Tuesday, September 30, 2014

Inflation in Japan (update/correction)

In this update, I'm also correcting the graph from this post. The error was mostly due to the kludgey way I put the new data points in (which was itself due to Mathematica's automatic date recognition choking on a Japanese excel file) so that instead of adding the dates to the end of the time series, it started just replacing earlier data points (which cascaded into other problems with e.g. the error bands).

Anyway, here are the corrected graphs for the normal model and the smoothed version (with a new data point for August on the end and extrapolation in the smoothed version):



Here's the reference for interpreting the graph. The orange line is the segment of the model that was fit to the data and the red line is the results using those same parameters (the light red line takes out the estimated fiscal component of Abenomics).

Monday, September 29, 2014

Jason versus the New York Fed DSGE model

A research group at the NY Fed has released some forecasting results [1] with their DSGE model, so here's a comparison with the information transfer model (ITM). The Fed results are in red (and observations in black), and the ITM is blue with gray error bands (in both cases the errors are the 70 and 90% confidence limits). These are both quarterly models/error estimates, but the data (shown in green) is the monthly data so the data should show a bit more spread. Here is the graph:


As a side note, the ITM is essentially linear extrapolation at this short scale. Well, short for the ITM. The prediction does turn into a shallower curve as it nears zero inflation rate (about 0.3%) as you head out towards 2050 (think Japan). However, the assumptions of log-linear extrapolation of NGDP and currency base will probably fail before then.

As for the relative simplicities of the two models, well there's this (on the left is the FRBNY DSGE model and on the right is the ITM):


Sorry, I couldn't help myself. But really ... 29 parameters [pdf] versus 2.

Update (2:45pm PDT): forgot a H/T to Mark Thoma.

[1] They are careful to say that these are not official Fed forecasts, but rather for research purposes.

Saturday, September 27, 2014

The economic combinatorial problem


I mentioned in this post that an economy is a combinatorial problem (along with some hints at entropy); let me sketch out the mathematical side of the argument.

I connected $\log M$ (where M is the currency supply) with $\beta = 1 / T$ in the partition function awhile ago (setting $k_{B} = 1$). If we take the thermodynamic definition of temperature:

$$
\frac{1}{T} = \frac{dS}{dE}
$$

as an analogy (where $S$ is entropy and $E$ is energy), we can write (putting in a constant $c_{0}$ that corresponds to the constant $\gamma M_{0}$ in the information transfer model):

$$
\log M/c_{0} = \frac{dS}{dN}
$$

where we've used the correspondence of the demand (NGDP, or $N$ -- i.e. aggregate demand AD) with the energy of the system. We don't know what the economic entropy is at this point. However, if we take (using the solution shown here):

$$
N = M^{1/k}
$$

Then we can write down

$$
k \log N/c_{0} = \frac{dS}{dN}
$$

So that, integrating both sides,

$$
S = k \; (N/c_{0}) (\log N/c_{0} - 1) + C
$$

Which, via Stirling's approximation, gives us (dropping the integration constant $C$)

$$
S \simeq k \log \; (N/c_{0})!
$$

If we compare this equation with Boltzmann's grave:

$$
S = k \log W
$$

We can identify $(N/c_{0})!$ with the number of microstates in the economy. The factorial $N!$ counts the number of permutations of $N$ objects and we can see that $c_{0}$ adjusts for the distinguishability of  given permutations -- all the permutations where dollars are moved around in the same company or industry are likely indistinguishable. This could lend itself to an interpretation of the constant $\gamma$ mentioned above: large economies are diverse and likely have the same relative size manufacturing sectors and service sectors, for example -- once you set the scale of the money supply $M_{0}$, the relative industry sizes (approximately the same in advanced economies) are set by $\gamma$.

This picture provides the analogy that a larger economy ($N$) has larger entropy (economic growth produces entropy) and lower temperature ($1/\log M$).

