Wednesday, August 24, 2016

Efficient markets and the Challenger disaster

Every the market boosters, Marginal Revolution has a new video out in its personal finance series that talks about the efficient markets hypothesis. Aside from the fact that it might be questionable to base financial decisions on an hypothesis. I haven't watched the video, but from the still it appears to reference the Challenger disaster. It's an interesting story propagated by believers in the wisdom of crowds. On the surface, the market appears to have discovered the problem was with the solid rocket boosters since Morton Thiokol's stock dropped more than the other NASA contractors involved with the shuttle program. Here's a paper [pdf] that investigates it. Now of course this could be attributable to larger exposure to NASA (Thiokol had about twice as much revenue from its shuttle program per the pdf) as well as Thiokol being a smaller, less diversified company than Lockheed, Marietta, or Rockwell at the time (see John Quiggin here). Here is a graph of stock prices from here:

But what does the information equilibrium model have to say?

The key piece of information comes from the study referenced above. Average daily returns for the previous three months was given in Table 1: Lockheed (0.07%), Marietta (0.14%), Rockwell (0.06%) and Thiokol (0.21%). If we assume all of these companies are information equilibrium with the same underlying process X, these differential growth rates imply different information transfer (IT) indices. For example, the IT index k -- well, actually it's k - 1 since log p ~ (k-1) log X -- is about three times higher for Thiokol than for Lockheed. This means that even given the same source of information, Thiokol will respond quite a bit more than Lockheed to the same shock. And some simulations bear this out; here's a typical example based on the growth and volatility in the paper cited above:

Note that the underlying process X is the same (a Wiener process with constant drift and volatility) but are different realized values. Here's a Monte Carlo with 100 throws per company:

In the information equilibrium model, the prices seem perfectly consistent with all four contractors being hit with the same information shock -- and  therefore there's no evidence the market figured out the cause within minutes of the disaster.


PS My grade school mascot was the Challenger shuttle (I grew up in the suburbs of Houston).

PPS I got to take a tour of the orbiter processing facility while NASA was preparing Discovery, Atlantis, and Endeavor were being prepared for the museums. Here's Discovery in the OPF with its aerodynamic engine cover before being flown to Washington, DC:

Tuesday, August 23, 2016

A vector of information equilibrium relationships

This is a mathematical interlude that looks at some geometric interpretations of an ensemble of information equilibrium relationships. It represents some notes for some future work.

Let's start with a vector of information equilibrium relationships between output in a given sector $y_{i}$ and the money supply $p_{i} : y_{i} \rightleftarrows m$ so that

\frac{dy_{i}}{dm} = A_{ij}(m) y_{j}

The solution to this differential equation is

y_{i}(m) = \left[ \exp \int_{m_{ref}}^{m} dm' A_{ij}(m') \right] y_{j}(m_{ref})

Let $A(m) = K/m$ so that

y_{i}(m) = \left[ \exp \left( K_{ij} \log \frac{m}{m_{ref}} \right) \right] y_{j}(m_{ref})

The volume spanned by these vectors (spanning the economic output space) is

V = \det \exp \left( K \log \frac{m}{m_{ref}} \right) \approx 1 + \log \frac{m}{m_{ref}} \;\text{tr}\; K

So that the infinitesimal volume added to the economy is

dV = \left( \log \frac{m}{m_{ref}} \right) \;\text{tr}\; K

Using maximum entropy to select one of multiple equilibria

Some time ago, I mentioned the idea [1] that maximum entropy could select a particular Arrow-Debreu equilibrium when there are many available; I thought I'd work through a specific example where that could work using a simple 2D Edgeworth box. Let's assume a utility function for agent 1 (borrowed from here [pdf]):

u1(g1, g2) = g1 - 0.125 g2-8

with g1 and g2 exchanged for the other agent. The offer curves (blue, yellow for the two agents) in the 2D Edgeworth box look like this (for an initial endowment given in the pdf link above):

These curves intersect in 4 points, three of which are very close to each other (and hard to see). Which equilibrium relative price (slope through the initial endowment and the point) does the market select? Traditional economics lacks a solution to this problem -- all three are viable equilibria. However, the point in the center of the triplet of points has higher entropy (consider the joint entropy of the distributions with a probability of finding an infinitesimal unit of good 1 with agent 1 versus agent 2 and likewise for good 2). You can see that if you zoom in on those points; I show an information entropy level curve (unconstrained maximum in the center of the Edgeworth box) as a dotted gray line.

