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1. 1. The problem Univariate Analysis Multivariable Analysis Conclusion How mathematicians predict the future? Mattia Zanella Group 5 Costanza Catalano, Angela Ciliberti, Goncalo S. Matos, Allan S. Nielsen, Olga Polikarpova, Mattia Zanella Instructor: Dr inz. Agnieszka Wyłomańska (Hugo Steinhaus Center) ˙ December 22, 2011How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
2. 2. The problem Univariate Analysis Multivariable Analysis ConclusionIntroduction and deﬁnitionsIntroduction SPOT RATE INFLATION RATE NOMINAL RATE REAL RATEHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
3. 3. The problem Univariate Analysis Multivariable Analysis ConclusionIntroduction and deﬁnitionsDatasHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
4. 4. The problem Univariate Analysis Multivariable Analysis ConclusionDetecting TrendsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
5. 5. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseOrnstein-Uhlenbeck Process Deﬁnition Let (Ω, F, P) a probability space and F = (Ft )t≥0 a ﬁltration satisfying the usual hypotheses. A stochastic process Xt is an Ornstein-Uhlenbeck process if it satisﬁes the following stochastic diﬀerential equation dXt = λ (µ − Xt ) dt + σdWt X0 = x 0 where λ ≥ 0, µ and σ ≥ 0 are parameters, (Wt )t≥0 is a Wiener process and X0 is deterministic. If (St )t≥0 is the process implied/real/nominal inﬂation we will in our model consider St = exp Xt ∀t ≥ 0.How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
6. 6. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseModel CalibrationMaximum Likelihood Estimation Let (Xt0 , ..., Xtn ) n + 1 − observations, the Likelihood Function of Xti |Xti −1 is n fi Xti ; λ, µ, σ|Xti −1 i=1How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
7. 7. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseModel CalibrationMaximum Likelihood Estimation Let (Xt0 , ..., Xtn ) n + 1 − observations, the Likelihood Function of Xti |Xti −1 is n fi Xti ; λ, µ, σ|Xti −1 i=1 The Log-Likelihood function is deﬁned as n L(X , λ, µ, σ) = log f (Xti ; λ, µ, σ|Xti −1 ). i=1How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
8. 8. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseModel CalibrationMaximum Likelihood Estimation Now we have to ﬁnd arg max L(X , λ, µ, σ) λ∈R,µ∈R,σ∈R+ putting conditions of the ﬁrst and second order.How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
9. 9. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseModel CalibrationResults λ=9.9241 µ = 2.8656 σ = 2.2687How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
10. 10. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseModel CalibrationResults λ=9.9241 µ = 2.8656 σ = 2.2687 λ=5.8952 µ = 4.4358 σ = 3.1919How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
11. 11. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseModel CalibrationResults λ=9.9241 µ = 2.8656 σ = 2.2687 λ=5.8952 µ = 4.4358 σ = 3.1919 λ=4.5916 µ = 1.5487 σ = 2.3572How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
12. 12. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical Approximations Consider a general SDE dXt = a(Xt )dt + b (Xt ) dWt , t ∈ [0, T ] and a partition of the time interval [0, T ] into n equal subintervals of width δ = Tn 0 = t0 < t1 < ... < tn = THow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
13. 13. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical ApproximationsMethods Euler-Maruyama scheme: Yi+1 = Yi + a (Yi ) δ + b (Yi ) ∆WiHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
14. 14. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical ApproximationsMethods Euler-Maruyama scheme: Yi+1 = Yi + a (Yi ) δ + b (Yi ) ∆Wi Millstein scheme: 1 Yi+1 = Yi +a (Yi ) δ+b (Yi ) ∆Wi + b (Yi ) b (Yi ) (∆Wi )2 − δ 2How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
15. 15. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical ResultsImplied InﬂationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
16. 16. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical ResultsNominal InﬂationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
17. 17. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical ResultsReal InﬂationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
18. 18. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseEmpirical DistributionsImplied InﬂationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
19. 19. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseEmpirical DistributionsNominal InﬂationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
20. 20. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseEmpirical DistributionReal InﬂationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
21. 21. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseAutoregressive ModelAR(p) Deﬁnition The AR(p) model is deﬁned as p Xt = c + ϕi Xt−i + εt i=1 where ϕ1 , ..., ϕp are the parameters of the model, c a constant and εt is normally distributed.How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
22. 22. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseAutoregressive ModelAutocorrelation Deﬁnition We deﬁne autocorrelation coeﬃcient of a random variable X observed at times t and s E [(Xt − µt ) (Xs − µs )] R(s, t) = . σs σt If R = 1: perfect correlation If R = −1: anti-correlation If R = 0: non correlatedHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
23. 23. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseAutoregressive ModelAutocorrelation PlotsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
24. 24. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseFirst Order Autoregressive ModelAR(1) Our model takes the form Xt+1 = c + ϕXt + εt .How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
25. 25. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseFirst Order Autoregressive ModelAR(1) Our model takes the form Xt+1 = c + ϕXt + εt . Or equivalently Xt+1 = µ + ϕ (Xt − µ) + N 0, σ 2 .How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
26. 26. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseFirst Order Autoregressive ModelNumerical Results Real Spot RateHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
27. 27. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CasePDF EvolutionReal Spot RateHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
28. 28. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseConﬁdence BandsEntire Data SetHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
29. 29. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseConﬁdence BandsPartial Data SetHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
30. 30. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionMultiple Regression       y1 1 x11 x12   ε1  y2   1 x21 x22  β0  ε2  . = . . .  β1  +  .        . . . . . . .  . . .  β2      yn 1 xn1 xn2 εnHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
31. 31. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionMultiple Regression       y1 1 x11 x12   ε1  y2   1 x21 x22  β0  ε2  . = . . .  β1  +  .        . . . . . . .  . . .  β2      yn 1 xn1 xn2 εn Or in equivalently y = Xβ + εHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
32. 32. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionRegressorsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
33. 33. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionAssumption on the Modely = Xβ + ε E (εi ) = 0 Var (εi ) = σ 2 ∀i = 1, . . . , n Cov(εi , εj ) = 0 ∀i = jHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
34. 34. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionLeast Square Estimationβ Coeﬃcients If X X is invertible the LSE of β is ˆ −1 β= XX XyHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
35. 35. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionLeast Square Estimationβ Coeﬃcients If X X is invertible the LSE of β is ˆ −1 β= XX Xy {β0 , β1 , β2 } = {−0.0068, −0.9869, 0.9935}How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
36. 36. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionLeast Square Estimationβ Coeﬃcients If X X is invertible the LSE of β is ˆ −1 β= XX Xy {β0 , β1 , β2 } = {−0.0068, −0.9869, 0.9935} Real = β0 + β1 Nominal + β2 Implied + εHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
37. 37. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionNumerical ResultsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
38. 38. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionAbout the NoiseHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
39. 39. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionConﬁdence BandsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
40. 40. The problem Univariate Analysis Multivariable Analysis ConclusionConclusion Ornstein-Uhlenbeck AR(1) RegressionHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
41. 41. The problem Univariate Analysis Multivariable Analysis ConclusionConclusion Ornstein-Uhlenbeck AR(1) Regression Validation of the classical Fisher hypothesis rr = rn − π e .How mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
42. 42. The problem Univariate Analysis Multivariable Analysis ConclusionThe end Thank you for attentionHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry