in April of 2010. I am not sure if that is an artifact of the way that I grabbed the data (effectively the first million entries in a CSV file) or a real event where over 60% of prices went up by about 10%. The major world events include a downgrade in the course of the Greek debt crisis and the Deepwater Horizon oil spill, but these come near the end of the month. Does anyone else know of an event in April of 2010 that would affect supermarket prices that could do this to the distribution:

It is possible that since I only have part of the data, it could be that the data I am looking at consists of just one store that decided to raise their prices by 10%. Anyway, I'm still investigating. Besides this single month of data, this seems to be a qualitative success of the "statistical equilibrium" approach.

The monetary base moves

There has been a big drop in the monetary base over the past two weeks:

I still don't know how I feel about the slow progress toward the information equilibrium in this model. Maybe it will be rejected.

Dynamic unemployment equilibrium (and sunspots)

I thought I'd add a bit more to a series of posts about defining a "dynamic" equilibrium in unemployment where we don't look at a natural level of unemployment U*, but rather a natural rate of decrease (dU/dt)*. This doesn't necessarily have anything to do with information equilibrium (except to say it looks like the US economy is usually in this particular unemployment equilibrium), but rather a particular parameterization of the data. The first five posts are:

1. Remarkable recovery regularity and other observations This is the first time I noted the regularity in dU/dt.
2. Unemployment equilibrium? Here I minimized an objective function based on entropy to find the best slope (dU/dt)*.
3. Did the ACA decrease unemployment? Here I used the parameterization of the data to motivate an interpretation of the data where the ACA (aka Obamacare) caused unemployment to fall faster than normal.
4. Eurozone unemployment equilibrium Here I showed the parameterization applied to the Eurozone.
5. Defining recessions? Here I used the parameterization to define the center of recessions and note that the definition matches (within about a month) the NBER midpoints. [Also added a bit on predicting recessions and Obamacare as a positive shock to employment on 7 Oct 2016.]

This post just adds a couple of other metrics. I show the deviation between the data and the model in the first graph, as well as how the derivative of the parameterization compares (red) to the derivative of the data (gray) in the second graph -- thereby showing the "recessions" as spikes, with the economy remaining in this "dynamic unemployment equilibrium" most of the time.

The second graph makes me think of sunspots in the solar cycle (and would lend itself to Steve Keen's ideas about nonlinear dynamical systems if only we had longer periods of data and or some kind of microfoundations).

Update 30 December 2016

Here's a post looking at multiple equilibria, and here's another post where I try to use this "model" to do a bit of forecasting.

Keen, chaos, and equilibrium

A brief back and forth on Twitter about Roger Farmer's very concise and critical response to Steve Keen quickly degenerated into squabbling among Keen, Noah Smith, and David Andolfatto. Here is the crux of Farmer's critique:

... are economic systems [described by chaotic nonlinear dynamics]? The answer is: we have no way of knowing given current data limitations. Physicists can generate potentially infinite amounts of data by experiment. Macroeconomists have a few hundred data points at most. ... Where does that leave non-linear theory and chaos theory in economics? Is the economic world chaotic? Perhaps. But there is currently not enough data to tell a low dimensional chaotic system apart from a linear model hit by random shocks. Until we have better data, Occam’s razor argues for the linear stochastic model.

Actually, as a physicist, I would say that even if the economy was a complex nonlinear chaotic system, linear stochastic models would still be its effective theory description. Regardless of what the quantum theory of gravity is, general relativity -- and even Newton's universal law of gravitation -- is still its long-distance effective theory.

Anyway, this prompted me to write something about Steve Keen's article in Forbes. Keen suffers from a problem that all public economists seem to suffer: asserting matters of opinion as matters of fact, and ongoing research programs as well-established frameworks. This will be made clear as we progress. Let's begin, shall we?

Equilibrium, but what kind?

