Is human agency Noah's big unchallenged assumption?

For me, it's hard not to read this (from Noah Smith):

But the real reason you have this tradeoff is because you have big huge unchallenged assumptions in the background governing your entire model-making process. By focusing on norms you ignore production costs, consumption utility, etc. You can tinker with the silly curve-fitting assumptions in the macro model all you like, but it won't do you any good, because you're using the wrong kind of model in the first place.

So when we see this kind of tradeoff popping up a lot, I think it's a sign that there are some big deep problems with the modeling framework. [Emphasis in original]

And think of this (from me):

Do we really know [that macroeconomics is the result of a whole bunch of little economic decisions by individuals and companies]? For a fact? To be specific, I'm not questioning the idea that an economy is made up of humans making decisions with money (of course it is) -- I'm questioning the idea that observed macroeconomic relationships (price level and money supply, RGDP and employment) are the result of humans making decisions with money.

There's a lot more on that front in the information transfer framework: utility maximization, Euler equations, price stickiness ... even money itself. It is possible that our decisions have only a limited impact most of the time. And when our decisions matter, it's mostly for the worse.

Scott Sumner's information equilibrium model

Earlier this month, Scott Sumner started a blog post off with an identity using the price level and the money supply:

$$
P = \frac{M}{M/P}
$$

and identified the numerator with the money supply and the denominator with real money demand allowing some conclusions to be teased out.

This is an information equilibrium model. We have the price of money being $1/P$ (a higher price level means money is less valuable -- see Sumner here), so this is inverted from our usual formulation:

$$
 \frac{dD}{dS} \equiv p = k \; \frac{D}{S}
$$

In this format, we'd write (with $k = 1$):

$$
 \frac{d(M/P)}{dM} \equiv \frac{1}{P} = \frac{M/P}{M}
$$

Note that the differential equation is the correct marginal thinking in traditional economics -- the price of money should be the marginal exchange rate of a unit of demand (i.e. a unit of utility) for a unit of supply. This equation appears in Irving Fisher's 1892 thesis, and the information equilibrium model is only a minor generalization of it (adding the $k$). 

This is also the simplest differential equation consistent with long run neutrality of money (homogeneity of degree zero in supply and demand), Bennett McCallum's definition of the quantity theory of money [1].

Now let's follow this trolley all the way to the end of the line.

First, let's replace $M/P$ with money demand $D$ and solve the differential equation (in general equilibrium where $D$ and $M$ vary):

$$
\log \left( \frac{D}{D_{ref}}\right) = \log \left( \frac{M}{M_{ref}}\right)
$$

$$
 \frac{D}{D_{ref}} = \frac{M}{M_{ref}}
$$

where the $ref$ identifies constants introduced in integrating the differential equation. If we substitute Sumner's form for $D = M/P$ (or just by substituting the above equation in the definition of the price), we can show

$$
 \frac{M}{P D_{ref}} = \frac{M}{M_{ref}}
$$

$$
 \frac{1}{P} = \frac{D_{ref}}{M_{ref}}
$$

$$
P = \frac{M_{ref}}{D_{ref}}
$$

i.e. the price level is constant in general equilibrium and there is no inflation. That's what we'd expect from $k = 1$. 

[Update 8/31/2015: There was a sign error in the equations and incorrect discussion in these last paragraphs which should have referred to movement of the supply and demand curves ($X_{0} \rightarrow X_{0} + \delta X$), not movement along them (changes in $ \Delta X = X - X_{ref} $). I marked the two changed sentences with an initial *. H/T Tom Brown in comments below.]

Next, let's check out partial equilibrium. We can solve the differential equation (constraining $D$ or $M$ alternately to be slowly varying around $D_{0}$ and $M_{0}$, respectively) to arrive at:

$$
P = \frac{M_{ref}}{D_{0}} \exp \left(+ \frac{D - D_{ref}}{D_{0}}\right)
$$

$$
P = \frac{M_{0}}{D_{ref}} \exp \left(- \frac{M - M_{ref}}{M_{0}}\right)
$$

which are supply and demand curves (this is essentially the same as the AD-AS model in the information equilibrium framework). *In partial equilibrium, an increase in demand for money (a shift in the demand curve, $D_{0} \rightarrow D_{0} + \delta D$) leads to a rise in the price of money ($1/P$) and a fall in the price level ($P$). *An increase in supply of money (a shift in the supply curve $M_{0} \rightarrow M_{0} + \delta M$) leads to a fall in the price of money ($1/P$) and a rise in the price level ($P$). In a sense, we only get inflation (or deflation) from changes in money demand and money supply. However, since $k = 1$, these should go away in the long run and inflation should be constant -- given the general equilibrium solution.

In Sumner's post, he says that $D$ is actually real GDP and "other stuff", thus we come to the conclusion that growth is deflationary.

