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Check out PDF for Equations
And Code to run yourself
Introduction:< o:p>
“This Time is different” by Reinhart and Rogoff is an empirical history of financial crises and panics. It describes many economic events such as; inflation, bubbles, and defaults.< o:p>
The theme that ties these events together is the idea that people have an unrealistic expectation of the future because they believe “This time is different”. With this new belief, they act in a way as to cause the next crisis.< o:p>
While the book dealt with the empirical investigation of the matter, it didn’t describe the theme itself in intricate detail. This post will attempt to fill that void by giving a mathematical interpretation of the theme. < o:p>
There are 3 aspects of the model I’d like to capture,< o:p>
< !--[if !supportLists]-->1. < !--[endif]-->prior beliefs act guide and tell us how to make decisions< o:p>
< !--[if !supportLists]-->2. < !--[endif]-->decisions have an effect on the future< o:p>
< !--[if !supportLists]-->3. < !--[endif]-->people update our beliefs when new information becomes available< o:p>
Economic Maximization Problem:< o:p>
I came up with a simple driving example that describes these properties. Suppose we’d like to maximize the speed of a car and minimize the probability of a crash (or maximize probability of not crashing). However, the probability of a car crashing is dependent on the speed. This can be described as a utility maximization problem described below, where we maximize subject to a constraint. < o:p>
Maximize:
Utility(Speed,Probability of Not Crashing) = (Speed^a)*(Probability of not crashing ^(1-a))
Utility(Speed,Probability of Not Crashing) = (Speed^a)*(Probability of not crashing ^(1-a))
Subject to:< o:p>
< !--[if gte msEquation 12]>< m:oMathPara>< m:oMath>< m:r>Probability< m:r> < m:r>Not< m:r> < m:r>Cras< m:r>h< m:r>ing< m:r>=< m:func>< m:funcPr>< m:ctrlPr>< m:fName>< m:r>< m:rPr>< m:scr m:val="roman"/>< m:sty m:val="p"/>β*log< m:e>< m:d>< m:dPr>< m:ctrlPr>< m:e>< m:f>< m:fPr>< m:ctrlPr>< m:num>< m:r>1< m:den>< m:r>Speed< m:r>–< m:r>1< m:r>+< m:r>C< ![endif]-->< !--[if !msEquation]-->< !--[endif]-->< o:p>
Probability of Not Crashing = B*log(1/speed-1) + C
Where B < !--[if gte msEquation 12]>< m:oMath>< m:r>C< m:r> < ![endif]-->< !--[if !msEquation]-->< !--[endif]-->and C < !--[if gte msEquation 12]>< m:oMath>< m:r>β< ![endif]-->< !--[if !msEquation]-->< !--[endif]--> are the constant and slope parameters of an inverse logistic function and a< !--[if gte msEquation 12]>< m:oMath>< m:r>α< ![endif]-->< !--[if !msEquation]-->< !--[endif]--> is tradeoff for utility function between the two goods (Probability of not crashing and Speed).< o:p>
(Look in PDF for optimal solution)
Bayesian Updating Problem:< o:p>
Described so far the problem is a fairly straightforward maximization problem. However, in this model, the parameters B< !--[if gte msEquation 12]>< m:oMath>< m:r>C< ![endif]-->< !--[if !msEquation]-->< !--[endif]--> and C< !--[if gte msEquation 12]>< m:oMath>< m:r>β< ![endif]-->< !--[if !msEquation]-->< !--[endif]--> are unknown to the individual. While there is a true function for the relationship between Speed and probability of not crashing < !--[if gte msEquation 12]>< m:oMath>< m:r>Probability< m:r> < m:r>of< m:r> < m:r>Not< m:r> < m:r>Cras< m:r>h< m:r>ing< ![endif]-->< !--[if !msEquation]-->< !--[endif]--> (that is a true B< !--[if gte msEquation 12]>< m:oMath>< m:r>C< ![endif]-->< !--[if !msEquation]-->< !--[endif]--> and C) the individual doesn’t know it. < o:p>
Bayesian statistics seems like a natural to model the individual’s beliefs over these parameters. With this prior belief over parameters, he maximizes his utility subject to it and makes a decision in each time period. When new information becomes he updates his prior information and uses this to make a new decision. In this sense, the system is dynamic and a time series can be produced. < o:p>
Unfortunately, I don’t believe there is a closed form solution to updating Bayesian logistic regression. Therefore, I used RSTAN to simulate from the posterior distributions (Code shown at end). < o:p>
I set the model with parameters of with, a=.05, B=1, and C=100. The initial prior probabilities are parameters C~N(100,5) and B~N(1,.05)< !--[if gte msEquation 12]>< m:oMath>< m:r>C< m:r>~ < m:r>N< m:d>< m:dPr>< m:ctrlPr>< m:e>< m:r>100,5< ![endif]-->< !--[if !msEquation]-->< !--[endif]--> , < !--[if gte msEquation 12]>< m:oMath>< m:r>β< m:r> ~ < m:r>N< m:d>< m:dPr>< m:ctrlPr>< m:e>< m:r>1,.05< ![endif]-->< !--[if !msEquation]-->< !--[endif]--> where N(x,y) is a normal distribution with mean x and stdev y. Below is one example of a simulated time series.< o:p>
< o:p> 
As you can see, the individual is learning the true parameters as he goes through time. When a crash occurs, he dramatically reduces his decision and if a crash doesn’t occur, he slightly increased his decision. This can be seen as a smoothed over “this time is different” where the actor adjusts his decision more optimistically until a negative event happen, in which case, he changes his decision more abruptly. < o:p>
The optimum decision is shown in the blue horizontal line, and while the actor doesn’t achieve it in this example, he will eventually (… I think) as this is the equilibrium.< o:p>
Conclusion:< o:p>
The purpose of this post was to describe a mathematical interpretation of “This Time is Different”. I did this by solving a maximization problem, with unknown parameters, and then updating those parameters using the Bayesian Framework.< o:p>
While the example I used was only relation to speed and car crashes, I don’t think its much of a stretch to use this model to describe economic events. An example of might be that a bank wants to have an optimal amount of amount of capital requirements vs event of bankruptcy for a bank. (The smaller capital requirements, the higher the expected profits, and higher probability of bankruptcy.)< o:p>
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