GARCH Processes – Probabilistic Properties (Part 1)

GARCH processes – probabilistic properties (Part 1)

Alexander Lindner

Centre of Mathematical Sciences Technical University of Munich D–85747 Garching Germany [email protected] http://www-m1.ma.tum.de/m4/pers/lindner/

Maphysto Workshop Non-Linear Time Series Modeling Copenhagen September 27 – 30, 2004

1 Contents

1. Deﬁnition of GARCH(p,q) processes 2. Markov property 3. Strict stationarity of GARCH(1,1) 4. Existence of 2nd moment of stationary solution 5. Tail behaviour, extremal behaviour 6. What can be done for the GARCH(p,q)? 7. GARCH is White Noise 8. ARMA representation of squared GARCH process 9. The EGARCH process and further processes

2 GARCH(p,q) on N0

(εt)t N0 i.i.d.,P (ε0 = 0) = 0 (1) ∈ α0 > 0, α1, . . . , αp 0, β1, . . . , βq 0. 2 2 ≥ ≥ (σ0, . . . , σmax(p,q) 1) independent of εt : t max(p, q) 1 − { ≥ − }

Yt = σtεt, (2) p q 2 2 2 σt = α0 + αiYt i + βjσt j, t max(p, q). (3) X − X − i=1 j=1 ≥ ARCH | {z } GARCH | {z } (Yt)t N0 GARCH(p,q) process ∈

3 GARCH(p,q) on Z (1), (2) and (3) for t Z, and ∈ εt independent of

Yt 1 := Ys : s t 1 . − { ≤ − }

ARCH(p): Engle (1981) GARCH(p,q): Bollerslev (1986) Generalized AutoRegressive Conditional Heteroscedasticity

4 Why conditional volatility?

2 Suppose σt Yt 1, ∈ − 2 or equivalently σt εt 1, t Z ∈ − ∀ ∈ (the process is “causal”)

2 Suppose Eεt = 0, Eε = 1, EYt < . Then t ∞

E(Yt Yt 1) = E(σtεt Yt 1) = σt E(εt Yt 1) = 0, − − − | 2 2| 2 2| 2 V (Yt Yt 1) = E(σt εt Yt 1) = σt E(εt Yt 1) = σt | − | − | − 2 Hence σt is the conditional variance

5 Markov property (a) GARCH(1,1):

2 (σt )t Z is a Markov process, since ∈ 2 2 2 σt = α0 + (α1εt 1 + β1)σt 1 − − 2 (σt ,Yt)t Z is Markov process, but (Yt)t Z is not. ∈ ∈

(b) GARCH(p,q), max(p, q) > 1

2 (σt )t Z is not Markov process ∈ 2 2 (σt , . . . , σt max(p,q)+1)t Z is Markov process. − ∈

6 Strict stationarity - GARCH(1,1) 2 2 2 2 σt = α0 + α1σt 1εt 1 + β1σt 1 2 − −2 − = (α1εt 1 + β1)σt 1 + α0 2 − − =: Atσt 1 + Bt (4) −

(At,Bt)t Z is i.i.d. sequence. ∈ (4) is called a random recurrence equation.

2 2 σt = Atσt 1 + Bt − 2 = AtAt 1σt 1 + AtBt 1 + Bt − − . − k 2 = At At kσt k 1 + At At i+1 Bt i . (5) ··· − − − X ··· − − i=0 =α0 |{z} Let k and hope that (5) converges. → ∞

7 Strict stationarity - continued

Suppose γ := E log A1 < 0

1 k (log At + log At 1 + ... + log At k+1) →∞ E log A1 a.s. k − − −→ random walk! | {z }

1 = a.s. ω n0(ω): log (At At k+1) γ/2 < 0 k n0(ω) ⇒ ∀ ∃ k ··· − ≤ ∀ ≥

kγ/2 = At At k+1 e k n0(ω) ⇒ | ··· − | ≤ ∀ ≥ Hence (5) converges almost surely.

8 Theorem: (Nelson, 1990) If 2 E log(α1ε1 + β1) < 0, (6) then GARCH(1,1) has a strictly stationary solution. Its marginal distribution is given by

∞ 2 2 α0 (α1ε 1 + β1) (α1ε i + β1) (7) X − − i=0 ···

If β1 > 0 (i.e. GARCH but not ARCH) and a strictly stationary solution exists, then (6) holds.

Remark: + If E log A1 < , then | | ∞ 1 γ = inf E log A0A 1 A n : n N n + 1 | − ··· − | ∈ is called the Lyapunov exponent of the sequence (An)n Z. ∈

9 When is EY 2 < ? t ∞ 2 ∞ 2 2 σ0 = α0 (α1ε 1 + β1) (α1ε i + β1) X − − i=0 ···

2 ∞ 2 i Eσ = α0 E(α1ε + β1) < 0 X 1 i=0 ∞ 2 α1Eε + β1 < 1 ⇐⇒ 1

2 2 2 EYt = Eσt Eεt

So stationary solution (Yt)t Z has ﬁnite variance iﬀ ∈ 2 α1Eε1 + β1 < 1

10 Tail behaviour of stochastic recurrence equations Theorem (Goldie 1991, Kesten 1973, Vervaat 1979) Suppose (Zt)t N satisﬁes the stochastic recurrence equation ∈ Zt = AtZt 1 + Bt, t N, − ∈ where ((At,Bt))t N,(A, B) i.i.d. sequences. Suppose κ > 0 such that ∈ ∃ (i) The law of log A , given A = 0, is not concentrated on rZ for any r > 0 | | | | 6 (ii) E A κ = 1 (iii) E| A| κ log+ A < (iv) E|B|κ < | | ∞ | | ∞ Then Z =d AZ +B, where Z independent of (A, B), has a unique solution in distribution which satisﬁes E[((AZ + B)+)κ ((AZ)+)κ] lim xκP (Z > x) = − =: C 0 x κE A κ log+ A →∞ ≥ | | | | (8)

11 Tail behaviour of GARCH(1,1)

Corollary: Suppose ε1 is continuous, P (ε1 > 0) > 0 and all of its moments exist. Further, suppose that 2 E log(α1ε1 + β1) < 0.

