Wednesday, June 18, 2014

"The" information transfer model

I thought I'd write down a reference post for "the" information transfer model. By putting "the" in quotes, I mean the specific collection of equations for macroeconomics. This is a particular set of solutions to the differential equations defined by the general information transfer framework.

In the following the notation $p:D \rightarrow S$ is shorthand for saying the demand $D$ transfers information to the supply $S$ that is detected by the price $p$.

The price level and the money market

The price level $P$ is given by the solution to the differential equation for endogenous (see here) $N$ and endogenous $M$ (i.e. the model sets them both together) in the market $P:N \rightarrow M$ where $N$ is NGDP and $M$ is the currency in circulation (empirically, see here) [link works now]. The solution to the differential equation give us $N \sim M^{1/\kappa}$ so that we come to:

$$P(N, M) = \frac{\alpha}{\kappa (N, M)} \left( \frac{M}{M_{0}} \right)^{1/\kappa(N,M) - 1}$$

The function $\kappa$ is taken to be

$$\kappa (N, M) = \frac{\log M/(\gamma M_{0})}{\log N/(\gamma M_{0})}$$

based on empirical results and some motivation from the underlying theory (see here, here). The parameters $\alpha$ and $M_{0}$ are fit to empirical data along with the parameter $\gamma$. However, if $\gamma$ is fit to the price level of one country and kept constant across other countries, all of the countries will be placed on the same two dimensional price level "surface" under a change of variables $P(N, M) \rightarrow P(\kappa(N, M), \sigma (M)) = P(\kappa, \sigma)$ where $\sigma \equiv M/M_{0}$ (see here).

Examples:

The other markets

The remaining markets are all "exogenous" (see here) demand and "exogenous" supply (information source and destination).

The labor market

The markets involving labor are $P:N \rightarrow L$ and $P:N \rightarrow U$ where L is the total number of employed people and U is the total number of unemployed people and result in the equations

$$P = \frac{1}{\kappa_{L}}\; \frac{N}{L}$$

$$P = \frac{1}{\kappa_{U}}\; \frac{N}{U}$$

The first one gives us a form of Okun's law (see here). The "natural rate" of unemployment -- the rate at which $U$ seems to fluctuate around when the above parameters $\kappa_{L}$ and $\kappa_{U}$ are fit to data -- is given by $u^{*} \simeq \kappa_{U}/\kappa_{L}$.

Examples:

[The last graph -- of the UK unemployment rate -- has messed up labels. The model is the blue line (the "natural rate"), the gray line is data and the axis is the unemployment rate, not price level. H/T Tom Brown.]

The interest rate market

The interest rate market is given by $r^{c}:N \rightarrow M_{r}$ where $c$ is a (currently unexplained, update 5/22/2015: explained) fudge factor relating the interest rate price $r^{c}$ to the actual market nominal interest rate $r$. The resuting equation for exogenous $N$ and $M_{r}$ is:

$$r^{c} = \frac{1}{\kappa_{r}}\; \frac{N}{M_{r}}$$

or

$$c \log r = \log \left( \frac{1}{\kappa_{r}}\; \frac{N}{M_{r}} \right)$$

The parameters $c$ and $\kappa_{r}$ are the same for both long and short term interest rates. One fits $c$ and $\kappa_{r}$ to the data for one interest rate market (the long term interest rate market uses currency in circulation for $M_{r_{long}}$ and the 10 year rate for $r = r_{long}$) and the same parameters work for the other market (i.e. the short term interest rate which uses the full monetary base including reserves for $M_{r_{short}}$ and the 3-month secondary market rate, the interbank rate or the effective Fed funds or other short term interest rate for $r_{short}$ depending on the country). When I do the fit, my current modus operandi is to simultaneously fit both markets with the same $c$ and $\kappa_{r}$. Note $M_{r_{long}} = M$ above in the price level/money market.

