Project for the European Consortium for Mathematics in Industry, ESSIM Summer School 2011 (Milan)

1. The Problem Univariate Analysis Multivariate Analysis Conclusion
How mathematicians predict the future?
Instructor: Agnieszka Wyloma´ska
n
Costanza Catalano, Angela Ciliberti, Gonçalo S. Matos, Allan S. Nielsen,
Olga Polikarpova, Mattia Zanella
European Summer School in Industrial Mathematics
Modelling Week
July 30, 2011
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2. The Problem Univariate Analysis Multivariate Analysis Conclusion
Supplied Data for Analysis
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3. The Problem Univariate Analysis Multivariate Analysis Conclusion
Supplied Data for Analysis
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4. The Problem Univariate Analysis Multivariate Analysis Conclusion
Our Approach
Univariate Analysis
Orstein-Unlenbeck Model
Autoregressive Model
Multivariate Analysis
Linear Regression
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5. The Problem Univariate Analysis Multivariate Analysis Conclusion
Orstein-Uhlenbeck process
Orstein-Uhlenbeck process
Orstein-Uhlenbeck process
The Orstein-Uhlenbeck process (or mean-reverting process) is
defined by the following equation:
dXt = θ(µ − Xt )dt + σdWt
Where Wt is a Wiener process, t ∈ T ⊆ R+ represents time and
θ > 0, µ and σ > 0 are time independent constants.
Here Xt = log(St ) is the logarithm of the implied/nominal/real
inflation St .
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6. The Problem Univariate Analysis Multivariate Analysis Conclusion
Orstein-Uhlenbeck process
Euler Maruyama method
Euler Maruyama method
The Euler Maruyama method is a method for the approximate
numerical solution of a stochastic differential equation. In our
case, for a partition of [t, t + 1] in n equal subintervals:
Xn+1 = Xn + θ(µ − Xn )δ + σ∆Wn
Where δ = 1/N is the length of the subintervals, and ∆Wn are
independent identically distributed random varibles with expected
value of 0 and a variance of δ.
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7. The Problem Univariate Analysis Multivariate Analysis Conclusion
Orstein-Uhlenbeck process
Empirical Distribution for 1 Step Prediction
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8. The Problem Univariate Analysis Multivariate Analysis Conclusion
AR(p)
Autoregressive model
Autoregressive model
The autoregressive model of order p, AR(p), is defined as:
p
Yt = a0 + ai Yt−i + εt
i=1
Where a0 , a1 , . . . , ap are the parameters of the model and εt is
independent identically distributed random variables.
Here Yt = St − St−1 is the backward difference of the
implied/nominal/real inflation St .
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9. The Problem Univariate Analysis Multivariate Analysis Conclusion
AR(p)
Autocorrelation Function of the Implied Inflation
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10. The Problem Univariate Analysis Multivariate Analysis Conclusion
AR(p)
Forecast
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11. The Problem Univariate Analysis Multivariate Analysis Conclusion
AR(p)
Evolution of Probability Distributions
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12. The Problem Univariate Analysis Multivariate Analysis Conclusion
Confidence Bands
Confidence Band (close up)
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13. The Problem Univariate Analysis Multivariate Analysis Conclusion
Confidence Bands
Confidence Band (all view)
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14. The Problem Univariate Analysis Multivariate Analysis Conclusion
Confidence Bands
Confidence Band (2 Years Data)
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15. The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Correlation between Time Series
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16. The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Linear Regression
The multivariate regression model is:
Y = XT β + ε
E(Y) = XT β
ΣY = σ 2 1
Where Y are the response variables and X are the explanatory
variables.
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17. The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Linear Regression Prediction
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18. The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Error in the Prediction
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19. The Problem Univariate Analysis Multivariate Analysis Conclusion
Linear Regression
Confidence Band
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20. The Problem Univariate Analysis Multivariate Analysis Conclusion
Conclusion
Final Remarks
• Summary:
Confidence Band and Spread control
Implied inflation, Real and Nominal seem to be correlated
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