The Great Stagnation: the information transfer non-story

I think one of the issues I have with the limited interest in the information transfer model among professional economists is a language barrier. I'm not fully versed in the language of economics and most economists aren't versed in the language of physics. In the post below I make references to "degrees of freedom" and "strongly coupled" mostly out of habit where an economist would say "agents" and "not in partial equilibrium". I probably need to shift a bit more toward the economists -- especially since I'm having a go at reinventing their entire field. However, in the long run, if this information transfer model (ITM) is correct (a big if), economists will have to learn some statistical mechanics.

That's because there's another issue: the idea of a "story". I think this is intimately linked with the degrees of freedom in the theory being humans as opposed to particles. Scott Sumner didn't see the story [1] in my attempt to explain how a falling exchange rate isn't necessarily a sign of inflation. Paul Krugman didn't see the story in Stephen Williamson's deflationary monetary expansion. In statistical mechanics, I don't try to come up with a story for why a molecule in an ideal gas decides to occupy a given energy state -- it occupies a given energy state because that is the most likely thing for it to do given the infinite number of possible energy states that's consistent with the macroscopic information I know (like pressure and temperature). The main insight of the information transfer model is that it doesn't really matter what people think (see e.g. here or here) ... there is no story.

With that throat-clearing out of the way, let me set about writing the information transfer "non-story" of the Great Stagnation.

I wrote a comment on Scott Sumner's post on the mysteriously low long run interest rates in the US, Canada, the EU and Japan (and earlier on Nick Rowe's Canadian-centric post on the same subject). I made the claim that maybe markets were coming to grips with a world of chronically under-shooting of inflation targets (Canada isn't doing this yet, but should soon if the model is correct). The picture you should have in your mind is this one:


The upper left graph shows that as economies grow (under a given definition of money), inflation slows down. The bottom left shows the same for NGDP: stagnation. On the right side are simulations based on 100 markets with random growth rates. That is the source of the story. However, this is not a story of technological stagnation, per Nick's comment on Scott's post. It's (an absence of) a story about the most likely transaction being facilitated by given dollar becoming a low growth market.

Let's tally up a set of random markets by growth rate at one instant in time. Each box represents one market [2]:


High growth markets (industries) are on the right and low growth (or declining) markets are on the left. Now any given market might move from where it is in this distribution from year to year -- a typical story would be an industry starts up at a high growth state, moves to lower and lower growth and might eventually collapse. The distribution doesn't change, though. When that industry moves from the high side to the low side, it's position on the high side is replaced by some other industry. If it collapses completely, it falls off the diagram and is replaced by some new industry. In the picture above, when the growth in the market represented by the box with the "X" slows down, moves to some new location in the picture below:


The two pictures are drawn from the same distribution (a normal distribution with the same mean and variance) -- industry "X" just went from high to low growth and some other industry took its place (although you can see it doesn't have to in order to keep the distribution the same).

This is where the key insight of the information transfer model comes in: that replacement happens for some random reason -- invention of the computer, a war causes oil prices to go up and oil companies make big profits, everyone starts a yoga class, everyone buys an iPhone and stops buying Nokia phones. Some companies are mismanaged. Some are well-managed [3]. Borrowing a picture from David Glasner, some plans are thwarted, others work out better than expected [4]. There are thousands of such stories in an economy and they all tend to cancel out (we muddle through) most of the time leaving the distribution unchanged .

Well, almost unchanged. Sometimes the changes in the locations of the boxes become correlated and you get a recession (plans that depend on each other get thwarted [4]). Over time the economy grows and the distribution shifts. How does it shift during growth? Like this schematic:


The smaller economy is the blue curve and the larger one is the purple curve. A larger economy is more likely to have its low growth rate states filled simply because there are more ways that an economy can be organized where this is true (given the details of the macro state -- e.g. NGDP, monetary base, price level). This is analogous to the molecule in the ideal gas. It is unlikely to find all of the high growth states occupied just like how it is unlikely to find an ideal gas where all of the energy is in a few molecules [5]. It's also unlikely to find all of the markets in the average growth state -- just like an ideal gas doesn't allocate an equal amount of energy to each molecule. 

In physics, we'd say a bigger economy has higher entropy: there are more possible states for each of the constituent markets to be in consistent with the macro information we know (like NGDP). We are missing more information about the exact microstate given the macrostate when the economy is larger (another way of saying it has higher entropy).

