A Stochastic Recurrence Equation Approach To financial Time Series Models

A stochastic recurrence equation approach to financial time series models
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Thomas Mikosch and Daniel Straumann
University of Copenhagen Laborarory of Actuarial Mathematics www.math.ku.dk/∼mikosch

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Rio de Janeiro, November 9, 2006 RiskMetrics
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The conditionally heteroscedastic model

Xt = σt Zt ,

2 2 2 σt = gθ (Xt−1, . . . , Xt−p, σt−1, . . . , σt−q ) .

2 (Zt)iid, EZ1 = 0, EZ1 = 1.

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Examples GARCH. Bollerslev (1986)
p 2 σt = α0 + i=1 2 αi Xt−i + j=1 q 2 βj σt−j .

AGARCH. Ding, Granger, Engle (1993)
p 2 σt = α0 + i=1 q

αi (|Xt−i| − γ Xt−i)2 +

2 βj σt−j , j=1

|γ| ≤ 1 .

EGARCH. Nelson (1991)
2 2 log σt = α + β log σt−1 + γZt−1 + δ|Zt−1| ,

α, γ, δ ∈ R , |β| < 1 .

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The GARCH

Yt =

      At =       

2 2 2 2 σt+1, . . . , σt−q+2, Xt , . . . , Xt−p+2 2 α1Zt

′

, ··· ··· ··· ... ··· ··· ··· ... 1 αp 0   0   .  .  0 ,  0   0  .  .  0 

+ β1 1 0 . . 0 2 Zt 0 . . 0

β2 0 1 . . 0 0 0 . . 0

Bt = (α0, 0, . . . , 0)′ . Then

· · · βq−1 ··· 0 ··· 0 . ... . ··· 1 ··· 0 ··· 0 . ... . ··· 0

βq 0 0 . . 0 0 0 . . 0

α2 0 0 . . 0 0 1 . . 0

α3 0 0 . . 0 0 0 . . ···

Yt= AtYt−1 + Bt = φt(Yt−1) ,

t ∈ Z.

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This is a stochastic recurrence equation. Notice: ((At , Bt)) and (φt) are iid.

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Stationarity and ergodicity. Bougerol and Picard (1992) There exists a unique causal stationary ergodic non-vanishing solution to the GARCH SRE if and only if α0 > 0 and the top Lyapunov exponent ρ = inf n−1 E log An · · · A1 The solution has form
t t−1

< 0,Yt =
i=−∞

At · · · Ai+1 Bi = Bt +
n

i=−∞

At · · · Ai+1 Bi ,

Example: GARCH(1,1). An · · · A1 =
2 and so ρ = E log(α1Z1 + β1). 2 (α1Zt + β1) , t=1

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Tails. Kesten (1973) Under general conditions, there exists a unique solution κ1 > 0 to the equation n and the following regular variation condition holds for all x = 0:
n→∞ u→∞

0 = lim

1

log E An · · · A1

κ

.lim uκ1 P ((x, Y1) > u) = w(x)

exists and is positive for all non-negative-valued vectors x = 0. Example: GARCH(1,1). Under general assumptions, the equation
2 1 = E(α1Z1 + β1)κ

has a unique positive solution κ1 and (x, Y1) is regularly varying with index κ1.

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If κ1 is not an integer, the finite-dimensional distributions of GARCH are regularly varying. • X ∈ Rd and its distributionare regularly varying with index α > 0: there exists Θ ∈ Sd−1 such that for any t > 0, S ⊂ Sd−1 with P (Θ ∈ ∂S) = 0,

(0.1)

x→∞

lim

P |X| > tx , X ∈ S P (|X| > x)

= t−α P (Θ ∈ S) ,

where x = x/|x| . (Weak convergence) • PΘ is the spectral measure of X.

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• Another way of writing (0.1) is
x→∞

P (|X| > x) P (|X| > tx) lim P X ∈ S = lim x→∞ P (|X| > x) x→∞ = t−α P (Θ ∈ S) .• (0.1) is equivalent to P (x−1X ∈ ·) v → µ(·) P (|X| > x)

lim

P |X| > tx , X ∈ S |X| > x

in R \{0}, for some non-null measure µ with µ(tA) = t−αµ(A).

d

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Mixing. Mokkadem (1990), Boussama (1998) Under general conditions, e.g. Z1 has positive density in neighborhood of the origin, the stationary solution (Yt) to

Yt = A(Zt)Yt−1 + Bt(Zt),

where A(Zt), B(Zt) has entrieswhich are polynomial functions of Zt, is β-mixing with geometric rate.

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Gaussian quasi-MLE for conditionally heteroscedastic models • Gaussian log-likelihood of X1, . . . , Xn from a general conditionally heteroscedastic model: n 1 2 −2 2 Ln(θ) = − [log(σt ) + σt Xt ] n t=1 n t=1 gθ,t • Assume θ lies in the stationarity area Θ of the model and θ0 ∈ Θ is the true parameter underlying thedata. • Maximization in the parameter space Θ yields the Gaussian quasi-maximum likelihood estimator (QMLE) θn of θ0. • Taylor expansion of L′ (θ) and ergodic theorem yield n θn − θ0 = −(L′′ (θn))−1 L′ (θ0) = C0 (1 + o(1)) L′ (θ0) , n n n = − 1
n

[log(gθ,t) +

2 gθ0,tZt

]

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where 1 ′ Ln(θ0) = n
n t=1

1 2 (Zt − 1) = gθ0,t n

′ gθ0,t

n t=1 2 Gt (Zt − 1) .

• Use the CLT...