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The problem Univariate Analysis Multivariable Analysis Conclusion
How mathematicians predict the future?
Mattia Zanella
Group 5Costanza 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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The problem Univariate Analysis Multivariable Analysis Conclusion
Introduction and definitions
Introduction
SPOT RATEINFLATION RATENOMINAL RATEREAL RATE
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The problem Univariate Analysis Multivariable Analysis Conclusion
Introduction and definitions
Datas
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The problem Univariate Analysis Multivariable Analysis Conclusion
Detecting Trends
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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 filtrationsatisfying the usual hypotheses. A stochastic process Xt is anOrnstein-Uhlenbeck process if it satisfies the following stochasticdifferential equation
dXt = λ (µ− Xt) dt + σdWt
X0 = x0
where λ ≥ 0, µ and σ ≥ 0 are parameters, (Wt)t≥0 is a Wienerprocess and X0 is deterministic.
If (St)t≥0 is the process implied/real/nominal inflation we will inour model consider St = expXt ∀t ≥ 0.
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationMaximum Likelihood Estimation
Let (Xt0 , ...,Xtn) n + 1− observations, the Likelihood Function ofXti |Xti−1 is
n∏i=1
fi(Xti ;λ, µ, σ|Xti−1
)
The Log-Likelihood function is defined as
L(X , λ, µ, σ) =n∑
i=1
log f (Xti ;λ, µ, σ|Xti−1).
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationMaximum Likelihood Estimation
Let (Xt0 , ...,Xtn) n + 1− observations, the Likelihood Function ofXti |Xti−1 is
n∏i=1
fi(Xti ;λ, µ, σ|Xti−1
)The Log-Likelihood function is defined as
L(X , λ, µ, σ) =n∑
i=1
log f (Xti ;λ, µ, σ|Xti−1).
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model CalibrationMaximum Likelihood Estimation
Now we have to find
arg maxλ∈R,µ∈R,σ∈R+
L(X , λ, µ, σ)
putting conditions of the first and second order.
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model 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 = 2.3572
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model 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 = 2.3572
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Model 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 = 2.3572
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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 subintervalsof width δ = T
n0 = t0 < t1 < ... < tn = T
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical ApproximationsMethods
Euler-Maruyama scheme:
Yi+1 = Yi + a (Yi ) δ + b (Yi ) ∆Wi
Millstein scheme:
Yi+1 = Yi+a (Yi ) δ+b (Yi ) ∆Wi+12b (Yi ) b′ (Yi )
((∆Wi )
2 − δ)
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical ApproximationsMethods
Euler-Maruyama scheme:
Yi+1 = Yi + a (Yi ) δ + b (Yi ) ∆Wi
Millstein scheme:
Yi+1 = Yi+a (Yi ) δ+b (Yi ) ∆Wi+12b (Yi ) b′ (Yi )
((∆Wi )
2 − δ)
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical ResultsImplied Inflation
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical ResultsNominal Inflation
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Numerical ResultsReal Inflation
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Empirical DistributionsImplied Inflation
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Empirical DistributionsNominal Inflation
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The problem Univariate Analysis Multivariable Analysis Conclusion
Continous Case
Empirical DistributionReal Inflation
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Autoregressive ModelAR(p)
Definition
The AR(p) model is defined as
Xt = c +
p∑i=1
ϕiXt−i + εt
where ϕ1, ..., ϕp are the parameters of the model, c a constant andεt is normally distributed.
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Autoregressive ModelAutocorrelation
Definition
We define autocorrelation coefficient of a random variable Xobserved at times t and s
R(s, t) =E [(Xt − µt) (Xs − µs)]
σsσt.
If R = 1: perfect correlationIf R = −1: anti-correlationIf R = 0: non correlated
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Autoregressive ModelAutocorrelation Plots
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
First Order Autoregressive ModelAR(1)
Our model takes the form
Xt+1 = c + ϕXt + εt .
Or equivalently
Xt+1 = µ+ ϕ (Xt − µ) + N(0, σ2) .
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
First Order Autoregressive ModelAR(1)
Our model takes the form
Xt+1 = c + ϕXt + εt .
Or equivalently
Xt+1 = µ+ ϕ (Xt − µ) + N(0, σ2) .
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
First Order Autoregressive ModelNumerical Results Real Spot Rate
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
PDF EvolutionReal Spot Rate
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Confidence BandsEntire Data Set
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The problem Univariate Analysis Multivariable Analysis Conclusion
Discrete Case
Confidence BandsPartial Data Set
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Multiple Regression
y1y2...yn
=
1 x11 x121 x21 x22...
......
1 xn1 xn2
β0
β1β2
+
ε1ε2...εn
Or in equivalentlyy = Xβ + ε
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Multiple Regression
y1y2...yn
=
1 x11 x121 x21 x22...
......
1 xn1 xn2
β0
β1β2
+
ε1ε2...εn
Or in equivalently
y = Xβ + ε
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Regressors
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Assumption on the Modely = Xβ + ε
E (εi ) = 0Var (εi ) = σ2 ∀i = 1, . . . , nCov(εi , εj) = 0 ∀i 6= j
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Least Square Estimationβ Coefficients
If X′X is invertible the LSE of β is
β =(X′X
)−1 X′y
β0, β1, β2 = −0.0068,−0.9869, 0.9935
Real = β0 + β1Nominal + β2Implied + ε
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Least Square Estimationβ Coefficients
If X′X is invertible the LSE of β is
β =(X′X
)−1 X′y
β0, β1, β2 = −0.0068,−0.9869, 0.9935
Real = β0 + β1Nominal + β2Implied + ε
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Least Square Estimationβ Coefficients
If X′X is invertible the LSE of β is
β =(X′X
)−1 X′y
β0, β1, β2 = −0.0068,−0.9869, 0.9935
Real = β0 + β1Nominal + β2Implied + ε
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Numerical Results
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
About the Noise
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The problem Univariate Analysis Multivariable Analysis Conclusion
Multiple Regression
Confidence Bands
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The problem Univariate Analysis Multivariable Analysis Conclusion
Conclusion
Ornstein-UhlenbeckAR(1)Regression
Validation of the classical Fisher hypothesis
rr = rn − πe .
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The problem Univariate Analysis Multivariable Analysis Conclusion
Conclusion
Ornstein-UhlenbeckAR(1)Regression
Validation of the classical Fisher hypothesis
rr = rn − πe .
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The problem Univariate Analysis Multivariable Analysis Conclusion
The end
Thank you for attention
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