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A 3-VARIABLE MODEL PREDICTING SUBPRIME CRISIS A POSTERIORI

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Préambule
Voici un premier jet de ma recherche sur un modèle mathématique de la crise des subprimes. Sa particularité est que je n'ai pas fixé les nonlinéarités (en fonction des relations de causalité qui devraient exister entre prix de l'immobilier, taux d'intért et dette des ména!e dans la réalité" mais j'ai laissé l'ordinateur choisir le meilleur fit, conduisant # des non$linéarités qui sont cependant asse% justes (voir fi!ure & et remarques subséquentes".
lorent ieterlen ) *$V)+)-/ 01/ P+/2345 S6-P+0/ 2+SS ) P1S3/+1+43+162314
7e present a simple model that predicts the subprime crisis of 899:$899; <ith onl= * variables (more specificall=, 8 variables and a parameter", su!!ested b= Sti!lit% (Sti!lit% 899;" > interest rates, housin! prices, and household debt.t means that <e don't inte!rate the effect of investment in the housin! mar?et.t is a d=namical s=stem model, that is to sa= coupled nonlinear differential equations. 3he most usual example of a d=namical s=stem is the predator pre= model > the rabbit eats !rass, !ro<s in number, and the number of foxes <ho eat rabbits !ro<s therefore also. 3hen <hen there are too man= foxes, the= eat a lot of rabbits <hose population reduces to a point <here foxes starve and reduce in number, then rabbits !ro< a!ain, etc. 3his is an example of oscillatin! d=namical s=stem, <here the increase of both populations (time derivatives on the left of the differential equations" depend on both populations numbers (a function to the ri!ht of the differential equation, <hich <e shall find <ith a non linear re!ression".7hat should be <ell understood is <e deal <ith variables interactin! to!ether, each one influencin! the others in a non linear <a=, and <ith couplin!s > for example, if r )4 p influence to!ether d. 3he choice of variables is ver= important, and should be done b= an economist that ?no<s all the causal influences bet<een potential variables. 3he choice should be done such as the variables chosen form a complete s=stem in mutual interactions. t should include a limited numbers of variables. 3his approach <ith fe< variables could be compared <ith thermod=namics (e.!. <ith pression, volume and temperature" versus methods li?e statistical mechanics that use a !reat deal of variables, except that our method is lin?ed to far from equilibrium thermod=namics, not equilibrium.3o compare it <ith statistics, the explainin! variables are r, p and d, and the explained variable is the s=stem in its <holeness, and the question is > does it become unstable @7e constructed a 8 variable d=namical s=stem for prices (p" and debt (d", <ith interest rates (r" as a parameter in the model, out of historical data for these variables and parameter from 8999 to 899: (see A data B and A references for the data B at the
end" > <e assume that the causalit= relationships bet<een them is the same in 899:$899; than in 8999$899;. 3his construction <as done usin! a method that enables to derive a s=stem of coupled nonlinear ordinar= differential equations out of historical data, usin! a nonlinear re!ression on the time derivatives. 3his method <as published in 899: in A )rchives des Sciences B, <here it <as validated and <here examples in different fields (ps=chiatr=, bacteria competition, demo!raph=, economics and finance" <ere treated (ieterlen 899:".
0/3C1
7e have used quarterl= data coverin! the time span from second quarter 8999 to second quarter 899:, allo<in! previsions from the third quarter of 899: on. 3he timela! of : =ears is arbitrar= here (it include nevertheless a period at the be!inin! <here the interest rates are hi!h". n the future, this timela! <ill have to be decided b= an economist, jud!in! a timela! <here the d=namics and the causal infuences <ere rou!hl= constant. the= should be the same in this timela!.7e didn't use mensual data, because the statistics for the fit <ere ver= bad. t probabl= means that some additional information adds up <hen one uses months, and that this additional information dro<ns the usefull information.7e then derived the time derivatives for p, r and d (derivatives on five points", and the nonlinearities > linear, square, cubic, square root, the * couplin!s of de!ree 8 and the D couplin!s of third de!ree.3hen <e did a linear re!ression of the time derivatives of p and d in function of all the linear and nonlinear terms. t is this re!ression <hich !ives the statistical properties of the model, essential= the explained variance and the !oodness of the fit. 1ne is at that point in front of a dilemna > either impose the causal influences bet<een the variables from an anl=sis of the ?no<n realit=, or let the computer do the step<ise re!ression that leads to the best combination of explained variance and !oodness of the fit. 1ne <ould li?e to do a combination of these t<o approaches.n the future, different re!ressions should be done, var=in! the number of nonlinesarities, and should be retained the ones that have hte best combination of explained variance, !oodness of the fit, and correctness of the mutual influences.7e didn't construct a differential equation for r, because its variations are human$ based, and therefore not A automaticall= B function of p and r.nstead, <e introduced r in the nonlinearities for p and d, so it became a parameter.7e therefore !ot a d=namical s=stem composed of 8 variables, p and d, and one parameter, r. See fi!ure & for the causalit= relations bet<een p, d and r.
