Tuesday, October 25, 2016

Parsing the macrohistory database: principal components

In my continuing adventures in parsing the macrohistory database [0], I noticed that most of the interest rate data looks a bit like the US interest rate time series. So I did a principal component analysis (specifically a Karhunen-Loeve decomposition).

It turns out there are three basic features of world interest rates in the macrohistory database: the US, Japan, and Germany. Here are the component vectors:

You can see the first three components are really the only ones with structure -- and they match up with the US, Germany (really only the Wiemar hyperinflation), and Japan:

As a side note, the US and Japan are the countries for which the interest rate model works the best. Additionally, if you leave out Germany [1], the US and Japan describe most of the variation of the data (i.e. the German component is mostly used to describe the Weimar hyperinflation in the German data itself).

Another way to put this is that most of the variation in the interest rate data in the macrohistory database is explained by the US data. This makes some sense: the US has been the largest economy for most of the time that the interesting variation in interest rates has happened (post-WWII). Before WWII, interest rates are relatively constant across with the low rates in the Great Depression being the only major variation [2]. Since the US interest rate captures most of the variation in the interest rate data, that means the US IT model explains most of the interest rate data -- since it captures most of the US interest rate data.



[0] Òscar Jordà, Moritz Schularick, and Alan M. Taylor. 2017. “Macrofinancial History and the New Business Cycle Facts.” in NBER Macroeconomics Annual 2016, volume 31, edited by Martin Eichenbaum and Jonathan A. Parker. Chicago: University of Chicago Press.

[1] Here is that analysis:

[2] Here is that analysis:

Monday, October 24, 2016

Parsing the macrohistory database: interest rates

Update 25 October 2016: check out this follow up using principal component analysis.

I was directed to this long time series macro data [1] by a reader via email, and so thought I'd try the IT interest rate model:

log(r) = a log(NGDP/MB) + b

Only a few series fit over the entire time period with one set of parameters {a, b}, but this should be expected because e.g. several European countries are now on the Euro. So I attempted to look for a natural way to break up the data using cluster analysis to find linear segments -- finding clusters associated by a given range of years, log(r) and log(NGDP/M0). Here are the clusters for the USA and DEU:

Note these are 2D representations of a 3D graph (imagine time going into the page).

I'm still looking at the fit methodology. Some of the data doesn't need to be broken up (particularly the Anglophone countries: USA, CAN, AUS, and GBR -- though I broke up USA and AUS in the graphs below). Many of the European countries do -- and the best place to break tends to be around World War II. Some preliminary results are listed below:

NOR and DEU (Norway and Germany) don't work very well unless you break up the data into many segments (about 4-5).

Nearly all of the countries that don't work well are currently on the Euro, or had pegged their currencies to some other currency at some point.

Anyway, this is a work in progress. I am going to try and map the break points to details of monetary history of the various countries.



[1] Òscar Jordà, Moritz Schularick, and Alan M. Taylor. 2017. “Macrofinancial History and the New Business Cycle Facts.” in NBER Macroeconomics Annual 2016, volume 31, edited by Martin Eichenbaum and Jonathan A. Parker. Chicago: University of Chicago Press.

Wednesday, October 19, 2016

Forecasting: IT versus all comers

On Twitter, John Handley asked about forecasting inflation with a constant 2% inflation as well as a simple AR process in response to my post about how the information transfer (IT) model holds its own against the Smets-Wouters DSGE model (see this link for the background). I responded on Twitter, but I put together a more detailed response here.

I compared the original four models (DSGE, BVAR, FOMC Greenbook, and the IT model) with four more: constant 2% (annual) inflation, an AR(1) process, and two LOESS smoothings of the data. The latter two aren't actually forecasts -- they're the actual data, just smoothed. I chose these because I wanted to see what the best possible model would achieve depending on how much "noise" you believe the data contains. I chose the smoothing parameter to be either 0.03 (achieving an  of about 0.7, per this) or 0.54 (in order to match the of the IT model one quarter ahead).  And here are what the four models look like forecasting one quarter ahead over the testing period (1992-2006):

So how about the performance metric (the of forecast vs realized inflation)? Here they are, sorted by average rank over the 1Q to 6Q horizon:

First, note that I reproduce the finding (per Noah Smith) that an AR process does better than the DSGE model. Actually, it does better than anything except what is practically data itself!

The IT model does almost exactly as well as a smoothing of the data (LOESS 0.54), which is what it is supposed to do: it is a model of the macroeconomic trend, not the fluctuations. In fact, it is only outperformed by an AR process (a pure model of the fluctuations) and a light smoothing of the data (LOESS 0.03). I was actually surprised by the almost identical performance for Q2 through Q6 of the LOESS smoothing and the IT model because I had only altered the smoothing parameter until I got (roughly) the same value as the IT model for Q1.