So what about that "other stuff"? Well, in the information equilibrium model, we put the other stuff in two places -- (exogenous) nominal shocks and a changing $k$. It's the latter that becomes Sumner's $V$ (velocity in the quantity theory of money). And the best place to see that model is in my draft paper available here.

Footnotes:

[1] Long-Run Monetary Neutrality and Contemporary Policy Analysis Bennett T. McCallum (2004)

Entropy is working for the weekend

Nick Rowe employs his ability for distillation in the task of explaining a monetary coordination failure that results in a recession. In his recent post, he mentions the coordination occurring on the weekend:

Every Saturday Canadian output and employment drop. And they drop again every Sunday. Every weekend, output and employment drop for two successive days. Are weekends mini recessions? I would say "no".

If you've been following along this blog -- in particular this post [1] -- you might ask: I thought you said coordination causes recessions?

I did say that. The important thing to understand is that it is coordination relative to the equilibrium distribution. Let's say here's what (the probability density of) output looks like during a typical week (I made this data up):

Call this distribution P. It's the equilibrium distribution. Some people (by no means everybody) have the opportunity to take weekends off. Now week to week the information loss measured by the KL divergence is zero:

D(P||P) = 0

But if suddenly this happens (call this distribution Q):

We get a KL-divergence that results in an information loss of:

D(P||Q) = 0.18 bit

It turns out that is a loss of about 6.6% relative to the information entropy of P ~ 2.78 bits. That would be a mini recession (if weeks were all the same except that one, it would be a recession of about 0.1% of output).

What would really be happening? What is the story behind this mini-recession? With more people off on days that they used to have on, they might go shopping. But with fewer people to stock the shelves and no one expecting a Thursday rush of long weekend consumers, less gets sold than in the status quo. ATM's might not have enough money in them for Wednesday being the new Friday and cash only establishments would miss out. Restaurants unexpectedly fill up for Friday brunch and people can't get a table.

We seasonally adjust data; that's an admissible procedure only because the seasonality represents the equilibrium distribution.

Now you may ask: What about holidays? Or the Stanley Cup?

Well, the first graph was actually a simplification. The actual distribution would be more complex, taking into account your country's public holidays, major sporting events and vacations. That would be the real P.

If part way through the year, 5% more people became unemployed so that your distribution of output changed (i.e. more the output would have been at the beginning of the year relative to an equilibrium year), then you would probably have a recession. That's a bad coordination.

Now I actually think the real coordination comes in the form of pessimism about asset prices and future sales, so the rise in unemployment is a symptom, not the cause. The entropy loss manifests as a fall in employment (and a rise in the number of people with zero wage change) as seen in the post above [1].

Rational expectations and information theory

KL Divergence using Gaussian distributions from Wikipedia.

Noah Smith once again deleted my comment on his blog, so I'll just have to preserve it (well, the gist of it) here.

He discussed an argument against rational expectations he'd never considered before. Since counterfactual universes are never realized, one can never explore the entire state space to learn the fundamental probability distribution from which macro observable are drawn. Let's call this probability distribution A. The best we can get is some approximation B.

Rational expectations is the assumption A ≈ B.

If this sounds familiar, it's exactly the way one would approach this with the information equilibrium model as I discussed several months ago.

In that post, I showed that the KL divergence measures information loss in the macroeconomy based on the difference between the distributions A and B.

D(A||B) = ΔI

That was the content of my comment on Noah's post. I go a bit further at the link and say that this information loss is measured by the difference between the price level and how much NGDP changes when the monetary base changes

ΔI ~ P - dN/dM = dN*/dM - dN/dM 

Which to me seems intuitive: it compares how much the economy should grow from an expansion of the money supply (ideally) to how much it actually does grow.

Just the aggregate ΔI is measured, however. Two different distributions B, B' and B'' can have the same KL divergence so this doesn't give us a way to estimate A better.

Now rational expectations are clearly wrong at some given level of accuracy, but then so are Newton's laws. The question of whether you can apply rational expectations depends on the size of ΔI. Since ΔI is roughly proportional to nominal shocks (the difference between where the economy is and where it should be based on growth of M alone [1]) and these nominal shocks are basically the size of the business cycle, it means rational expectations are not a useful approximation to make when analyzing the business cycle.

As far as I know, this is the first macroeconomic estimate of the limits of the rational expectations assumption that doesn't compare it to a different model of expectations (e.g. bounded rationality, adaptive expectations). (There are lots of estimates for micro.)

Footnotes:

[1] In case anyone was curious, this also illustrates the apparent inconsistency between e.g. this post where nominal shocks are negative and e.g. this post where they are positive. It depends on whether you apply them before including the effect of inflation or after. Specifically

0 = dN*/dM - (dN/dM + σ) = (dN*/dM - σ) - dN/dM

Sticky wages?