There isn't a reason or a story behind this. By random chance you are more likely to find an economy with markets occupying a distribution of growth states with an average that gets smaller as the economy gets larger. If you follow a dollar in the economy, as the economy grows larger, you are more and more likely to find that dollar being used in a low growth industry.

Maybe a better way to put it is this: because there isn't a reason for the markets in an economy to be in any particular growth state (no one is coordinating a market economy), you treat all possible states are equally likely and the result is a distribution where the average growth rate decreases with the size of the economy. 

This is the "Great Stagnation" (the supply-side version) or "secular stagnation" (the demand-side version). Supply and demand are strongly coupled in the ITM (i.e. not in partial equilibrium) so reduced demand growth is reduced supply growth and vice versa. It's not because all the easy things have been invented or the easy gains from the inventions that happened during WWII have been realized. It's not slowing growth of the working age population. It is quite literally a combinatorial problem with Dollars (or Euros or Yen, etc) and units of NGDP. And it happens because there is no story. It happens because the economy isn't being coordinated by anyone -- we just find it in its most likely state. That most likely state is one that grows more slowly as the economy expands.

How do we solve this problem? One way is to coordinate the economy, like in WWII (or communist economies) -- but the coordination problem is hard to solve [6] and the economy would probably collapse eventually. Another way is to change the combinatorial problem by redefining money through monetary regime change or hyperinflation. A third way is to leave it alone and provide better welfare programs to handle economic shocks [7]. Secular stagnation essentially renders the central bank impotent to help against the shocks. This third option seems preferable to me: it reduces the influence of an un-elected group on the economy (e.g. the ECB or FRB). The lack of inflation will be harder on people who borrow money, but hey, interest rates fall!

Footnotes:

[1] In the original comment, Sumner was saying that he didn't see the story where inflation doesn't lead to currency depreciation. However, in that case, the story is a traditional economics story -- in exchange rates, ceteris doesn't seem to be paribus (supply and demand shifts are strongly coupled) and an expansion of the currency supply is always accompanied by an increase in demand to grab it (at least empirically).

It's actually a similar story to "loose" money leading to high interest rates -- there is a short run drop due to the liquidity effect, but inflation and income effects cause rates to rise in the longer run. In fact, it is governed by the same equation (except that in the case of interest rates the information transfer index varies causing the relationship to change slowly over time).

[2] I'm assuming all the markets are the same size right now, but that is not a big deal. Fast growing markets will get big with slow growing (or shrinking) markets getting smaller relative to the other markets. As these markets move around the distribution, their average growth rate will be the average of the distribution.

[3] Note that there is little evidence that a CEO has a significant effect on the company's stock price which tends to follow the industry average (or the SP500).


[4] I borrowed this picture from David Glasner who describes an economy in terms of the coordination of plans:
The Stockholm method seems to me exactly the right way to explain business-cycle downturns. In normal times, there is a rough – certainly not perfect, but good enough — correspondence of expectations among agents. That correspondence of expectations implies that the individual plans contingent on those expectations will be more or less compatible with one another. Surprises happen; here and there people are disappointed and regret past decisions, but, on the whole, they are able to adjust as needed to muddle through. There is usually enough flexibility in a system to allow most people to adjust their plans in response to unforeseen circumstances, so that the disappointment of some expectations doesn't become contagious, causing a systemic crisis. 
But when there is some sort of major shock – and it can only be a shock if it is unforeseen – the system may not be able to adjust. Instead, the disappointment of expectations becomes contagious. If my customers aren't able to sell their products, I may not be able to sell mine. Expectations are like networks. If there is a breakdown at some point in the network, the whole network may collapse or malfunction. Because expectations and plans fit together in interlocking networks, it is possible that even a disturbance at one point in the network can cascade over an increasingly wide group of agents, leading to something like a system-wide breakdown, a financial crisis or a depression.
[5] This isn't always true -- a laser works by creating a population inversion where the high energy states are occupied. 

[6] I'm so glad I get a chance to link to what I consider to be the greatest blog post of all time anywhere.

[7] Shocks -- unmitigated by monetary policy -- are the major drawback of secular stagnation.

