Session II - Estimation methods and accuracy Li-Chun Zhang Discussion: Sess...
Ecmi presentation
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, 2011
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2. The problem Univariate Analysis Multivariable Analysis Conclusion
Introduction and definitions
Introduction
SPOT RATE
INFLATION RATE
NOMINAL RATE
REAL RATE
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3. The problem Univariate Analysis Multivariable Analysis Conclusion
Introduction and definitions
Datas
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4. The problem Univariate Analysis Multivariable Analysis Conclusion
Detecting Trends
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5. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Ornstein-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.
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6. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model Calibration
Maximum 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
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7. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model Calibration
Maximum 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=1
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8. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model Calibration
Maximum 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.
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9. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model Calibration
Results
λ=9.9241 µ = 2.8656 σ = 2.2687
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10. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model Calibration
Results
λ=9.9241 µ = 2.8656 σ = 2.2687
λ=5.8952 µ = 4.4358 σ = 3.1919
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11. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model Calibration
Results
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 = 2.3572
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12. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical 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 = T
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13. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical Approximations
Methods
Euler-Maruyama scheme:
Yi+1 = Yi + a (Yi ) δ + b (Yi ) ∆Wi
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14. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical Approximations
Methods
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
2
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15. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical Results
Implied Inflation
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16. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical Results
Nominal Inflation
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17. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical Results
Real Inflation
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18. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Empirical Distributions
Implied Inflation
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19. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Empirical Distributions
Nominal Inflation
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20. The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Empirical Distribution
Real Inflation
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21. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Autoregressive Model
AR(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.
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22. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Autoregressive Model
Autocorrelation
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 correlated
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23. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Autoregressive Model
Autocorrelation Plots
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24. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
First Order Autoregressive Model
AR(1)
Our model takes the form
Xt+1 = c + ϕXt + εt .
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25. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
First Order Autoregressive Model
AR(1)
Our model takes the form
Xt+1 = c + ϕXt + εt .
Or equivalently
Xt+1 = µ + ϕ (Xt − µ) + N 0, σ 2 .
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26. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
First Order Autoregressive Model
Numerical Results Real Spot Rate
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27. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
PDF Evolution
Real Spot Rate
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28. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Confidence Bands
Entire Data Set
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29. The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Confidence Bands
Partial Data Set
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30. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Multiple Regression
y1 1 x11 x12 ε1
y2 1 x21 x22 β0 ε2
. = . . . β1 + .
.
. .
. .
. .
. .
.
β2
yn 1 xn1 xn2 εn
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31. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Multiple Regression
y1 1 x11 x12 ε1
y2 1 x21 x22 β0 ε2
. = . . . β1 + .
.
. .
. .
. .
. .
.
β2
yn 1 xn1 xn2 εn
Or in equivalently
y = Xβ + ε
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32. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Regressors
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33. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Assumption on the Model
y = Xβ + ε
E (εi ) = 0
Var (εi ) = σ 2 ∀i = 1, . . . , n
Cov(εi , εj ) = 0 ∀i = j
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34. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Least Square Estimation
β Coefficients
If X X is invertible the LSE of β is
ˆ −1
β= XX Xy
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35. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Least Square Estimation
β 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}
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36. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Least Square Estimation
β 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 + ε
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37. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Numerical Results
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38. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
About the Noise
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39. The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Confidence Bands
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40. The problem Univariate Analysis Multivariable Analysis Conclusion
Conclusion
Ornstein-Uhlenbeck
AR(1)
Regression
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41. The problem Univariate Analysis Multivariable Analysis Conclusion
Conclusion
Ornstein-Uhlenbeck
AR(1)
Regression
Validation of the classical Fisher hypothesis
rr = rn − π e .
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42. The problem Univariate Analysis Multivariable Analysis Conclusion
The end
Thank you for attention
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