I'm not sure I understand how economists (including Mark Thoma) can look at this data:

... and say:

Taken at face value, this analysis suggests the presence of some amount of wage rigidity.

24% of people are reporting nominal hourly wage declines. Only 20% are reporting the same wage. That means something like 80% of people are reporting wage changes. They should check out the section of my paper on entropic forces and nominal rigidity. Or this post on macro rigidity and micro flexibility.

Information equilibrium as an economic principle

I have finished the first public draft of the information equilibrium paper I started to write back in February. Here is a link (please let me know if it doesn't work -- I think I've set my Google Drive settings properly) and the outline below:

[Now a pre-print on arXiv. Updated the draft paper with edits from Peter Fielitz, Guenter Borchardt and Tom Brown. Version from 9/26/2015 is available here. Previous version was 8/21/2015 and is available here.]

Information equilibrium as an economic principle
http://arxiv.org/abs/1510.02435 [q-fin.EC]
http://econpapers.repec.org/RePEc:arx:papers:1510.02435
http://ssrn.com/abstract=2894072

1 Introduction
2 Information equilibrium
   2.1 Supply and demand
   2.2 Physical analogy
   2.3 Alternative motivation
3 Macroeconomics
   3.1 AD-AS model
   3.2 Labor market and Okun's law
   3.3 IS-LM model and interest rates
   3.4 Price level and inflation
   3.5 Solow-Swan growth model
   3.6 A note on constructing models
   3.7 Summary
4 Statistical economics
   4.1 Entropic forces and emergent properties
5 Summary and conclusion

Don't forget the VAT

Update: Mark Sadowski (in comments below) correctly points out that there was no VAT change between 2008-2009. I have corrected my error in the figures and results. There is a tiny change in the conclusion (where the "VAT increase" is instead referred to as a "spike in inflation" in the second to last paragraph of the + 2 hour update)

Scott Sumner is linking to Mark Sadowski again and showing us that inflation as measured by the GDP deflator has risen in Japan since 'Abenomics' went into effect. However, the graph shown fails to take into account three two VAT increases -- which Scott Sumner explicitly pointed out might fool people -- that show up visibly in the GDP deflator data. Here's Sumner:

Japan will be hit by an adverse supply shock next year (higher sales tax rates) which will boost inflation–making it look like they will hit their 2% target. Don’t be fooled.

And here's the data and a version with the VAT increases filtered out ...

If you remove these VAT increases, the change is less dramatic and the change appears to start before the monetary policy changes were announced.

I make a pretty good case here that the change in direction of the price level (core CPI) is mostly due to the fiscal policy component. The information equilibrium model seems to get the data about right without any monetary policy effects.

...

Update + 2 hours:

If you reduce the size of the averaging window from 10 quarters to 4 (i.e. 1 year), you can see that the effect of the VAT doesn't stick around in 2015 ... much like the effect of the VAT increase in 2008 and 1997:

Also using a single year averaging means that the start of the increase in inflation is pushed all the way back to 2010 (that year is no longer impacted by the VAT increase spike in 2008). Heck, here's using only a single quarter 'average':

The trend towards increased inflation appears to start after the financial crisis; Abe appears in the middle of it.

The Chinese unemployment rate

I wrote awhile ago about Chinese economic statistics (CPI and NGDP) not seeming terribly suspect. Alex Tabarrok points to the stillness of the Chinese unemployment rate as a sign that the statistics can't be trusted. And I tentatively agree -- there appears to be something wrong with the reported Chinese unemployment rate.

Combining the China model linked above with this post on employment growth, I thought I'd try to estimate the Chinese unemployment rate.

Here are the nominal shocks (see the post on employment growth):

And here is an estimate of the resulting unemployment rate (making an assumption of 5% for the "natural rate" and a 50% employment-population ratio):

The yellow line is the 'official' rate (from here).

This result is consistent with the unemployment rate rising to 11% in 2002 in the estimate presented in Tabarrok's post. That analysis says that unemployment stays high through 2009. However, my model seems to think that the unemployment rate dropped to 4-5% before 2009. My model is more consistent with the burst of NGDP (and RGDP) growth between 2005 and 2010 (using e.g. Okun's law). And it puts unemployment nearer to 4% before the recent economic trouble.

This is not to say unemployment isn't high today in China (the last NGDP data I have is from 2014). And China does not appear to be reporting the unemployment spikes from recessions. This of  course could be a difference due to the economic systems. While it seems to operate a large capitalist economy nowadays, the country is officially communist. Involuntary unemployment during recessions may not be the same thing there as it is in the US.

Employment doesn't depend on inflation

Robert Waldmann and Simon Wren-Lewis, in discussing Paul Romer's history of macro 1977-1982, bring up the Phillips curve. I've also written about it on occasion.