Examples:

Shifts

I look at the effect of shifts in the variables by taking:

$$M_{r_{long}} \rightarrow M_{r_{long}} + \delta M \text{ and } N \rightarrow N \frac{P(N, M_{r_{long}} + \delta M)}{P(N, M_{r_{long}})}$$

if M is the currency in circulation for a purely monetary shift and

$$M_{r_{long}} \rightarrow M_{r_{long}} \text{ and } N \rightarrow N + \delta N$$

for a purely exogenous NGDP shift (including fiscal expansion or exogenous shocks).

If we use the monetary base (i.e in the short term interest rate market) $MB = M_{r_{short}}$, then in this baseline version of the information transfer model there is no effect on the price level or NGDP. This doesn't rule out an impact on output via the IS-LM model (see e.g. here), in that case the shift "re-appears" in the model above as an exogenous NGDP shock (I will devote a future blog post to explaining that better). However, I am not as confident in that conclusion so I'm leaving the details out of this reference post

Examples:

29 comments:

1. Jason, near the top you write "(empirically, see here)" but there's no link.

1. Thanks for catching that. It's supposed to link to the post "Models and Metrics" -- I'll fix it.

2. Jason, are your labor market plots all OK?

Upper left: You have the model (blue) and the "Price Level" (green). But they're both plotting the same thing, right? So is green the Price Level data and blue the Price Level from the model? What are the units on the y-axis?

Lower left: What's the blue curve? What's the green curve?

Upper right: What are the units on the y-axis?

Lower right: You write "gray" and "green" in the title, but there's no green curve. Plus, is the y-axis really "Price Level?" That doesn't match the title. Units on y-aixs?

1. You're right; that graph of the unemployment rate in the UK is messed up. The natural rate from the model is blue and data is gray and the y-axis is unemployment rate.

The units on the y-axis in the noisy plot are percent if you multiply by 100. It plots the growth rate of rgdp and the growth rate of the labor supply.

The two graphs on the left are the model of the price level given employment. One line is P and the other is ~ N/L (top) and ~ N/U (bottom).

I'll have to fix them when I get a chance.

3. I'm running out of steam tonight... but a lot of info here if you follow all the links like I tried to do. I'll try to pick up the rest tomorrow. This stuff if really interesting.

4. O/T: Jason, what do you know about agent based models (ABMs)? I got into a brief discussion here about it, but I really know nothing:

http://noahpinionblog.blogspot.com/2014/06/big-ideas-in-macroeconomics-book-review_8748.html?showComment=1403135047457#c7400240500037529335

Based on the name (ha!)... I'm guessing a giant simulation with lots of "agents" (rather than one or two or six). Like a simulation of an ant colony or something. Is it in danger of being "ad hoc" (as rosserjb points out) because all those ants have lots of parameters to fiddle with so you can get any result you want? ... Well I guess I should really look it up in Wikipedia before asking you and wasting your time, ha!

1. That is basically it however there are times when the agent models are solved analytically.

It's not in any more danger of being ad hoc than a macro model ... From what I've seen macro models have about as many parameters for an economy as a given agent.

And the agent ones sometimes base the parameters on micro research ... Overall it seems a wash.

My opinion is that the details of the agents don't matter at the macro scale so in a sense maybe agent based models are in less danger of being ad hoc :)

2. In the comments on Noah's I express surprise that the DSGEers wouldn't be delighted to compare notes with the ABMers. How about yourself? Wouldn't it be nice to have some AMB type simulations back up your opinion that "the details of the agents don't matter at the macro scale" and that furthermore you can capture the macro results with your model over a whole class of specific agent models?

3. "back up" was to strong. How about "test?"