The DSGE model, on the other hand, is only slightly better than constant inflation, the worst model of the bunch.

Monday, October 17, 2016

What should we expect from an economic theory?

Really now ... what were you expecting?

I got into a back and forth with Srinivas T on Twitter after my comment on an Evonomics tweet. As an aside, Noah Smith has a good piece on frequent contributor at Evonomics David Sloan Wilson (my own takes are here, here, here). I'll come back to this later. Anyway, Evonomics put up a tweet quoting an economist saying "we need to behave like scientists" and abandon neoclassical economics.

My comment was that this isn't how scientists actually behave. Newton's laws are wrong, but we don't abandon them. They remain a valid approximation when the system isn't quantum or relativistic. Srinivas took exception, saying surely I don't believe neoclassical economics is wrong in the same way Newton is wrong.

I think this gets at an important issue: what should we expect of an economic theory?

I claimed that the neoclassical approximation was good to ~10% -- showing my graph of employment. Neoclassical economics predicts 100% employment; the current value is 95%. That's a pretty good first order estimate. While Newton has much better accuracy (< 1%) at nonrelativistic speeds, this is in principle no different. But what kind of error should we expect of a social science theory? Is that 10% social science error commensurate with a < 1% "hard science" error?

I'd personally say yes, but you'd have to admit that it isn't at the "throw it out" level -- at least for a social science theory. What do you expect?

Now I imagine an argument could be made that the error isn't really 10%, but rather much, much bigger. That is hard to imagine. Supply and demand isn't completely wrong, and the neoclassical growth model (Solow) isn't completely inconsistent with the Kaldor factsThis experiment shows supply and demand functions at the micro level.

Basically, neoclassical economics hasn't really been rejected in the same way Newton is rejected for relativistic or quantum systems. Sure, there are some places where it (neoclassical economics) is wrong, but any eventual economic theory of everything is going to look just like neoclassical economics where it is right. Just like Einstein's theories reduce to Newton's for v << c (the speed of light setting the scale of the theory), the final theory of economics is going to reduce to neoclassical economics in some limit relative to some scale.

But there lies an important point, and where I sympathize with Srinivas' comment. What limit is that? Economics does not set scope conditions, so we don't know where neoclassical economics fails (like we do for Newton: v ~ c) except by trial and error.

This is where economic methodology leaves us wanting -- Paul Romer replied to a tweet of mine agreeing about this point ...
As a theoretical physicist, I try to demonstrate by example how theoretical economics should be approached with my own information transfer framework. The ITF does in fact reduce to neoclassical economics (see my paper). But it also tells us something about scope. Neoclassical economics is at best a bound on what we should observe empirically, and holds in the market where A is traded for B as long as

I(A) ≈ I(B)

i.e. when the information entropy of the distribution of A is approximately equal to the information entropy of the distribution of B. This matches up with the neoclassical idea of no excess supply or demand (treating A as demand for B).

Now it is true that you could say I'm defending neoclassical economics because my theory matches up with it in some limit. But really causality goes the other way: I set up my theory to match up with supply and demand. Supply and demand has been shown to operate as a good first order understanding in the average case -- and even in macroeconomics the AD-AS model is a descent staring point.

Throwing neoclassical economics out is rejecting a theory because it fails to explain things that are out of scope. That is not how scientists behave. We don't throw out Newton because it fails for v ~ c.

It seems to me to be analogous to driving Keynesian economics out of macroeconomics. Keynesians did not fail to describe stagflation and e.g. the ISLM model represents a descent first order theory of macro when inflation is low. Maybe it's just that there's more appetite for criticisms of economics (per Noah Smith above):
There's a new website called Evonomics devoted to critiquing the economics discipline. ... The site appears to be attracting a ton of traffic - there's a large appetite out there for critiques of economics. ... Anyway, I like that Wilson is thinking about economics, and saying provocative, challenging things. There's really very little downside to saying provocative, challenging things, as long as you're not saying them into the ear of a credulous policymaker.
Driving out neoclassical economics seems to be politically motivated in the same way Keynesian economics was driven from macro. It's definitely not because of new data. Neoclassical economics has been just as wrong as it was in Keynes time. However, it has also been just as right.

We have to be careful about why we reject theories. We shouldn't reject 50% solutions because they aren't 100% solutions. Heck, we shouldn't reject 10% solutions because they aren't 100% solutions.