I thought I'd have a look at the Phillips using the DSGE form of the information equilibrium model. Turns out it results in something really cool ... Here are the relevant equations from the DSGE form link:

$$
\text{(1) } n_{t} =  \sigma_{t} + \left( \frac{1}{\kappa} - 1 \right) (m_{t} - m_{t-1}) + n_{t-1}
$$
$$
\text{(2) } \pi_{t} = \left( \frac{1}{\kappa} - 1 \right) (m_{t} + m^{*}) + c_{m}
$$
$$
\text{(4) } \ell_{t} = n_{t} - \pi_{t} + c_{\ell}
$$

Here $n$ is nominal output, $m$ is base money (minus reserves), $\pi$ is the price level, and $\ell$ is the total employed. The symbol $\sigma$ represents 'nominal shocks'. They are the stochastic part of the model, and they're typically positive. They represent the difference between where $n$ is at time $t$ and where it should be based on the change in $m$ from time $t-1$ to time $t$ alone -- essentially equation (1). The rest of the symbols are constants, with $\kappa$ being the IT index (approximately constant over the short run).

I had hoped to show some kind of relationship between changes in the total employed ($\ell_{t} - \ell_{t-1}$) -- and thus changes in unemployment -- and inflation ($\pi_{t} - \pi_{t-1}$). But the math led me to something I didn't expect. With a bit of algebra, you can show that labor growth is given by:

$$
\ell_{t} - \ell_{t - 1} = \sigma_{t}
$$

regardless of the information transfer index $\kappa$. Those nominal shocks I've talked about since this post? They are basically changes in the number of employed. That's why they're typically positive and typically around a few percent.

Effectively, employment growth is the part of nominal output (NGDP) that is left over after accounting for inflation. Thus there shouldn't be any relationship between inflation and unemployment -- i.e. the Phillips curve isn't real. This even applies to a version of the model consistent with expectations, since we could easily write:

$$
E_{t} \ell_{t+1} - \ell_{t} = E_{t} \sigma_{t+1}
$$

That is to say expected changes in unemployment are the expected shocks to the economy after accounting for inflation.

How does this look empirically? Pretty good ($\sigma$ in blue, $\ell$ in yellow):

Comparison of interest rate predictions

The decline [in long term interest rates] has come largely as a surprise. Financial markets and professional forecasters alike consistently failed to predict the secular shift, focusing too much on cyclical factors.

The above graph and quote are from this blog post by Obstfeld and Tesar (H/T John Cochrane) from July of this year.

I've been doing some time series forecasting using the information equilibrium (IE) model recently; how does the IE model compare to the Blue Chip Economic Indicators (BCEI)?

With the exception of 2005 (which is when the Fed embarked on its tightening before [1] the great recession) where the prediction is uncertain, the IE model does a much better job of forecasting 10-year nominal interest rates in the far future. The bands in the following are the 1-sigma errors (68% confidence limits).

1996

In this case, the IE model is a bit low in the near term, but better in the long term -- the opposite of the BCEI predictions.

2000 (1999)

In this case, I used 1999 as the prediction year instead of 2000 because the spike in base money due to fears of the 1999/2000 transition throws the model off. Even this does better than the (March of) 2000 BCEI predictions.

2005

This is the aforementioned case where the IE model becomes uncertain of the path of future rates and the Fed guides the economy between the high (initially) and low end (after the crisis) of the prediction bands.

2010

In this one, the IE model is spot on and far better than the BCEI prediction.

2015

We don't have the data yet, but I'd hazard a guess that the IE model predictions will be better than the BCEI predictions. If the Fed goes through with its projected short term rate increase in September-ish time period, it could impact this projection, but only if the Fed unwinds a large fraction of the QE in the short term.

PS

The information equilibrium model for interest rates is described here and here. In the notation I've used (described at the second link), we have the model (r → p) : N → M where N is NGDP, M is base money (minus reserves), r is the interest rate and p is the 'price of money'. In words, that model reads:

Aggregate demand (N) is in information equilibrium with base money (M). The quantity information flow from aggregate demand to aggregate supply (mediated by money) is measured by the price of money (p). That price is in information equilibrium with the nominal interest rate (r).

Update 8/19/2015:

The model (green line) shown in all of the graphs above is actually the model fit using all of the data from 1960 to 2015. However, the projections only use the model fit to the data from 1960 to the projection year (1996, 1999, 2005, 2010, 2015). This is visible in some of the projections above as the start point of the projection is noticeably off of the green line (2005 is the most visible, 1999 is the second most visible).

The model fit for 2015 is:

log r = k₁ log(N/M) - k₂

with

k₁ = 2.8

k₂ = 6.4

Using FRED series M = MBCURRCIR [converted from millions to billions of dollars], N = GDP and r = GS10.

You can have a look at the model here (with fit parameters):

https://research.stlouisfed.org/fred2/graph/?g=1EJp

Footnotes:

[1] I wanted to put "precipitating" instead of "before" because of some other considerations of the IE model, but went with the more neutral version.