I propose to ban the discussion of 'equilibrium' in economics unless it is accompanied by an adjective describing it. Olivier Blanchard says that "Macroeconomics is about general equilibrium ... ". General equilibrium (which doesn't violate my new rule) is the idea that there is a price vector that eliminates excess supply and excess demand -- and that an economy in general equilibrium has neither. As Keen notes, this has been expanded from referring to excess supply and demand at one point in time to an intertemporal version that sees e.g. the market for blueberries in spring of next year as a separate market from currently available blueberries.

This idea encapsulates two hundred years of economic wisdom. It passes cursory inspection as not to be entirely false. If I go to the grocery store, it is not overflowing with blueberries -- or if it is, they are on sale. I frequently like to post this graph as a joke:

It's 1-U where U is the unemployment rate. General equilibrium tells us there should be no excess supply of labor, i.e. that 1 - U ≈ 1. Presently 1 - U = 0.951. This physicist would say that's a jolly good first order approximation! Even at the height of a worst recessions in the US, 1 - U ~ 0.8 to 0.9. And this holds over multiple time periods. Another way of looking at unemployment equilibrium is as a dynamic equilibrium where dU/dt is constant over long periods of time punctuated by recessions.

The information equilibrium framework couches this idea of general equilibrium in terms of the information content of events drawn from the probability distributions of supply and demand -- but notes that it appears to hold only approximately in empirical data.

However, Keen wants to throw this out and assert (without evidence) "[m]acroeconomics is about complexity". I am of the opinion that asserting complexity in macroeconomics is the equivalent of Krugman's "very serious people" asserting whatever conventional wisdom they're asserting. Heads nod, and chins are stroked. Yes, complexity is important. (But what kind?) Keen says:

In the mid-20th century, other modelling disciplines developed the concept of “complex systems”, along with the mathematical and computing techniques needed to handle them. These developments led them to the realisation that these systems were normally never in equilibrium — but they were nonetheless general models of their relevant fields. ... Economics needs to embrace the reality that, even more so than the weather, the economy is a complex system, and it is never in equilibrium.

Emphasis in the original. There is no evidence that this approach actually captures economic phenomena, and in graphs later in the article Keen shows things that look nothing like an actual economy (which Noah Smith points out in the Twitter conversation linked above, saying "This oscillatory employment picture hardly matches anything in the actual economy" and "This picture also fails to match reality pretty starkly" with accompanying graphs). And even if it was the case that these graphs looked even remotely like a real economy, Keen's use of 'equilibrium' in the previous quote is completely wrong.

David Andolfatto asks of Keen's use of the word equilibrium: "Reading the first page of Keen's reply to [Blanchard], it seems he's confusing 'equilibrium' with 'a state of rest?'" to which Farmer replies "I'll let [Keen] answer that. I suspect he does understand that equilibria can be non-stationary". Keen does kind of reply by referring to "dynamic equilibrium", but in his article he does use it to mean 'state of rest', or at least a stationary point -- which he points out as the center points in the Lorenz model. This is where I started pulling out my hair.

Global weather is a system that isn't in global thermodynamic equilibrium at maximum entropy. Full stop. That's what is meant by "never in equilibrium". The Earth is differentially heated and has two different fluids (air and water) that can move this heat around. However, it is in local thermodynamic equilibrium -- because otherwise the idea that it will be 60 °F in Seattle today would not make any sense. And to a good approximation, it is always in local thermodynamic equilibrium. And if any of you have ever been outside on a nice day, you can see that the weather can be in local thermodynamic equilibrium over a fairly wide area. There are some places where the wind seems to blow constantly, but it's only because the Earth is so big and we are so small that the tiny pressure imbalances can knock houses down. [And generally, the Earth's atmosphere well described by a mechanical pressure equilibrium.] But we only say the Earth's weather is not in global thermodynamic equilibrium because we developed a concept of thermodynamic equilibrium that is relevant to the microfoundations of the weather system, i.e. physics.

Let me illustrate these points using a weather example. Here is the barometric pressure for Seattle for 2016 so far:

You can actually see the complex emergent structures (high/low pressure systems) passing through with a quasi-cyclic frequency as well as the "great moderation" that is summer in the Pacific Northwest. But we only know this because we've studied weather from thousands of weather stations all over the Earth and we understand the basic physics (thermodynamics) involved.