4. Yes, that would be great. It appears that many people who do agent based modeling do so out a belief that agents are important -- so I can't imagine them being very receptive to results that say they aren't :)

5. O/T: What do you think of this comment:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2014/06/regression-towards-the-mean.html?cid=6a00d83451688169e201a511d0be15970c#comment-6a00d83451688169e201a511d0be15970c

1. Economics tends to set a pretty low bar for significance, however the commenter makes an assumption that results about economics are drawn from a distribution that has a 10:1 ratio of "right" and "wrong" results ... there is a "study" bias that pre-selects for things that have more significance (you don't test hypothesis that the relative number of macs to PCs determines the tax rate). The initial set of 1000 hypotheses in this case would be more likely to be true than a "randomly selected hypothesis" from the 1000 about the data. That is it would be more likely to draw from the 100 true hypotheses than the 900 false ones for initial study.

In the final result, the 5% significance publication bias only impacts the results near 5% significance (there are fewer "4%" and more "6%" significance results than would otherwise occur by random). However the publication bias due to ta 5% significance level doesn't impact e.g. 1% significance papers.

http://www.washingtonmonthly.com/archives/individual/2006_09/009531.php

http://crookedtimber.org/2006/09/19/attractive-models/

2. Those are interesting. Thanks Jason. You brought up 5% "significance" but what does that mean exactly? The commenter on Rowe's blog, Avon Barksdale (do you recognize the name?), brought up a 5% false alarm (false positive) rate. I'm not sure how 5% false positive relates to 5% significance or how either relate to the 1.96 "z-score" threshold they mention in your 1st link. The significance and the z-score at least seem to have numbers which run the same direction, i.e. the bigger the better. While the false alarm rate is better when it's smaller. I have a feeling I just need to do some reading about statistics to fill in the gaps here.... but can you read a significance percentage from Avon's comment? Was the "5%" you mention just coincidentally the same numerical percentage he referred to (but you were talking about completely unrelated things)?

Thanks.

3. All of this is related to a p-value of 0.05, which is sometimes referred to as "significant at the 5% level" or in my sloppy language "5% significance". It is also related to the probability that we could obtain the given data given the null hypothesis is true (a kind of false positive the commenter referred to).

http://en.wikipedia.org/wiki/P-value#Misunderstandings

z-score and p-value are somewhat related and 1.96 corresponds to the same "5%" level but with a different distribution.

http://en.wikipedia.org/wiki/Z-test

4. Thanks, that helps a lot.

6. Jason, O/T: You made an analogy before between the electrostatic solution to electromagnetic differential equations, and your "exogenous" price level solution. Likewise, you made an analogy between the electrodynamic solution and your endogenous solution. I don't know how far we can push that analogy (probably not very far), but it occurs to me that if we're modeling a radio transmitter, we might chose to exclude the radio operator, whom ultimately modulates the radio frequency carrier signal described by the electrodynamic equations. So there's a potential for an "exogenous" element (external "forcing function?") to the electrodynamic solution, which is entirely different from the electrostatic solution, isn't there? OK, if by some miracle there's room in the analogy for this, what might logically correspond to (i.e. be analogous to) the exogenous source of radio modulation here? CB leadership deciding to pursue one target or another at various times?

1. Also, I get the impression from your endogenous models, that if the CB targets an inflation rate, or does NGDPLT, or perhaps targets price levels, that ultimately the economy may well evolve to the point where kappa = 1, and the CB can no longer successfully hit it's target. Is that fair? However, the CB could escape this circumstance by switching to a strategy which ... uh... I'm not sure how to put it: exploits? excites? ... an exogenous price level solution, perhaps resulting in hyperinflation, but nonetheless, lowering the kappa value closer to 1/2 again, true?

Now is there someway a skillful CB might do all that but w/o causing much damage: either from wallowing in ineffectiveness when kappa=1, or from the hyperinflation?

What would a masterful CB dance look like which smoothly switched from endogenous to exogenous solutions and perhaps back again, to minimize overall economic pain (by some measure). Can such equations be set up and solved?

Or do we need another control lever to pull on here in general (fiscal policy?). Does the CB "communication strategy" play any kind of role here at all?

