Monday, March 7, 2016

Information equilibrium and time series


I have been dealing with time series for quite awhile now, but I thought I'd formalize a couple of properties. First, let's rewrite the information equilibrium condition. If $A = A(t)$ and $B = B(t)$, then

$$
\frac{dA}{A} = k \frac{dB}{B}
$$

becomes

$$
\frac{1}{A}\; \frac{\partial A}{\partial t} dt = k \frac{1}{B}\; \frac{\partial B}{\partial t} dt
$$

$$
 \frac{\partial \log A}{\partial t} = k \frac{\partial \log B}{\partial t}
$$

This simplifies for two relevant classes of functions: waves and exponential growth. 

Waves

For waves, we have (if $A(t) = A_{0} \exp i \omega_{A} t$ and $B$ is analogous):

$$
\omega_{A} = k \omega_{B}
$$

and the abstract price $P \equiv dA/dB = k A/B$ is

$$
P = k\frac{A_{0}}{B_{0}} \exp i \left(\omega_{A} - \omega_{B} \right)t =k \frac{A_{0}}{B_{0}} \exp i (k - 1)\omega_{B}t
$$

Here is a graph of an example:


Here's another with changing frequencies:


Exponential growth

For an exponential growth system, we have (if $A(t) = A_{0} \exp \gamma_{A} t$ and $B$ is analogous)

$$
\gamma_{A} = k \gamma_{B}
$$

Basically, the growth rates are proportional to each other. The abstract price $P \equiv dA/dB = k A/B$ is

$$
P = k\frac{A_{0}}{ B_{0}} \exp \left(\gamma_{A} - \gamma_{B} \right)t = k\frac{A_{0}}{B_{0}} \exp (k - 1)\gamma_{B}t
$$

Here is a graph of an example:


Or in terms of growth rates:


As I said above, the growth rates are proportional to each other. However, the idea that growth rates are proportional to each other is really only useful if the growth rates change. For example:


And it's even better to see this with real data (the quantity theory of labor model):


PS The growth rate graphs should say growth rate on the y-axis, but I'm too lazy to change them right now.

...

Update + 2 hours

I figured I'd eventually get the question: what happens if you combine waves and growth. However in thinking about it, I came the the realization that the waves picture might not be useful for economics (it could be useful for communication theory). That's because if you have a damped oscillator, you'd actually have to have the economy completely collapsing and going negative. This doesn't really make a lot of sense.

What you really have in macroeconomics is fluctuations in growth rates, not some growth rate plus a oscillating component.

I did put together a picture to illustrate this, and found something interesting. Here is the growth rate $\gamma_{B}$ -- it falls but does not go to zero and there are some oscillations:


And here are the (continuously compounded) growth rates of the three components ($A, B, P$):


I was interested in the shape of the second set of curves -- they appear to fall and then rise again. The source growth rate $\gamma_{B}$, however, just sort of smoothly falls towards its steady state. Since this was supposed to be a cartoon/toy model of the US economy (see above), it made me thinking that it was possible our current low growth and inflation rates might rebound like this -- that current low growth was just such a point.

Maybe that is why Japan has hints of a rebound?

I could have also made some kind of calculation error. Consider this highly speculative.

7 comments:

  1. It seems to me that in economics, borrowing is always an impulse. (An impulse is an unanticipated, high frequency event.)

    We can have repeated, chaotic borrowing which becomes somewhat predictable statistically. I don't think economic borrowing becomes as predictable as gas theory because of low sampling rates in economics.

    As I write this, my 'impulse' is to suggest that the rules that govern economics change within time periods. The longer the time period, the more likely that there has been a rule change within that period.

    Changing the economic rules in a running system is like changing the valves in a steam generating plant---the system will behave in a different fashion.

    The latest wrinkle in rule changing may be "negative interest rates". I have my own theory of first effects (from negative rates) and I see a lot of other theories. I would model negative rates as a tax that would benefit government. The initial effect would be an impulse. Later, the economy should reach a stable equilibrium which included negative rates as a stable known factor.

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  2. So, if Japan does rebound, how will we prevent the market monetarists from declaring an unfair victory (ignoring the fact that they have already declared premature victory)?

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    Replies
    1. As it is unfalsifiable, it is perennially victorious. We should instead look not to the quantity of victories, but to the quality of the victories.

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    2. I really think someone just needs to go about explaining why movements in assets prices on the announcement of monetary expansion at the ZLB do not mean that monetary expansion at the ZLB is effective. I would think it'd be obvious, considering the lack of inflation, but evidently the only reason inflation hasn't gone up because of QE is that the Fed didn't want inflation to be higher.

      I guess you're right, it is unfalsifiable. I wonder if anyone has ever challenged Nick Rowe for a way to falsify is -- Sumner doesn't seem to have any desire to be scientifically rigorous (his epistemology seems to consist of a very strong form of rationalism in which he has deduced the validity of MMism), but Nick might at least try to answer...

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    3. John, I asked Rowe once "What evidence would convince you that you're wrong?" He said something about it being hard to determine if there are unit roots. Basically he said the data wasn't good enough to convince him he's wrong. That's my recollection. It was a one liner on Rowe's part. I'll try to find it.

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    4. I really think someone needs to challenger both of them on the failure of QE to do anything in particular besides make wall street happy for a day or two upon announcement. I guess they'd still argue that the Fed failed to create inflation because it caused expectations of tight money in the future, but, at least with QE the tautology of their definition of tight money (I.e., money is tight whenever a recession happens) does become slightly more incredulous (although I find it laughable to begin with, but apparently neither Scott nor Nick agree).

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