$
dp
E Esquare root $
r
i! &$a > influences from r and p on d proposed b= the step<ise re!ression. E and F mean positive or ne!ative influences, double arro<s mean de!ree 8.)cordin! to <i?ipedia sites on subprime crisis, p influence d ne!ativel=, <hich is the case (se in the references S02 p.8$*".)s for interest rates influencin! debt, it is ne!ative in 8998$899G (see 2S, lo< rates, <hich corresponds to lo< de!rees in influence, <hich is the case here", and positive after 899G, <hich correspond to the second de!ree term here (see S02".
$ $$square root
d E p $ E Es.r.
$E $sq. rootE E$ $
Er
i! &$b > influences from r and d on p proposed b= the step<ise re!ression. E and F mean positive or ne!ative influences, double arro<s mean de!ree 8.7i?ipedia articles on the subprime crisis sa= that <hen debt is hi!h (to<ards the end of the period", it ma?es prices !o do<n, <hich is the case here > the double arrro<, the most active for hi!h values, is ne!ative, so are the coupled influences from p )4 d (S02 p.;, S2-". 3he prices should increase b= their o<n (p influences p" positivel= for lo< prices (S02 p.;", before 899D (square root positive", and ne!ativel= for hi!h prices $ unsold homes, etc. , see S02$ (here p influences positivel= p". 7e have another factor to the square, positive that <e don't explain.3he interest rate influences prices positivel=, <hich is accepted (S02 p.:", as hi!h rates and hi!h prices lo<er the price (coupled terms p )4 r".
+/S63S
3hen <e derived the stabilit= (see 5lansdorff and Pri!o!ine &H:&" for the 8 differentail equations, and entered the values for r, p and d for dates from 8999 to 899;. See fi!ure 8 for the results. n future models of prevision, one should enter several scenarii.
Instabilité
-10-50510152005q2 05q3 05q4 06q1 06q2 06q3 06q4 07q1 07q2 07q3 07q4 08q1 08q2
an
Lambda-1Lambda-2
i!ure 83he s=stem is stable in 899I$D, metastable in 899: and unstable in 899;.1ne can find a critic to this result > <e used the information !iven b= data from 8999 till 899: to find the stabilit= before 899:. 3he critic is valid, because in real life, one has to do the model for the =ears preceedin! the prediction. 3herefore the main result of our model is that the s=stem !ets unstable at the end of 899: and in 899;, <ith the help of data ran!in! from 8999 to 899: > <e predicted the subprime crisis of 899:$899;.
21426S14
)s a main result, <e persented a !raphic (fi! 8" sho<in! the stabilit= variations from 899: to 899;. t sho<s a decisive increase in instabilit= in 899;, startin! in 899:, coincidin! <ith the subprime cisis. t follo<s a period of stabilit= in 899I$899D, <hich is questionable for it <as derived from data ran!in! from 8999 to 899:.3his sharp transition from stable to metastable to unstable beteen 899I and 899; could have alerted the financial communit=, had this model been available at that time, and had it been updated ever= quarter. Co<ever, this could have been done if one predicted the next values of r, p and d. 1ne possibilit= is to !et several scenarii for p and d, and then see ho< r could lead to an instabilit=, therefore to a crisis.)ccordin! to this scenario, this method can be applied to other financial bubbles (li?e the housin! bubble in S<it%erland or the !old bubble" in the future, to prevent other crisis. +//+/42/S2S J <i?ipedia A 2rise des subprimes B sept$89&*/3/+/4, . 899: 0ettre les sciences humaines en équations, )rchives des Sciences D&, *I$IG 5)4S1+ P, P+5154/ . &H:&.3hermod=namic theor= of structure, stabilit= and fluctuation. Kohn7ile= and Sons, ondon. S2- J <i?ipedia A subprime bac?!round information B sept. 89&*

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