Also, do you think that NGDPLT has any advantage over inflation rate targeting (say when kappa is closer to 1/2 than to 1)?

It may be hard to see the connection, but David Andolfatto's post today got me thinking about this:

http://andolfatto.blogspot.com/2014/06/excess-reserves-and-inflation-risk.html

In the final paragraph he's asking for other kinds of stories to tell. I thought you might have one to offer him (which may not involve answering any of the questions I pose to you above.... so if you had time to do just one (all else equal) I'd response to David and forget about Tom's questions if I were you :)

2. Hi Tom,

Yes, I'd imagine that an external source of money that is pumped into the economy could be like a radio transmitter in that analogy.

Andolfatto's post is nicely written. I'll have to see if I can provide a story.

Regarding your central bank strategy, I think that could work -- I've imagined such a scenario in some post a ways back (I'll try and find the link as I can't right now). However, I think it requires a larger inflation target in order to do that smoothly. Or another way: once you've reached 2% inflation you've gone too far into the liquidity trap.

(I don't know if NGDP targeting would work better or not. It seems like it would, but that gets into details that aren't well spelled out in the model.)

There is of course the old idea that fiscal policy could regulate the economy, and we just stay near kappa = 1 forever ... maybe the idea that monetary policy should guide the economy is a holdover from an era when it could. Maybe a mature economy has kappa = 1, and developing economies have kappa < 1? Don't forget that kappa < 1 comes with high/rising interest rates.

That's just speculation -- it's strongly dependent on the model being right.

3. FYI, here is a picture of the UK shifting the liquidity trap line via wartime hyperinflation:

http://informationtransfereconomics.blogspot.com/2013/11/the-long-run-in-uk.html

4. "That's just speculation -- it's strongly dependent on the model being right."

That's why I like you Jason!... it's seems there's entirely too much certainty out there in the econ world.... a world, I might add, entirely undeserving of any.

But yes, I was implicitly assuming your model is right, and perhaps I was getting *way* too far ahead of the game with my ideas of solving your equations for optimal CB/fiscal strategies. :D

BTW, I watched a BBC documentary last night about Japan's low birth rate (and very low immigration numbers too). They touched on the Japanese economy and how the low birth rate might have an effect on that. It was kind of interesting. It's on youtube, and is called something like "Sex? No thanks. We're Japanese." (seriously!).

5. I picked out a couple of your posts and made a pitch for you having a different story to tell to David. He didn't say much except to say he'd have a look. In the meantime both Cullen Roche and Nick Rowe have responded to David's post (Nick also comments).

7. Nick Rowe makes the following statement in this comment:

"It makes perfect sense to say that banks will lend out excess reserves and the result will be inflationary. You just have to define "excess reserves" in the economically correct (non-US) sense. Standard textbook stuff."

http://andolfatto.blogspot.com/2014/06/excess-reserves-and-inflation-risk.html?showComment=1403606329577&m=1#c5411660641128578141

By the "US-sense" I think he means legalistic sense. My circular reasoning gut check goes off when I read that. Does my gut need a tune up, or do you have a similar reaction?

How do we know when excess reserves (US-sense) transition to being excess reserves (non-US-sense)? When inflation begins.

Is the concept of excess reserves non-US sense in danger of being one of those logically consistent, always consistent with the data concepts which actually explains nothing (that you've pointed out before)?

1. John Carney had a similar impression about Nick's definition of excess reserves here.

2. I can see what you mean ... if excess reserves are defined relative to desired reserves, then you can always come up with a "desired" that allows your model of "excess" to work. That is (excess reserves) = (reserves) - (desired reserves) with the last term being unobservable.

