What is a Good Approximation ?

What is a good approximation ?
The word good is important but very general. One (long long ago) I pulled a hair out of my scalp and told Elisabetta Addis (we’re married) “look my first white hair.” She replied “you have a very good wife and a very bad mirror.” Odd the same event, failure to point out my many older white hairs, can be both good and bad.
The same thing goes for approximations.
Click the link for my latest anti economic theory diatribe.

What is a good approximation ?
The word good is important but very general. One (long long ago) I pulled a hair out of my scalp and told Elisabetta Addis (we’re married) “look my first white hair.” She replied “you have a very good wife and a very bad mirror.” Odd the same event, failure to point out my many older white hairs, can be both good and bad.
The same thing goes for approximations.
Click the link for my latest anti economic theory diatribe.

Rajiv Sethi: The Invincible Markets Hypothesis: .

There has been a lot of impassioned debate over the efficient markets hypothesis recently, but some of the disagreement has been semantic rather than substantive, based on a failure to distinguish clearly between informational efficiency and allocative efficiency. Roughly speaking, informational efficiency states that active management strategies that seek to identify mispriced securities cannot succeed systematically…. Allocative efficiency requires more… is satisfied when the price of an asset accurately reflects the (appropriately discounted) stream of earnings…. If markets fail to satisfy this latter condition, then resource allocation decisions (such as residential construction or even career choices) that are based on price signals can result in significant economic inefficiencies.
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The critics concede that informational efficiency is a reasonable approximation, at least with respect to short-term price forecasts, but deny that prices consistently provide “accurate signals for resource allocation.”

This post is long so I will try to put the punch-line up here. Economists make terrible errors when they say a statement “a reasonable approximation” of reality. Two very different meanings are conflated. One is that direct evaluation of the statement shows that rejection is subtle so the statement is approximately true. The other is that all claims which would be true if the statement were absolutely true are approximately true.

There is a general reliance on a sort of smoothness assumption, so that a model based on approximately true assumptions must have approximately true implications.

It is absolutely known and positively proven that this idea is totally false. The result that it is totally false is old and very well established (google “epsilon equilibrium”) People who study the implications of the assumptions must know this. Yet the idea that there is some general smoothness property in the mapping from assumptions to conclusions absolutely dominates economic theory.

Yes I just said that economic theory is based on assuming something which is known to be a false statement in mathematics. The reason is simple. Without the demonstrably false assumption that approximately true assumptions imply approximately true conclusions, economic theory would lead to no conclusions. The results of theories would be mere hypotheses to be rejected if they didn’t fit the data.

Economists absolutely claim that this is how they use theory. This is absolutely false. Standard models have implications which are rejected by the data and yet they remain standard models.

OK the one and only efficient markets hypothesis.

I’d say that, in a standard general equilibrium model, informational efficiency does imply allocational efficiency. So to the extent that one accepts such models as guides, one should believe that, for practical purposes informational efficiency implies allocational efficiency.

The problem is that approximate informational efficiency does not imply approximate allocational efficiency.

This is a well known theoretical result. When the assumption of informational efficiency is used to draw conclusions (e.g. of allocational efficiency) it is necessary to assume that you can never beat the market. Predictions about the present absolutely require that assumption that the market can be beaten at 10:00 AM EST on March 12 AD 1 million. This hypothesis can’t be tested now obviously.

Anomalies in risk adjusted returns on the order of 1% per year can’t be detected. We can’t be sure of exactly how to adjust for risk. However, they can make the difference between allocative efficiency and gross inefficiency.

For policy makers there is a huge huge difference between “markets are approximately informational efficient” and “markets are informational efficient.” The second claim (plus standard false assumptions) implies that markets are allocatively efficient. The second implies nothing about allocative efficiency.

What does the phrase “a reasonable approximation” mean ? Does it mean A) rejection of the claim is subtle and controversial or B) rejection of all implications of the claim are subtle ?

I think standard usage would be B) a reasonable approximation is one that doesn’t lead us astray. In economics the phrase is used in sense A, that is, only some implications of the claim can be used to test the claim because … because we said so.

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