Now pretend that this is the only data we have [1]. One city with approximately hourly measurements over part of a year -- about 8000 data points. And we don't have a well-established theory of physics underlying it all. Can we really tell the difference between the complex nonlinear system of pressure systems and seasons (including the entire theory of physics that entails) ... and a pressure equilibrium of about 1013 mbar subjected to stochastic shocks? No, and that is Farmer's point. My joke graph in this case looks like this:

[The three lines are the average pressure at sea level, alongside the maximum and minimum pressures ever measured.]

[Update 11 Oct 2016] I hope it wasn't lost because I wasn't explicit with what I meant by confounding the nonlinear dynamics of pressure systems and the linear stochastic view. In one case, if you view the Earth as a nonlinear dynamical system (which you should), you can see high and low pressure systems in the pressure time series data being generated and moved by differential heating, coriolis forces, etc. Seattle can be affected by Canadian highs and Pacific lows coming from various directions (at various speeds).

However, if you only have data from Seattle, then a lot of the information about speed and direction of a particular pressure system is lost. Two "shocks" that are farther apart and different amplitudes because the low pressure system moved slower and passed to the south will be exactly equivalent to two shocks with random timing and amplitude (e.g. an AR process). [end update]

Whither microeconomics?

There is one piece of Keen's article that I half agree with. He quotes physicist Philip Anderson:

As [Philip Anderson] put it “Psychology is not applied biology, nor is biology applied chemistry”. The obvious implication for economists is that macroeconomics is not applied microeconomics.

I completely agree that we should not assume that macroeconomics cannot be studied as a field unto itself, separate from microeconomics, as the microfoundations purists would have us believe. But I also don't believe that this means we should assume macroeconomics is completely independent of microeconomics. In fact, I think Keen should pay more attention to microfoundations. The understanding of weather systems Keen points to as an example to be followed critically depends on understanding the underlying physics (the microfoundations) -- and given the physics, even with the paucity of data in the weather example above, we could accept the complex nonlinear system explanation.

Basically, while there isn't enough data to assert the macroeconomy is a complex nonlinear system, if Keen were to develop a convincing microfoundation for his nonlinear models that get some aspects of human behavior correct, that would go a long way towards making a convincing case. Keen wants to have it both ways: to be unhindered by the constraints of microfoundations and to have insufficient data to reject his models.

Chaos!

Keen: I didn't even use the word 'chaos'!

Andolfatto: Steve, you used 'complexity' ... by which we took be nonlinear dynamics. Don't split hairs.

Keen's article also contained several paragraphs with references to the most famous chaotic system resulting from a simple system of differential equations.

Three-equation Monte

Keen:

Let me illustrate this with a macroeconomic model which in essence is even simpler than Lorenz’s, since it can be derived from three macroeconomic definitions that no macroeconomist can dispute:

• The employment rate is the ratio of the number of people with a job to the total population;

• The wages share of output is the total wage bill divided by GDP; and

• The private debt ratio is the ratio of private debt to GDP

Yes, those are three definitions. Keen continues:

Differentiate those three definitions with respect to time. That generates the following three statements ... which, like the definitions themselves, have to be accepted by all macroeconomists simply because they are true by definition:

• The employment rate will rise if economic growth exceeds the sum of population and labour productivity growth;

• The wages share of output will rise if wages rise faster than labour productivity; and

• The debt ratio will rise if debt grows faster than GDP

You mean differentiation and sneaking in the concept of labor productivity.

Bravery

In the end, Keen thinks macroeconomics is complex, calls general equilibrium primitive, and makes a case for a research program based on nonlinear dynamics. He presents the first two opinions as facts. As Farmer points out, there is insufficient data to say macroeconomic cycles (of which there have only been a dozen or so in the post-war period from which there is quality data) are the result of nonlinear dynamics or a linear model subject to shocks -- therefore declaring the cycles to be evidence of nonlinear dynamics is an opinion.