The Dungeons and Dragons approach to economics

Dice roll with six-sided and twenty-sided dice.

Editor's note: This post has been languishing in the drafts folder for several months, so I gave it a quick finish and posted it.

In a footnote to the previous post I mentioned flipping a coin versus rolling a twenty sided die as a different way of thinking about the solution to paradox of value -- why diamonds cost more than water (we need water to survive, yet water is cheaper than diamonds).

The 19th century solution is marginal utility. And it works well.

If we look at the picture above, we need a total of about 33 bits to specify this particular result

$$
11 \cdot \log_{2} 6 + 1 \cdot \log_{2} 20 \simeq 32.8 \text{ bits}
$$

Each of the six-sided dice reveals about 2.6 bits, while the 20-sided die reveals about 4.3 bits.

In equilibrium, the supply effectively knows the demand's "roll". 'Quanta' of 4.3 bits are flowing back and forth in the diamond market and quanta of 2.6 bits are flowing back and forth in the water market. However if there is a change, a change in the 20-sided die in the demand's roll, it requires a flow of 4.3 bits to the supply side. A change in one of the six-sided dice requires a flow of 2.6 bits.

Therefore the information transfer index $k_{d}$ for diamonds is going to be larger than the index for water $k_{w}$. If there is something that functions as money (see here and here, Ed. note: these are later posts that more clearly demonstrate my point), then the information transfer index in terms of money will also be larger. Therefore, the price of diamonds will grow much faster than the price of water since for some $m$:

$$
p_{d} \sim m^{k_{d} - 1} > p_{w} \sim m^{k_{w} - 1}
$$

and therefore

$$
\frac{d}{dm} p_{d}  > \frac{d}{dm} p_{w}
$$

if $k_{d} > k_{w}$.

The absolute price is not knowable in the information transfer model, but by simply being rarer (i.e. lower probability so that more information is revealed by specifying its allocation) the price will grow much faster. Thus eventually, regardless of the starting point, diamonds will be more valuable than water. The one caveat (assumption) is that both things must continue to have a market.

...

Update: fixed typos of 4.6 bits where it should read 4.3 bits (H/T Tom Brown in comments below).

Interest rates and predictions

I've been playing around with the time series forecasting tools with my home-use copy of Mathematica 10.2. Below is an animation of  the 10-year forecast of the information equilibrium model (or see here; update: added links) starting from the 1960s to today. A couple of the frames failed to fit using the default settings for TimeSeriesModelFit (for the extrapolations) and NonlinearModelFit (to fit the model), so instead of fixing what was wrong I took the lazy way out and just deleted them. Here's the animation:

There are three key things about the data we'd hope a model could see before they happened: 1) the 'great inflation', 2) the end of the great inflation and 3) the 'great moderation' and the trend toward lower interest rates. I'd argue the IE model sees all three of these things ...

1) The great inflation

It's a bit early, but also is extrapolated from only 10 years or so of data.

2) End of the great inflation

Note that at the point where this prediction was made, interest rates were still rising.

3) Trend towards lower rates

(The above graph is actually a 20-year prediction)

Additionally, the DSGE-form of the model gives a nice way to organize how this works.

Also, a side note -- I discovered that smoothing the monetary base data in the model, but not NGDP data gives a better fit to 10-year interest rates (first figure ... as opposed to e.g. smoothing both, second figure):

Update + 6 hours:

In a remarkable coincidence, John Cochrane discusses the downward trend in 10-year rates as well and links to a study by the Council of Economic Advisers.

Update + 9 hours:

So does Stephen Williamson.

Giant molecules and representative agents

A good quote from Kevin Hoover via Lars Syll via Brad DeLong:

The reasoning of the representative-agent modelers would be analogous to a physicist attempting to model the macro-behavior of a gas by treating it as single, room-size molecule.

A single molecule cannot tell us about entropic effects and the idea of a gas taking up "volume" is also emergent as molecules are generally treated as having zero volume themselves so that the volume is zero individually, but the aggregate expected value is the volume of an ideal gas.

Are higher interest rates inflationary?

Scott Sumner writes "higher interest rates are inflationary, as they raise velocity". Is this true? Generally? Is the mechanism true?

