Dienstag, 26. Januar 2016

Wann "Falken" "Tauben" werden

"I’m often asked about why I changed my mind about the appropriate stance of monetary policy in 2012.  I think that this question is typically aimed at trying to get at something personal - and I’ll attempt such an answer at a later date.  But in this post, I’ll use a more technical approach to addressing the question.  I’ll demonstrate through an example how even small amounts of evidence can lead a Bayesian updater to large changes in their beliefs about the validity of extreme models.  

Imagine a coin being flipped.  A decision-maker has two models of the coin.  In the first model, the probability of Heads is 0.99.   In the second model, the probability of Heads is 0.5. 

The first model is an abstract way to represent the perspective of someone who believes (as I did in 2009) that aggregate demand factors only matter for extremely short-run behavior in the macro-economy.   The second model is an abstract way to represent the perspective of someone who believes (as I do now) that both aggregate demand and supply matter over extended periods of time.   

In the background, we can imagine a decision between an option D (dovish) and an option H (hawkish).  It would be easy to set up some kind of decision framework in which D is optimal if (and only if) the posterior belief on model 1 falls below 0.5.

I think of Model 1 as being an extreme model, in the sense that it predicts that the vast majority of observations should be Heads.  What we will see is that relatively small amounts of information can lead a Bayesian to update away from extreme models.  

With that motivation in mind, suppose the coin is flipped and we get a Tail.  How would a Bayesian update between the two models?   Analytically, it is easy to show that if one’s prior assigns a probability q to the extreme Model 1, then one’s posterior after a Tail is given by q/(50-49q).  

This updating function is highly nonlinear.  It implies that, even for close to dogmatic priors, a Bayesian responds aggressively to new information.   For example, suppose q = 0.99, so that the initial prior puts only one chance in 100 on Model 2 being true.  After only one Tail, the posterior belief falls to (about) 2/3.  After a second consecutive Tail, the posterior belief falls to less than 4%!   In other words, after two Tails, a Bayesian switches from a strong Hawk to nearly as strong a Dove.  

Intuitively, getting two consecutive Tails is much less likely if Model 1 is true than if Model 2 is true.   So, after seeing two Tails, even a relatively dogmatic Bayesian has to switch from Model 1 toward Model 2.  

It may seem that I’ve been unfair in making the (Hawkish) model 1 so extreme and the (Dovish) model 2 so balanced. So, what happens if we assumed that model 2 was more extreme so that it said that the probability of Tails was, say, 0.75 instead of 0.5?  A Bayesian updater would actually become even more sensitive to the data.  Now, a Bayesian Hawk with a prior weight of 0.99 on Model 1 would have a posterior of less than 2% on Model 1 after two consecutive Tails. 

Does this example represent an absurdly simplistic way to model the complex decision-making that is involved in monetary policy?  Absolutely.  But I believe that it helps give a little bit of a feel for why even small amounts of data can lead to large changes in the beliefs of a Bayesian updater."

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