His case for nonlinear dynamics as a framework (no doubt using his Minksy software) is not convincing (to me, but that is just my opinion). If it was just being presented as a research program, that would be fine. But it isn't:

This is not a complete model either of course ... But it is a lot closer to reality than DSGE models were before the crisis, and much more realistic than they can ever hope to be in the future, because they cling to a false modelling choice that forces equilibrium onto a manifestly non-equilibrium system.

So the road to a better, more realistic macroeconomics begins with a step away from the equilibrium and micro-foundations pillars on which it has to date been constructed. Will economists be brave enough to take that step? Only time will tell.

The thing is that this statement comes right after this graph [2]:

This is supposed to characterize this data (from the US):

Closer to reality? More realistic than DSGE models? More realistic macroeconomics? Here's a DSGE forecast of inflation for 2014 to 2019 (red):

You would have to be brave to abandon that DSGE model for Keen's graph!

...

Update 6 October 2016

Narayana Kocherlakota follows up with a piece that is more lenient with regard to Farmer's point about it being difficult to distinguish between nonlinear models and linear ones with stochastic shocks (he makes a few other points as well and his piece is well worth reading). I did like his description of the process for how the mainstream stays the mainstream:

1. Mainstream consists of a class of models C.
2. Someone proposes new class of models C’.  It is clear to most that the answers to many questions of interest will be different if C’ is true rather than C.
3. It’s argued (usually by a theorist!) that we can’t readily distinguish C’ from C using existing data.
4. One response to (3) is: let’s figure out, and undertake, a class of (possibly very expensive!) experiments that would allow us to distinguish C’ from C.  I don’t know much about the hard sciences but this response seems common there. 
5. The standard approach to (3) in macro is: let’s abandon C’ and keep C.  Since (3) is a common issue, C tends to remain the mainstream.

The best argument in favor of Keen's proposed C' comes from a place of fairness because C (the DSGE paradigm) isn't really preferred by the data either. We can give the horses a try after all the King's men have had a go.

However in all sciences it should be the power to explain the data that is the ultimate arbiter. The issue is not just that Keen's C' will be indistinguishable from C given the existing data, but that his C' does not appear to explain any existing data that I've seen. If it was such a good model you'd think there'd be some theoretical curves going through some data that are easy to find on the internet, but there aren't any. The only one I've come up with is my own graph! Happy to be corrected.

Additionally, Kocherlakota seems to suggest that there could be data we could obtain now to distinguish C from C'. However the issue is that we need hundreds of years of data to nail down a difference between nonlinear cycles and linear models with stochastic shocks without established microfoundations (with established microfoundations, you wouldn't need as much data).

I should emphasize that I have no problem with anyone studying any given C' -- maybe there will be a breakthrough. I actually think Keen's nonlinear equations could be used to constrain DSGE models (i.e. a specific DSGE model is a perturbative expansion of the nonlinear theory near a particular point in the "dynamic equilibrium"). It is possible that Keen's C' could establish relationships between the different coefficients in models in C that could be rejected given the data. But his attitude is one of disdain, so I doubt he'll try to write out a linear approximation to his model in DSGE form [3].

But we should always consider whether model explains the empirical data first.

...

Footnotes:

[1] Actually, this is not a bad proxy! Look at US NGDP growth data:

[2] Also note the time scale in this graph: 100 years. We don't have 100 years of decent data yet.

[3]  Note that I go out of my way to connect to mainstream economic theory: here or here for DSGE specifically. That's the burden you have bringing something from outside the mainstream. Keen doesn't seem to want to do the dirty work of showing how his models compare and contrast to DSGE theoretically. As an aside, this seems to be a common failure of outside the mainstream approaches -- no one wants to do the work of learning the mainstream because their approach is oh so much better and we can't dirty it by bringing it to the level of the mainstream. Sean Carroll has a good summary of this approach you should take. My point here is his point #2.