The point in the middle is the maximum entropy point, subject to all the constraints in the utility maximization problem.



[1] I also mentioned it here. At that link, I also mentioned a potential solution to the so-called aggregation problem where one looks at traces (differential volume elements) -- those volume elements would be related to the state space volumes I use to look at maximum entropy. This footnote is intended mostly as a note to myself.

Monday, August 22, 2016

IE vs NY Fed DSGE model update

I haven't updated the head-to-head with the NY Fed DSGE model -- by that I mean the 2014 vintage of that model. The model and the forecast has been changed several times since 2014, including in May of this year a month after the last update from me. The model now only projects a year ahead (as opposed to the nearly 4 years of the original vintage 2014 model).

And the saddest part? The original 2014 vintage of the model is doing an amazing job! The core PCE inflation data has been heavily revised [1] and what previously looked like a blow out for the IE model has turned into a slight advantage for the vintage 2014 DSGE model.

Still, we'll probably have to wait until the beginning of 2017 to know which model is better. This is consistent with the expected performance of the IE model: observation times shorter than a few years are dominated by irreducible measurement error.


Update 23 August 2016

Is the NY Fed DSGE model following a ringing oscillation from the financial crisis?



[1] These revisions have been almost enough to make me reconsider rejecting this lag model

A trend towards lower inflation in Australia (IE prediction)

This recent post by John Quiggin reminded me of my prediction of a trend towards undershooting inflation in Australia (here and here). A commenter on this post going by Anti said unique predictions that fit empirical evidence are "the real question"; I'd say this is a unique prediction of the information equilibrium model that fits the empirical evidence:

Saturday, August 20, 2016

Did the ACA decrease unemployment?

Scott Sumner repeated his unsupported claim that the expiration of unemployment insurance in 2014 decreased unemployment. It was picked up by John Cochrane and Tyler Cowen. I looked at the data a year ago and showed that a sizable chunk of it could be explained by increasing job openings (and assuming a matching model) in the health care sector brought on by the ACA going into effect in 2014:

Additional jobs would be created via a Keynesian multiplier. I tweeted about this and was asked how much higher unemployment would have been without the ACA and estimated 0.5 percentage points higher.

That estimate was loosely based on this model of employment equilibrium; however I thought I'd look at it a bit more rigorously. I re-ran the model fitting only to the data before 2014 and found that the impact was even larger at 1.3 percentage points:

It is true that a lot of things went into effect at the same time, but using a typical Keynesian multiplier of 1.5 accounts for about half of the boom in the total number of jobs and the biggest increase in openings was in health care. That's a pretty consistent story.

Japan (lack of) inflation update

I haven't updated the price level model for Japan in awhile (last update here, and here is the link to all the ongoing forecasts for other countries and indicators), so here you go:

Friday, August 19, 2016

DSGE, part 5 (summary)

I've just finished deriving a version of the three-equation New Keynesian DSGE model from a series of information equilibrium relationships and a maximum entropy condition. We have

\Pi & \rightleftarrows N \;\text{ with IT index } \alpha\\
X & \rightleftarrows C \;\text{ with IT index }1/\sigma\\
R & \rightleftarrows \Pi_{t+1} \;\text{ with IT index }\lambda_{\pi}\\
R & \rightleftarrows X_{t+1} \;\text{ with IT index }\lambda_{x}

along with a maximum entropy condition on the intertemporal consumption $\{ C_{t} \}$ subject to a budget constraint:

C_{t+1} = R_{t} C_{t}

We can represent these graphically

These stand for information equilibrium relationships between the price level $\Pi$ and nominal output $N$, real output gap $X$ and consumption $C$, nominal interest rate $R$ and the price level, and the nominal interest rate and the output gap $X$. These yield

r_{t} & = \lambda_{\pi} \; E_{I} \pi_{t+1} + \lambda_{x} \; E_{I} x_{t+1} + c\\
x_{t} & = -\frac{1}{\sigma} \left( r_{t} - E_{I} \pi_{t+1}\right) + E_{t} x_{t+1} + \nu_{t}\\
\pi_{t} & = E_{I} \pi_{t+1} + \frac{\alpha}{1-\alpha}x_{t} + \mu_{t}

with information equilibrium rational (i.e. model-consistent) expectations $E_{I}$ and "stochastic innovation" terms $\nu$ and $\mu$ (the latter has a bias towards closing the output gap -- i.e. the IE version has a different distribution for its random variables). With the exception of a lack of a coefficient for the first term on the RHS of the last equation, this is essentially the three equation New Keynesian DSGE model: Taylor rule, IS curve, and Philips curve (respectively).