Maybe

excess = reserves - max(legal, desired)

3. David has acknowledged that Nick makes some good points, and he's updated his thoughts with a new post on the subject, this time with at model:

http://andolfatto.blogspot.com/2014/06/excess-reserves-and-inflation-risk-model.html?m=1

8. Jason,

This person "dtoh" is always pestering Scott that he should be talking about "the base minus ER" rather than just the base. That's close to your "currency component of the base" most of the time (with reserve requirements at a low percentage). I've pointed out to him the similarity, and gave him a link to one of your posts, but I don't think he followed it. Here's a typical comment:

http://www.themoneyillusion.com/?p=26941#comment-355085

What I'm wondering is have you ever done much of an analysis on why it is that M0 works better than MB? I know you selected M0 primarily because it fit the data better, and I know you've written a couple of sentences speculating on why that might be, but have you actually looked at it in any greater depth? (Plus I forgot you two sentence explanation).

1. The short answer is no, I haven't looked at it in too much depth.

The hints come from the interest rate market: MB affects short term interest rates (e.g. 3-month) while M0 affects long term interest rates (e.g. 10-year).

Sumner says that basically MB - M0 ("MB minus M0") is expected to be temporary. That suggests that MB is around for, say, the 3 months, but not for 10 years.

However, that is inconsistent with the fact that the large value of MB - M0 has been around for almost 5 years now; you'd imagine expectations to slowly converge on a significant probability that MB - M0 will be around for 10 years.

The 3-year rate is right between the two (in log interest rate space), and has always been. With an expectations argument, you'd kind of expect it to be getting closer to the 3-month rate as MB - M0 shold have been expected to have stuck around for 3 years (since it has for almost 5 years). Instead it is getting closer to the 10-year rate. Maybe that means MB - M0 is expected to be more temporary (i.e. not impacting the 3-year or 10-year rates) now than it was in 2012.

http://research.stlouisfed.org/fred2/graph/?g=Eig

Additioally, MZM works better than M0 in the 10-year rate

http://informationtransfereconomics.blogspot.com/2014/04/broad-money-narrow-money-and-interest.html

MZM does fit the interest rate better, but is worse at the CPI (even including food and energy).

This is why I don't think the idea of temporary vs permanent is a solid explanation for why M0 is used in the price level (inflation).

It seems to be because a printed dollar bill or minted quarter explicitly pins the information content of a dollar to the physical world in the way other aggregates don't. (This reminds me of Douglas Adams: "The ships hung in the sky in much the same way that bricks don't.")

I've been trying to come up with a more intuitive explanation using 3D printers and goods. A piece of physical currency is like a printed 3D object. Because of copyright (i.e. because counterfeit is illegal), only owners of a particular copyright can print a particular good. Reserves are like claims on print orders -- but a store (bank) isn't going to ask for all its print orders to be printed if it can't sell them. In this case, it seems in the real world that only actually printed objects are going to affect their price on the market (i.e. the price level). Claims on print orders will have another price based on future demand (i.e. an interest rate).

2. Thanks Jason. Regarding your 3D printer analogy... I'm difficult for me to square that with Nick Rowe's response to David Andolfatto's piece, but that doesn't surprise me too much:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2014/06/repeat-after-me-people-cannot-and-do-not-spend-money.html#more

3. The 3D printer analogy was only an attempt at finding out why physical currency matters for the price level. Nick Rowe appears to be answering a different question. Banks lend reserves (and currency, etc) creating new "MZM" via fractional reserve banking. However the jump in MB since 2009 hasn't lead to a comparable jump in MZM. That means the idea that "banks don't lend reserves" is an incorrect reason for the lack of impact of QE on the price level (the reason for Nick's post).

In the information transfer model, MZM appears to react to M0 rather directly:

http://informationtransfereconomics.blogspot.com/2014/05/do-monetary-aggregates-measure-money.html

which implies it has some stable relationship with the price level, but it fluctuates wildly meaning it's not a good predictor of the price level like M0 is:

http://informationtransfereconomics.blogspot.com/2014/04/broad-money-narrow-money-and-interest.html

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