ecmi presentation

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

How mathematicians predict the future? ECMI

European Consortium for Mathematics in Industry


Introduction and definitions

Introduction

SPOT RATEINFLATION RATENOMINAL RATEREAL RATE




Introduction and definitions

Datas




Detecting Trends




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.




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).




Continous Case


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).




Continous Case


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.




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




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




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)




Continous Case

Numerical ResultsImplied Inflation




Continous Case

Numerical ResultsNominal Inflation




Continous Case

Numerical ResultsReal Inflation




Continous Case

Empirical DistributionsImplied Inflation




Continous Case

Empirical DistributionsNominal Inflation




Continous Case

Empirical DistributionReal Inflation




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.




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




Discrete Case

Autoregressive ModelAutocorrelation Plots




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) .




Discrete Case

First Order Autoregressive ModelNumerical Results Real Spot Rate




Discrete Case

PDF EvolutionReal Spot Rate




Discrete Case

Confidence BandsEntire Data Set




Discrete Case

Confidence BandsPartial Data Set




Multiple Regression

Multiple Regression

y1y2...yn

=

1 x11 x121 x21 x22...

......

1 xn1 xn2

β0

β1β2

+

ε1ε2...εn

Or in equivalentlyy = Xβ + ε




Multiple Regression

Multiple Regression

y1y2...yn

=

1 x11 x121 x21 x22...

......

1 xn1 xn2

β0

β1β2

+

ε1ε2...εn

Or in equivalently

y = Xβ + ε




Multiple Regression

Regressors




Multiple Regression

Assumption on the Modely = Xβ + ε

E (εi ) = 0Var (εi ) = σ2 ∀i = 1, . . . , nCov(εi , εj) = 0 ∀i 6= j




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




Multiple Regression

Numerical Results




Multiple Regression

About the Noise




Multiple Regression

Confidence Bands




Conclusion

Ornstein-UhlenbeckAR(1)Regression

Validation of the classical Fisher hypothesis

rr = rn − πe .




The end

Thank you for attention



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