I had forgotten exactly how useful the log-linearized "DSGE form" of the information equilibrium model was to answer this kind of question. Ubiquitous caveat: this is a model-dependent result. The conclusions are true assuming the information equilibrium model (and my math is right). Anyway, here's are the (relevant) equations from this post (and ignore the difference between monetary base and monetary base minus reserves for right now):

$$
\text{(1) } \; n_{t} =  \sigma_{t} + \left( \frac{1}{\kappa} - 1 \right) (m_{t} - m_{t-1}) + n_{t-1}
$$
$$
\text{(2) } \; \pi_{t} = \left( \frac{1}{\kappa} - 1 \right) (m_{t} + m^{*}) + c_{m}
$$
$$
\text{(3) } \; r_{t} = c_{1} (n_{t} - m_{t} - m^{*}) + c_{2}
$$

If we take $\kappa$ = 0.5, i.e. the quantity theory of money, then we can show:

$$
\text{(4) } \; r_{t} - r_{t-1} = c_{1} \sigma_{t}
$$

The interest rate at time $t$ is the interest rate at time $t-1$ plus a 'nominal shock' that is part of NGDP growth -- $\sigma_{t}$ in equation (1) -- that is generally positive (think population growth or TFP growth) but is negative when recessions hit (negative shocks tend to cause recessions and may actually be the same thing as a recession).

But if we look at $\pi_{t} - \pi_{t-1}$, with $\kappa$ = 0.5 we find:
$$
\pi_{t} - \pi_{t-1} = n_{t} - n_{t-1} - \sigma_{t}
$$
or
$$
\pi_{t} - \pi_{t-1} = m_{t} - m_{t-1}
$$

So when  $\kappa$ = 0.5, interest rates tend to rise because of positive nominal shocks, while the price level tends to  rise because of NGDP growth minus the nominal shocks -- i.e. just plain money growth. The rising interest rates are not related to rising price level.

Now what happens when $\kappa$ is close to 1? That is the liquidity trap/IS-LM regime ... Well, first:

$$
\pi_{t} - \pi_{t-1} \simeq 0
$$

There is approximately zero inflation when $\kappa \simeq 1$. And for interest rates we get an extra term in equation (4):

$$
\text{(5) } \; r_{t} - r_{t-1} = c_{1} \sigma_{t} - c_{1}  (m_{t} - m_{t-1})
$$

Monetary expansion lowers interest rates if it is larger than the nominal shocks. Note that when $\kappa \simeq 1$, we also have

$$
n_{t} - n_{t-1} \simeq \sigma_{t}
$$

i.e. NGDP growth is mostly due to nominal shocks. You can see this point alongside some graphs of the nominal shocks here (also shown below).

So when $\kappa \simeq 1$, monetary expansion (e.g. base growth) that is greater than NGDP growth makes interest rates fall and monetary expansion that is slower than NGDP growth makes interest rates rise.

Summary:

• $\kappa \simeq 0.5$: Inflation is due to monetary expansion (NGDP growth without nominal shocks) and interest rates tend to rise due to nominal shocks. These are separate processes. Rate rises are dominated by what might be called the 'income effect' arising from higher nominal output.
• $\kappa \simeq 1.0$ Inflation is approximately zero and interest rates tend to rise if monetary expansion is slower than NGDP growth and fall if monetary expansion is faster than NGDP growth. NGDP growth is entirely due to nominal shocks. Here the income and liquidity effects are acting in opposite directions with the liquidity effect (the new second term in the interest rate in equation 5) coming to dominate.

So the general picture of the US should show rising interest rates and price level, the latter of which causes $\kappa$ to rise (since $\kappa$ goes as log M/log N, the denominator reaches a point of slower growth before the numerator), leading to a turn-over and falling interest rates and a flattening of the price level. This is precisely the set of effects described here.

Update 8/18/2015: Updated summary with some economic language from the link at the very end.

Explicit implicit models

Not sure why I am doing this, but I thought it might be helpful to see "explicit" implicit models and how they frame the data. This is in regard to this (ongoing) discussion with Mark Sadowski.

Let's take (the log of the) monetary base (MB) data from 2009 to 2015 and fit it to two theoretical functions. One is a line (dashed) and the other is a series of three Fermi-Dirac distribution step functions (solid):

The first difference of the data (yellow), theoretical line (dashed) and theoretical steps (solid, filled) are shown in this plot:

If we expect a linear model, we can see the data as fluctuations around a constant level. If we expect the step model, we can see the data as fluctuations around three "pulses". It's not super-obvious from inspection that either of these is the better model of Δ log MB. The Spearman test for correlation [1] of the first differences is -0.07 (p = 0.53) for the line and 0.51 (p = 2e-6) for the steps.  The steps win this test. However if you use the data instead of the theoretical curves to compare to other variables, you can't actually conduct this test so you don't know which model of Δ log MB is best. 

Now let's assume a linear model between the (log of the) price level P and log MB and fit the two theories:

Again, the first differences (data = yellow, line theory = dashed, step theory = filled):

Although it wasn't obvious from the difference data which model of Δ log MB was better, it's now super-obvious which model of Δ log MB is the better model of Δ log P (hint: it's the line). The Spearman test for correlation of the first differences is 0.23 (p = 0.04) for the line and 0.006 (p = 0.95) for the steps (i.e. the line is correlated with the data). This would imply that:

• If you believe the linear theory of log MB, then log MB and log P have a relationship.
• If you believe the step theory of log MB, then log MB and log P don't have a relationship.

