Saturday, April 16, 2016

Angels dancing at the end of time

This should be considered some notes for a future blog post/paper. I've strung together a series of papers to form an argument. The basic idea is that if expectations far out in the future matter in your model, and you use a model where the actual future value enters (or at least the actual future model value -- model consistent expectations, e.g. rational expectations), then your model violates the central limit theorem (in the sense that it requires you to be able to invert the process). In order for the expected future value in the model to lead to a specific present state, you'd have to be able to distinguish which present distribution lead to the distribution of future expected values. However since a random walk derived from any of the distributions with the same well-defined mean and variance lead to a single normal distribution (universality of the normal distribution), this is impossible.

This is a bit more general form of the indeterminacy problem (lots of different paths are consistent with some future expected value of e.g. inflation). Here, lots of different distributions from which a random walk is generated lead to the same future distribution of values.

But really, this means that the expectations operator -- if it is any way forward looking -- is the inverse of a non-invertible operator (the forward-propagating smoothing operator defined by the central limit theorem -- see the pictures below).


...


One thing to keep in mind when using mathematics to describe physical reality is that you have to be very careful about taking limits. In pure mathematics it's generally acceptable to send a variable off to infinity ... . If you are using mathematics to describe physical reality, then [your variable] might just have 'dimensions' (aka 'units') so sending a dimensionful number off to dimensionless infinity (or zero) can give you weird results. ... the only way you can send it off to infinity or zero is to have another scale ... to compare it to. ... In pure math, [sending two time scales to infinity simultaneously] produces an issue of almost uniform convergence; however, if you're using math to describe physical reality then it is nonsense to send ... two dimensionful scales off to dimensionless infinity simultaneously.


The discount rate [one time scale] ... should be a deep parameter in the Lucas sense (based on how human behavior discounts the future) and the growth rate scale [the second time scale] should be a deep parameter of the economy. So we should find the expected value ... depends primarily on the temporal shape of expectations ... . It should therefore primarily depend on the temporal shape of expectations and discounting when exposed to shocks, since temporary shocks shouldn't change deep parameters ... For example, I can cause the representative household's expected utility of consumption to fall in half by switching from [one discount factor to another]. The difference comes from times ... far out [in the future] ... 


... information ... about the future can potentially propagate backwards into the past. All you need to do is couple [a future time to the present using expectations]. ... But I have an additional critique beyond the basic structure of expectations-based macroeconomic theories that might seem a bit strange and it's related to the causality problems ... What keeps expectations in the future? What prevents expectations of [the future] from becoming the value of [the present]? ... In a sense, expectations can travel from a point the future to the present instantaneously ...
Now what does that backward time translation (i.e. expectations) operator do? Well, it should translate the information in the future normal distribution [of the expected value of a variable] into the present distribution -- but there are an infinite number of different present distributions consistent with a future normal distribution [because of the central limit theorem]. How do you choose? You can't. In a sense, [expectations at infinity] requires you to un-mix the cream and the coffee!

2 comments:

  1. I'm liking your titles these days!

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    Replies
    1. It was a title I wanted to use for the second link but didn't think of in time :)

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