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  • 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. 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. The problem Univariate Analysis Multivariable Analysis ConclusionIntroduction and definitionsDatasHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 4. The problem Univariate Analysis Multivariable Analysis ConclusionDetecting TrendsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical ResultsImplied InflationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 16. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical ResultsNominal InflationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 17. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseNumerical ResultsReal InflationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 18. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseEmpirical DistributionsImplied InflationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 19. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseEmpirical DistributionsNominal InflationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 20. The problem Univariate Analysis Multivariable Analysis ConclusionContinous CaseEmpirical DistributionReal InflationHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 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. 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. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseAutoregressive ModelAutocorrelation PlotsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 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. 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. 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. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CasePDF EvolutionReal Spot RateHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 28. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseConfidence BandsEntire Data SetHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 29. The problem Univariate Analysis Multivariable Analysis ConclusionDiscrete CaseConfidence BandsPartial Data SetHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 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. 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. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionRegressorsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 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. 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. 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. 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. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionNumerical ResultsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 38. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionAbout the NoiseHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 39. The problem Univariate Analysis Multivariable Analysis ConclusionMultiple RegressionConfidence BandsHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 40. The problem Univariate Analysis Multivariable Analysis ConclusionConclusion Ornstein-Uhlenbeck AR(1) RegressionHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry
  • 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. The problem Univariate Analysis Multivariable Analysis ConclusionThe end Thank you for attentionHow mathematicians predict the future? ECMIEuropean Consortium for Mathematics in Industry

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