This is what I mean by model dependence introduced by the underlying theory. If you think log MB is log-linear, you can tease a relationship out of the data.

Now if you go through this process with (the log of) short term interest rates (3-month secondary market rate), you end up with something pretty inconclusive on its face:

You might conclude (as Mark Sadowski does corrected; see comment below) that short term interest rates and the monetary base don't have a relationship. The Spearman test for correlation of the first differences says otherwise; it gives us -0.09 (p = 0.44) for the line and 0.29 (p = 0.01) for the steps (i.e. the steps are correlated with the data).

...

However, Mark left off the first part of QE1 in his investigation -- he started with Dec 2008.  So what happens if we include that data? It's the same as before, except now we use 4 Fermi-Dirac step functions for the step model:

Note that the linear model already looks worse ... here are the first differences:

The Spearman test for correlation of the first differences is -0.03 (p = 0.77) for the line and 0.59 (p = 6e-10) for the steps (i.e. the steps are correlated with the data).

The step theory (filled) captures many more of the features of the data (yellow) than the linear model (dashed). The price level first differences are pretty obviously the line, and pretty obviously not the step:

The Spearman test for correlation of the first differences says both are uncorrelated with the data; it gives -0.01 (p = 0.93) for the line and 0.04 (p = 0.72) for the steps (i.e. neither are correlated with the data).

But the really interesting part is in the (log of the) short term interest rates:

In the first differences, you can see the downward pulses associated with each step of QE:

The Spearman test for correlation of the first differences is -0.1 (p = 0.36) for the line and 0.25 (p = 0.02) for the steps (i.e. the steps are correlated with the data).  Actually, in the plot above there seems to be a market over-reaction to each step of QE -- rates fall too far, but then rise back up. The linear theory just says its all noise.

So the results of the 4 step model from 2008 to 2015?

• log MB is not related to log P
• log MB is related to log r

But remember -- all of these results are model-dependent (linear vs steps).



Footnotes:

[1] I used Spearman because Pearson's expects Gaussian errors and on some of the data, the errors weren't Gaussian. Mathematica automatically selected Spearman for most of the tests, so I decided to be consistent.

A trivial maximum entropy model and the quantity theory

This is a trivial maximum entropy model, but I still think the result is interesting.

Let's take an economy with d = 100 dimensions subjected to a budget constraint Σ m = M(t) where t is time, m is the (randomly chosen) allocation of the total amount of money M to each dimension (these could be "markets", "firms" or even "agents"). This is effectively this picture except d = 3 here (the Ci are the dimensions):

If we take the derivative of the ensemble average d/dt log 〈m〉(assuming a uniform distribution), we get something (blue paths) that roughly follows d/dt log NGDP (yellow path):

The paths obviously converge to d/dt log M (black path) ... it doesn't even depend on d because 〈m〉= α M with constant α even if we don't have d >> 1 since

d/dt log α M = d/dt (log M + log α) = d/dt log M

If d >> 1 then we have α ~ d/(d + 1) → 1, but this isn't terribly important. What this does show is that

d/dt log M ~ d/dt log NGDP

k M ~ NGDP

and we have a simplistic quantity theory of money based on randomly allocating money to different markets.

Comparing methodologies (monetary base and short term interest rates)

In the process of setting up the Granger-causality test of the relationship between the monetary base (St. Louis Adjusted monetary base) and short term interest rates (3 month secondary market rate) mentioned here, but I thought there should be a quick aside on something Mark Sadowski said:

... if a time series model’s results are reported without routine references their statistical significance, then they should always be viewed with deep skepticism. ...

... an argument consisting of little more than a line graph depicting the time series of values of a quantity should never be accepted as a proof of anything, except that the person who is arguing that it proves something is not familiar with what constitutes acceptable empirical evidence, is incapable of understanding what constitutes acceptable empirical evidence, or is simply willfully ignoring what constitutes acceptable empirical evidence because it contradicts their preferred model.

Almost zero theoretical physics papers pay attention to statistical significance in this way. Obviously physics is "doing it wrong" and that's why the field is in a state of existential methodological crisis. Oh, wait. That's economics.

Personally, I'd say any statistical results presented without reference to an explicit model means that an implicit model has been used -- and you treat any implicit results with deep skepticism.

As Paul Krugman says:

Any time you make any kind of causal statement about economics, you are at least implicitly using a model of how the economy works. And when you refuse to be explicit about that model, you almost always end up – whether you know it or not – de facto using models that are much more simplistic than the crossing curves or whatever your intellectual opponents are using.

Here's an example where you can see this at work ... first setting up the data

The next bit looks for the best single lag to compare the first differences of the two time series (in both directions) ...

This is a basic fit with a single lag ...

It works pretty well for an economic model (changes short term interest rates cause changes in the monetary base: rates go down, base goes up ...)