Wednesday, September 24, 2014

Does Canada know about the information transfer model?

Nick Rowe put up a post yesterday and to simplify the discussion, let me just quote him:
Either the bond market thinks it will take a very long time for the Bank of Canada to get back to the neutral rate [1-2% real rate], or else the bond market thinks that the neutral rate is lower than the Bank of Canada thinks it is.
First, let me say that it is probably impossible to get anything out of market fluctuations of this size. The graph at the end of Nick's post is inside the green square in this picture:


These are really tiny movements.

Now the information transfer model (ITM) is probably the best model of interest rates in existence [1] (the results for Canada are also shown in the graph above). Again, the movements Nick is talking about are swamped not only by market fluctuations, but by model error.

However, there is another possibility Nick doesn't mention in his either/or quoted above: the bond market could believe Canada will undershoot its inflation target. Why would they believe this? Maybe some people read this post and believed the model. The ITM predicts the average inflation rate going forward 30 years is about 1.6% in Canada, which means that the nominal neutral rate [2], instead of being 3% to 4%, is actually 2.6% to 3.6% -- exactly where the data is. I show both these ranges on the graph above as well (in gray and blue, respectively).

Nick will probably apoplectic about the Bank of Canada being unable to meet its 2% inflation target (if he even notices). Personally, I'd subscribe to the "it's within the model error" view.

Footnotes:

[1] That's a joke. Or hubris. Take your pick :)

[2] Add inflation to the real rate of 1-2%

PS Graphs of the inflation model projection:



Tuesday, September 23, 2014

Information transfer prediction aggregation

Here are a list of the predictions I've made with the information transfer model (ITM). This is mostly to help me keep track of what predictions I've made, but should also help keep me honest.

These are all inflation predictions for the US in a head to head with the Federal Reserve. The ITM says the same thing for all of them -- a slow downward drift in inflation over the medium term:

http://informationtransfereconomics.blogspot.com/2014/09/jason-versus-fed-update.html
http://informationtransfereconomics.blogspot.com/2014/09/jason-versus-fed-presidents.html
http://informationtransfereconomics.blogspot.com/2014/07/us-inflation-predictions.html

This has the same inflation prediction as above, but includes a comparison between the ITM and David Beckworth's claim that the Fed is targeting 1-2% PCE inflation

http://informationtransfereconomics.blogspot.com/2014/08/smooth-move.html

Here is the most recent update of a more complete version of the model that incorporates a guess for the path of monetary policy and predicts YoY inflation, RGDP growth, interest rates and unemployment:

http://informationtransfereconomics.blogspot.com/2014/08/prediction-update-not-bad-for-five.html

Here is a prediction for Canada; I am hoping to test the hypothesis that a central bank can always achieve an inflation target it wants to (per e.g. Nick Rowe) versus a hypothesis that Canada will undershoot based on the ITM:

http://informationtransfereconomics.blogspot.com/2014/07/worthwhile-canadian-prediction.html

Japan is another interesting venue for testing the ITM as Japan is deep in the liquidity trap and things like deflationary monetary expansion become a possibility. This isn't so much of a prediction (I haven't projected the inflation rate for Japan) but rather following the model as new NGDP and currency data becomes available:

http://informationtransfereconomics.blogspot.com/2014/09/update-on-japanese-inflation.html
Update/correction 9/30/2014:
http://informationtransfereconomics.blogspot.com/2014/09/inflation-in-japan-updatecorrection.html

I have some older predictions on the site, however they happened before I figured out the relationship between the monetary base (including reserves) and the currency component. I'd consider them moot at this point.

Jason versus the Fed (update)

I realized I forgot to apply the same adjustment to the error I applied here [1] to the result here [2]. In the graph at the second link [2], I show the expected error in the monthly PCE inflation values -- but the FRB/US model is quarterly. So here is the same graph with the expected quarterly error:


This somewhat decreases the chance we'll fail to reject either model. The overall picture is that the Fed models (and predictions in [1] above) expect that the US economy will return to 2% PCE inflation, while the information transfer model predicts the economy will continue to undershoot that value.