Sticky "information", i.e. knowledge, and emergence

I spent some more time pondering an paper Noah Smith tweeted out the other day (and Tom Brown also retweeted to me) and wrote a blog post about a bit later (see his post for all the background). The article and Noah both emphasize "emergence", so of course it is of interest to this blog. However I am not sure that is an appropriate interpretation.

First, let's see if I can understand the paper itself. I am going to make a vocabulary change however -- let's use the word knowledge instead of information because this has little to do with information theory.

With that vocabulary change, the paper makes several points:

1. Sticky knowledge (information) and noisy knowledge (information) models lead to (equivalent) dependence of the forecast error on forecast revisions FE = β FR + n where β = β(λ) (knowledge  stickiness) or β = β(G) (noisy knowledge Kalman gain) and n is a noise term.
2. Actual forecast errors from a variety of sources indicate β ~ 1, meaning λ ~ 0.5 or G ~ 0.5.
3. Additional tests to show this isn't due to some other factors, and applies across consumers, professionals, academics, and internationally.
4. Forecasts can be useful estimates of non-rational expectations
5. Non-rational expectations can "emerge" from aggregating rational agents

Let me set up a toy model version of what is happening here. (Aside: this is a specific example where an unrealistic model is useful for understanding a result about the real world.) Let's take our observable O (inflation, growth rate, whatever) to be a Wiener process (Brownian motion) with zero drift. In this case the "rational expectations" forecast (REF) for any future price is the present price (i.e. the EMH for a random walk). If we look at forecasts that are updated every integer unit of time T = n, our sticky knowledge forecast (SKF) with stickiness λ for time T = 3 will be

SKF(3, T=0) = (1 - λ) REF(0, T = 3) + λ REF(-1, T = 3) at time T = 0
SKF(3, T=1) = (1 - λ) REF(1, T = 3) + λ REF(0, T = 3) at time T = 1
SKF(3, T=2) = (1 - λ) REF(2, T = 3) + λ REF(1, T = 3) at time T = 2
etc

Think of 1-λ as a probability of an individual agent updating their forecast. Also note that

REF(i, T = i + k) = O(i)

so this simplifies a bit

SKF(3, T=0) = (1 - λ) O(0) + λ O(-1)
SKF(3, T=1) = (1 - λ) O(1) + λ O(0)
SKF(3, T=2) = (1 - λ) O(2) + λ O(1)
etc

We then look at

FR = SKF(3, T=i) - SKF(3, T=i-1)

versus

FE = O(3) - SKF(3, T=i)

i.e. forecast revisions versus forecast error. For λ = 0, we get the rational expectations result --  the forecast error and the forecast revisions are uncorrelated:

For λ = 0.5, we get correlation:

In the above pictures, I show a "typical" path alongside several hundred simulation results. Effectively, the SKF(t, T = t-k) is a biased estimator of the future value of O(t) while O(t-k) = REF(t, T = t-k) is an unbiased estimator.

This pretty much covers points 1-4 above. I would like to add that sticky knowledge ("information") models can be interpreted as agents ignoring most outside information, making them uncorrelated. Agents that all updated their forecasts at the same time to exactly the same data would end up very correlated -- which leads to a failure of information equilibrium (which is bad and probably behind recessions).

Now let's talk emergence (point 5). It's a bit easier to explain in the case of sticky "information". At time t, individual agents either forecast REF(t-1, T = t+k) or REF(t, T = t+k) -- which are both "rational" expectations (one is fresh and one is stale) and unbiased estimators of O(t+k). The average agent, however, forecasts

λ REF(t, T = t+k) + (1 - λ) REF(t-1, T = t+k)

But it seems to me a bit cheeky to say the bias of forecast revisions is really emergent. Individual agents either revise or don't revise, so it doesn't make sense to talk about the bias of forecast revisions for agents that don't revise. It's not a revision of zero, it's no revision. The concept of forecast revisions is itself somewhat emergent -- you are averaging a set of revision numbers with a bunch of NaNs you take to be zero.

Additionally, this exact same result can be obtained if we think of λ as a weight instead of a probability and apply it to every agent.