And to bring it back to my critique of David Sloan Wilson that I linked to above: we shouldn't plunge head first into completely new theories unless those theories have demonstrated themselves to be at least similarly effective as our current 50% solution. Wilson's evolutionary approach to economics hasn't even shown a single empirical success. It can't even explain why unemployment is on average between 5 and 10% in the US over the entire post-war period instead of, say, 80%. It can't explain why grocery stores work.

Neoclassical economics at least tells us that grocery stores work and unemployment is going to be closer to 0% than 100%. That's pretty good compared to anything else out there**.

** Gonna need explicit examples if you want to dispute this.

Thursday, October 13, 2016

Forecasting: IT versus DSGE

I was reading Simon Wren-Lewis's recent post, which took me to another one of his posts, which then linked to Noah Smith's May 2013 post on how poor DSGE models are at forecasting. Noah cites a paper by Edge and Gurkaynak (2011) [pdf] that gave some quantitative results comparing three different models (well, two models and a judgement call). Noah presents their conclusion:
But as Rochelle Edge and Refet Gurkaynak show in their seminal 2010 paper, even the best DSGE models have very low forecasting power.
Since this gave a relatively simple procedure (regressing forecasts with actual results), I thought I'd subject the IT model (the information equilibrium monetary model [1]) to an apples-to-apples comparison with the three models. The data is from the time period 1992 to 2006, and the models are the Smets-Wouters model (the best in class DSGE model), a Bayesian Vector Autoregression (BVAR), and the Fed "greenbook" forecast (i.e. the "judgement call" of the Fed).

Apples-to-apples is a relative term here for a couple reasons. First is the number of parameters. The DSGE model has 36 (17 of which are in the ARMA process for the stochastic inputs) and the BVAR model would have over 200 (7 variables with 4 lags). The greenbook "model" doesn't really have parameters per se (or possibly the individual forecasts from the FOMC are based on different models with different numbers of parameters). The IT model has 9, 4 of which are in the two AR(1) processes. It is vastly simpler [1].

The second reason is that the IT model's forecasting sweet spot is in the 5-15 year time frame. The IT model is about the long run trends. Edge and Gurkaynak are looking at the 1-6 quarter time frame -- i.e. the short term fluctuations.

However, despite the fact that it shouldn't hold up against these other forecasts of short run fluctuations, the scrappy IT model -- if I was to add a stochastic component to the interest rate (which I will show in a subsequent post [Ed. actually below]) -- is pretty much top dog in the average case. The DSGE model does a bit better than the IT model on RGDP growth further out, but is the worst at inflation. The Greenbook forecast does better at Q1 inflation, but that performance falls off rather quickly.

Note that even if the slope or intercept are off, a good  indicates that the forecast could be adjusted with a linear transformation (much like many of us turn our oven to e.g. 360 °F instead of 350 °F to adjust for a constant bias of 10 °F) -- meaning the model is still useful.

Anyway, here are the table summaries of the performance for inflation, RGDP growth, and interest rates:

[Ed. this final one is updated with the one appearing at the bottom of this post.]

I also collected the  values for stoplight charts for inflation and RGDP growth [Ed. plus the new interest rate graph from the update]:

Here is what that R² = 0.21 result looks like for RGDP growth for 1 quarter out

This is still a work in progress, but the simplest form of the IT model holds up against the best in class Smets-Wouters DSGE model. The IT model neglects the short run variation of the interest rate [Ed. update now includes] as well as the nominal shocks described by the labor supply. Anyway, I plan to update the interest rate model [Ed. added and updated] as well as do this comparison using the quantity theory of labor and capital IT model.


Update + 40 min

Here is that interest rate model comparison adding in an ARIMA process:

And the stoplight chart (added it above as well):

I do want to add that the FOMC Greenbook forecast is a bit unfair here -- because the FOMC sets the interest rates. They should be able to forecast the interest rate they set slightly more often than quarterly out 1Q no problem, right?



[1] The IT model I used here is

(d/dt) log NGDP = AR(1)
(d/dt) log MB = AR(1)

log r = c log NGDP/MB + b

log P = (k[NGDP, MB] - 1) log MB/m0 + log k[NGDP, MB] + log p0

k[NGDP, MB] = log(NGDP/c0)/log(MB/c0)

The parameters are c0, p0, m0, c, and b while two AR(1) processes have two parameters each for a total of nine. In a future post, I will add a third AR process to the interest rate model [Ed. added and updated].

Tuesday, October 11, 2016


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).

Monday, October 10, 2016

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 envelop 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 naive 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 finacial 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 indistiguishable 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.


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.

Sunday, October 9, 2016

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:

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:
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.


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.