One thing I'd like to emphasize is that although this model exists as a set of information equilibrium relationships, they are not the best set of relationships. For example, the typical model I use here (here are some others) that relates some of the same variables is

\Pi : N & \rightleftarrows M0 \;\text{ with IT index } k\\
r_{M} & \rightleftarrows p_{M} \;\text{ with IT index } c_{1}\\
p_{M} : N & \rightleftarrows M \;\text{ with IT index } c_{2}\\
\Pi : N & \rightleftarrows L \;\text{ with IT index } c_{3}\\

where M0 is the monetary base without reserves and $M =$ M0 or MB (the monetary base with reserves) and $r_{M0}$ is the long term interest rate (e.g. 10-year treasuries) and $r_{MB}$ is the short term interest rate (e.g 3-month treasuries). Additionally, the stochastic innovation term in the first relationship is directly related to changes in the employment level $L$. In part 1 of this series, I related this model to the Taylor rule; the last IE relationship is effectively Okun's law (in terms of hours worked here or added with capital to the Solow model here -- making this model a kind of weird hybrid of a RBC model deriving from Solow and a monetary/quantity theory of money model).

Here is the completed series for reference:
DSGE, part 1 [Taylor rule] 
DSGE, part 2 [IS curve] 
DSGE, part 3 (stochastic interlude) [relates $E_{I}$ and stochastic terms] 
DSGE, part 4 [Phillips curve]
DSGE, part 5 [the current post]

DSGE, part 4

In the fourth installment, I am going to build one version of the final piece of the New Keynesian DSGE model in terms of information equilibrium: the NK Phillips curve. In the first three installments I built (1) a Taylor rule, (2) the NK IS curve, and (3) related expected values and information equilibrium values to the stochastic piece of DSGE models. I'm not 100% happy with the result -- the stochastic piece has a deterministic component -- but then the NK DSGE model isn't very empirically accurate.

Let's start with the information equilibrium relationship between nominal output and the price level $\Pi \rightleftarrows N$ so that we can say (with information transfer index $\alpha$, and using the definition of the information equilibrium expectation operators from here)

E_{I} \pi_{t+1}- E_{I} \pi_{t} = \alpha \left( E_{I} n_{t+1}- E_{I} n_{t} \right)

Using the following substitutions (defining the information equilibrium value in terms of an observed value and a stochastic component, defining the output gap $x$, and defining real output)

E_{I} a_{t} & \equiv a_{t} - \nu_{t}^{a}\\
x_{t} & \equiv E_{I} y_{t} - y_{t}\\
n_{t} & \equiv y_{t} + \pi_{t}

and a little bit of algebra, we find

\pi_{t} & = E_{I} \pi_{t+1} + \frac{\alpha}{1-\alpha} x_{t} + \mu_{t}\\
\mu_{t} & \equiv \nu_{t}^{\pi} - \frac{\alpha}{1-\alpha} \nu_{t}^{y} -\frac{\alpha}{1-\alpha} (E_{I} y_{t+1} - E_{I} y_{t})

The first equation is essentially the NK Phillips curve; the second is the "stochastic" piece. One difference from the standard result is that there is no discount factor applied to future information equilibrium inflation (the first term of the first equation). A second difference is that the stochastic piece actually contains information equilibrium real growth (the last term). In a sense, it is a biased random walk towards reducing the output gap.

Anyway, this is just one way to construct a NK Phillips curve. I'm not 100% satisfied with this derivation because of those two differences; maybe a better one will come along in a later update.

Wednesday, August 17, 2016

Is information equilibrium silly?