But the biggest problem is that a stale (i.e. not updated with probability 1 - λ) rational expectation from time t - 1 at time t isn't technically a rational expectation at time t. The aggregating process for a forecast at time t is aggregating λ N agents with rational expectations and  (1 - λ) N agents with non-rational (i.e. stale) expectations. The aggregation of rational agents and non-rational agents being non-rational isn't really "emergence". It's kind of like saying an average over the whole numbers {0, 0, 0, 0, 0, 1, 1, 1, 1, 1} isn't 0 and the average 0.5 (a fraction) is "emergent". I'd call that a stretch.

It's not the worst thing to do in a paper and sometimes a little flair is good to draw attention. As there's no hard and fast definition of "emergence", so there's no technical flaw with what the authors say.

...

PS As a side note, the models presented in the paper fail for "complete stickiness" λ = 1 (or Kalman gain G = 0). In that case the coefficient of the forecast revision is singular, which should make sense to us all: if we never update our forecast with new knowledge, then how could you have revisions? This creates a "scope" for these models: λ < 1. This means agents must update their forecasts with rational expectations at some point. If agents never update their forecasts with true rational expectations, then these results don't apply to the real world -- even though the data appears to be described by the sticky knowledge (information) model!

Invariance under inversion

Commenter M left a comment that made me look at the symmetry principles that leave the information equilibrium (IE) condition:

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

invariant. This may seem like an esoteric point, but in physics it is an incredibly powerful principle that in fact leads to everything we know. I could say special relativity ... or I could say Poincare invariance.  Quantum electrodynamics is basically a Poincare invariant theory with a local $U(1)$ gauge group symmetry. The standard model (core theory) is the gauge group $SU(3) \times SU(2) \times U(1)$ plus general covariance (general relativity).

Anyway, I've previously noted the scale invariance of the IE condition where $A \rightarrow \alpha A$ leaves the equation unchanged. This manifests in the real life invariance to changing the number of zeros on the price of something in different currencies (e.g. about 100 Yen versus 1 dollar versus about 6-7 Francs when they existed). We could suddenly talk about everything in "cents" in the US and it shouldn't change the economic theory. In a sense, this scale invariance is related to the homogeneity of degree zero of supply and demand functions in economics (which Bennett McCallum relates to long run neutrality and the quantity theory of money).

I have a bad habit of referring to this scale invariance as a conformal symmetry (due to dealing with approximately conformal field theories like QCD in the large N expansion where quantum corrections are suppressed [see here]). Scale invariance is just a subset of conformal symmetry. But M's comment made me see if the IE condition was invariant under more of the conformal symmetry group.

It turns out it is, and as you could probably tell from the title, it is invariant under inversion where $A \rightarrow 1/A$ and $B \rightarrow 1/B$:

$$
\begin{align}
\frac{d(1/A)}{d(1/B)} & = k \; \frac{1/A}{1/B}\\
\frac{-1/A^{2}}{-1/B^{2}}\; \frac{dA}{dB} & = k \; \frac{B}{A}\\
\frac{dA}{dB} & = k \; \frac{B}{A} \;\frac{A^{2}}{B^{2}}\\
\frac{dA}{dB} & = k \; \frac{A}{B}
\end{align}
$$

This may end up being more consequential than the scale invariance, but other than T-duality, I can't think of an immediate example of relevance in physics. In fact, see this Wikipedia paragraph on inversion symmetry. What this symmetry says is that we could talk about 1/NGDP (one over aggregate demand) or 1/L where L is the labor supply and that would be the same theory (with the same, not inverted, price). Also, zero and infinity are essentially the same thing. The equation isn't invariant under translations ($A \rightarrow A + c$), so it won't be invariant under special conformal transformations.

In any case, this give me something to ponder.

...

Update 4 October 2016

Commenter M expanded this to transformations of the form

$$
\begin{align}
A & \rightarrow \alpha A^{\gamma}\\
B & \rightarrow \beta B^{\gamma}
\end{align}
$$

with $\alpha$, $\beta$, and $\gamma$ nonzero. Inversion is the special case of $\gamma = -1$.