How about the same thing but looking at levels? Well the traditional stats test is going to be garbage because of spurious correlation ... but the model looks much better than the statistically "proper" version ...

Does a slavish devotion to statistical purity lead us away from real understanding?

Granger causality is an information equilibrium model

Information equilibrium relationship between Yt and its lagged values Yt-i and the lagged values of the second variable Xt.

In the continuing discussion with Mark Sadowski, I've realized that Granger causality between log-linear variables is equivalent to an information equilibrium model. Let's posit the equilibrium (so direction of the arrow doesn't matter) markets:

$$
Y_{t} \rightarrow Y_{t - i}
$$
$$
Y_{t} \rightarrow X_{t - j}
$$

So that in general equilibrium

$$
Y_{t} = \prod_{i = 1}^{m} \left( \frac{Y_{t-i}}{Y_{0}^{(i)}} \right)^{a_{i}} \prod_{j = p}^{q} \left( \frac{X_{t-j}}{X_{0}^{(j)}} \right)^{b_{j}}
$$

This is the information transfer model depicted in the diagram at the top of this post. If we log-linearize we have:

$$
\log Y_{t} = \sum_{i = 1}^{m} a_{i} \log Y_{t-i} - a_{i} \log Y_{0}^{(i)} + \sum_{j = p}^{q} b_{j} \log X_{t-j} - b_{j} \log X_{0}^{(j)}
$$

$$
y_{t} = a^{(0)} + \sum_{i = 1}^{m} a_{i} y_{t-i} + \sum_{j = p}^{q} b_{j} x_{t-j}
$$

where $y_{t} = \log Y_{t}$ and $a^{(0)}$ represents the collected constant terms. The method of a Granger causality test is then first fitting the models $Y_{t} \rightarrow Y_{t - i}$, and then using the results to fit the combined models $Y_{t} \rightarrow Y_{t - i}, Y_{t} \rightarrow X_{t - j}$ (as well as looking at the reverse structure with $X_{t}$ as the information source).

Mark Sadowski's series of posts (all linked here) can then be summarized as:

• monetary base → price level
• monetary base → industrial production index
• monetary base → TIPS (5-year breakeven)
• monetary base → DJIA
• monetary base → ten-year rate
• ten-year rate → industrial production index
• monetary base → exchange rates
• monetary base → deposits
• monetary base → credit

using the notation source → destination. The markets monetary base → price level ($MB \rightarrow P$), monetary base → TIPS ($MB \rightarrow (d/dt) \log P$) and monetary base → ten-year rate ($MB \rightarrow r_{10y}$) both make the mistake of looking at a market as supply transferring information to the price instead of the price functioning as the detector of information transfer and are better represented as the markets:

$$
P : NGDP \rightarrow MB
$$
$$
(r_{10y} \rightarrow p) : NGDP \rightarrow MB
$$

where $p$ is the price of money. The price level is the price in the market for nominal output and the interest rate is in information equilibrium with the price of money in the market for nominal output. The residuals are even lower if $M0$ is used instead of $MB$ in the above markets. The monetary base is best used in the short term (e.g. 3 month) interest rate market:

$$
(r_{3m}\rightarrow p) : NGDP \rightarrow MB
$$

Here are the results for the long and short interest rates for the US and UK, for example:

So when Mark says:

But as we have seen here there is really is no monetary transmission channel that works primarily, much less exclusively, through expected short-term interest rates.

(even though he does not present an analysis of the monetary base and short term interest rates -- which will be forthcoming from me), we can take the model $(r_{3m}\rightarrow p) : NGDP \rightarrow MB$  as evidence of the large class of models [1] excluded from his analysis. 

Additionally, the market monetary base → exchange rates is also incorrect as exchange rates are better represented straightforwardly as the ratio of the prices of money for the two countries in the markets 

$$
p_{1} : NGDP_{1} \rightarrow M0
$$
$$
p_{2} : NGDP_{2} \rightarrow M0
$$
$$
X_{1,2} = \frac{p_{1}}{p_{2}}
$$

The monetary aggregate can be various aggregates like M1 or M2 as the data does not really pick one over the other.

Maybe the information transfer framework is the wrong way to think about economics. However, if it turns out to be a correct theory of macroeconomics, then most of Mark's analysis represents spurious correlation as he has set up the flow of information incorrectly.

...

Update 8/10/2015

Corrected sign error and changed figure for the US per Tom Brown's comment below. Thanks for catching that!

Footnotes:

[1] Funnily enough, Scott Sumner's model of expectations is also excluded because if the reason the price level hasn't risen massively due to QE is that monetary base is expected to vanish, MB should have no impact on other variables -- changes in MB should not Granger-cause anything. If changes in MB does Granger-cause other variables and we take that model seriously, then Scott Sumner's expectations theory is wrong.