Tom Brown sent me a link to a highly critical comment from TallDave on Scott Sumner's blog the other day. I think it contains a decent critique, but also misunderstands the project. Here is TallDave's comment:
I think the problem with Jason’s math is that when translated into words you get assertions like what is called “demand” in economics is essentially a source of information that is being transmitted to the “supply”, a receiver, and the thing measuring the information transfer is what we call the “price” which are kind of silly on their face. Modelling economics as a function of information transfer is a bit like modelling the digestive process on the basis of food’s color when it enters and exits — it just doesn’t capture enough of the process to be a useful exercise.
Emphasis in the original. It is true that naively applying the language of communications channels to economics in this way would seem like an exercise in modeling by elaborate analogy. However, the information equilibrium approach really is just a generalization of the idea of supply meeting demand. Imagine the distribution of blueberries as a function of time and space. During the spring, they are mostly distributed near the farms where they are grown. During the summer, they are distributed among many grocery stores. Much like in the Arrow-Debreu formulation of general equilibrium, we have a blueberry at a point in space at a particular time that represents a blueberry "supply event". Let's say that probability distribution P(B) looks something like this:

Now a blueberry consumer has a property we call demand for blueberries. It changes in space and time as well. In the same way we have supply events, we have demand events (I have money for blueberries at the grocery store near my house at a given time today). In an ideal world, the distribution of blueberry supply events and the distribution of blueberry demand events [call it P(A)] would be identical:

These supply events and demand events together would form a joint distribution of "transaction events" where money was traded for blueberries:

This situation where the distribution of supply events and the distribution of demand events are the same is what we call information equilibrium. Information? If you check out any given Wikipedia page for a probability distribution (e.g. the normal distribution), you will see an entry in the box on the right-hand side for "Entropy" that links to the information entropy page.

Any probability distribution (e.g. our supply and demand distributions above) can be quantified in terms of its information entropy.

That's well and good for two identical distributions that don't change, but what happens if we infinitesimally wiggle one distribution [P(A)]? How much does the other distribution [P(B)] have to wiggle in order to maintain information equilibrium? The simplest answer to that question for uniform distributions gives us the information equilibrium condition (see e.g. here, except I used D and S instead of A and B):

The information in that wiggle δP(A) must have flowed (was transferred) to P(B). (Note that the P in the equation above is not the probability distribution, but the price which I will talk about below.) That's where the communication channel interpretation comes in. We have come complex multi-dimensional demand distribution and some multi-dimensional supply distribution with the information in the fluctuations of the demand distribution being transmitted through some channel and received by the supply distribution. (In a sense, Shannon's theory comes about from wanting the distribution of messages at one end to be identical to the distribution of messages at the other end.) This gives us the standard picture of a communication channel:

What about the price? I just defined the price in the equation above as the derivative dA/dB -- this is actually an abstract price and should really be considered an exchange rate for an infinitesimal unit of A for an infinitesimal unit of B. Does this make any sense? Yes, it does. For example, check out Irving Fisher's 1892 thesis:

The information equilibrium condition is just a minor generalization of the equation Fisher writes down relating the exchange of gallons of A for bushels of B. But there is more -- in fact, if you define the LHS of the information equilibrium condition as the price, you can use that equation to derive supply and demand curves (see my paper or this blog post).

For more theoretical motivation, I'd also recommend you check out my slides on the connection between information equilibrium and Gary Becker's paper Irrational Behavior and Economic Theory. For physicists, there's another theoretical motivation in terms of effective field theory (here, here).

There is a decent critique contained in TallDave's comment, though:
Modelling economics as a function of information transfer is a bit like modelling the digestive process on the basis of food’s color when it enters and exits — it just doesn’t capture enough of the process to be a useful exercise.
It is definitely possible that the information in the wiggles δP(A) are not received by the distribution P(B) -- information is lost. It could be the case that P(A) is a complex multi-dimensional distribution and P(B) is ... less complex. In that case (for uniform distributions), the best we can say is that information equilibrium is a bound on the information transfer

and we have what we call non-ideal information transfer. But does information equilibrium capture enough of the process to be useful? This should primarily be an empirical question, but I'd say yes for two reasons:

Therefore, I'd say there's really no reason to consider information equilibrium prima facie "silly". If information equilibrium is silly, so is supply and demand since they are formally identical. That may well be true -- but then economics in general would be silly.