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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

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