Then κ > 0 and C1 > 0,C2 > 0 such that for the stationary ∃ 2 solutions of (σt ) and (Yt): κ 2 lim x P (σ0 > x) = C1 x →∞ 2κ lim x P (Y0 > x) = C2 x →∞ Mikosch, St˘aric˘a(2000): GARCH(1,1) de Haan, Resnick, Rootz´en,de Vries: ARCH(1)

12 Extremal behaviour of GARCH(1,1) Mikosch and St˘aric˘a(2000) and de Haan et al. (1989) gave ex- treme value theory for GARCH(1,1) and ARCH(1).

Suppose a stationary solution (Yt)t N0 exists. ∈ d Let (Yt)t N0 be an i.i.d. sequence with Y0 = Y0. ∈ e e Then under the previous assumptions, θ (0, 1) such that for ∃ ∈ a certain sequence ct > 0, t N: ∈ 2κ lim P max Yi ctx = exp( θx− ), x R t i=1,...,t →∞ ≥ − ∈ 2κ lim P max Yi ctx = exp( x− ), x R. t i=1,...,t ≥ − ∈ →∞ e The GARCH(1,1) process has an extremal index θ, i.e. ex- ceedances over large thresholds occur in clusters, with an average cluster length of 1/θ

13 What can be done for GARCH(p,q)? 2 p 1 τt := (β1 + α1εt , β2, . . . , βp 1) R − 2 p 1 − ∈ ξt := (εt , 0,..., 0) R − ∈ q 2 α := (α2, . . . , αq 1) R − − ∈ Ip 1:(p 1) (p 1) identity matrix − − × − Mt:(p + q 1) (p + q 1) matrix − × − τt βp α αq   Ip 1 0 0 0 Mt := −  ξt 0 0 0     0 0 Iq 2 0  − p+q 1 N := (α0, 0,..., 0)0 R − 2 2 ∈ 2 2 Xt := (σt+1, . . . , σt p+2,Yt ,...,Yt q+2)0 − − Then (Yt)t Z solves the GARCH(p,q) equation if and only if ∈ Xt+1 = Mt+1Xt + N, t Z (9) ∈

14 GARCH(p,q) - continued (9) is a random recurrence equation. Theory for existence of stationary solutions can be applied.

2 For example, if Eε1 = 0, Eε1 = 1, then a necessary and suﬃcient condition for existence of a strictly stationary solution with ﬁnite second moments is p q α + β < 1. (10) X i X j i=1 j=1 (Bougerol and Picard (1992), Bollerslev (1986)) But stationary solutions can exist also if p q α + β = 1. X 1 X j i=1 j=1

A characterization for the existence of stationary solutions is achieved via the Lyapunov exponent (whether it is strictly negative or not).

15 GARCH(p,q) is White Noise

2 Suppose Eεt = 0, Eε = 1 and (10). Then for h 1, t ≥ EYt = E(σtεt) = E(σtE(εt Yt 1)) = 0 | − E(YtYt+h) = E(σtσt+hεt E(εt+h Yt+h 1)) = 0 | − =0 2 | {z } But (Yt )t Z is not White Noise. ∈

2 2 2 2 ut := Y σ = σ (ε 1) t − t t t − Suppose Eu2 < and let h 1: t ∞ ≥ Eut = 0 2 2 2 2 E(utut+h) = E(σ (ε 1)σ ) E(ε 1) = 0 t t − t+h t+h − Hence (ut)t Z is White Noise ∈

16 2 The ARMA representation of Yt

p q 2 2 2 σt = α0 + αiYt i + βjσt j X − X − i=1 j=1 2 2 Substitute σ = Y ut. Then t t −

p q q 2 2 2 Yt αiYt i βjYt j = α0 + ut βjut j X − X − X − − i=1 − j=1 − j=1

2 Hence (Yt )t Z satisﬁes an ARMA(max(p, q), q) equation. ∈ 2 Hence autocorrelation function and spectral density of (Yt )t Z is that of ARMA(max(p, q), q) process with the given parameters.∈

17 Drawbacks of GARCH

Black (1976): Volatility tends to rise in response to “bad • news” and to fall in response to “good news” The volatility in the GARCH process is determined only by • the magnitude of the previous return and shock, not by its sign. The parameters in GARCH are restricted to be positive to • 2 ensure positivity of σt . When estimating, however, sometimes best ﬁts are achieved for negative parameters.

18 Exponential GARCH (EGARCH) (Nelson, 1991)

(εt)t Z i.i.d.(0, 1) ∈ Yt = σtεt p q 2 2 log(σt ) = α0 + αig(εt i) + βj log(σt j) X − X − i=1 j=1 2 2 g(εt) = θεt + γ( εt E εt ), θ + γ = 0 | | − | | 6 Most often, εt i.i.d. normal. EGARCH models often ﬁt the data nicely, but

2 log(σt ) has tails not much heavier than Gaussian tails (like • c dx2 x e− , x , c, d > 0) → ∞ tails of Yt are approximately like • (log x)2 log x P (Y0 > x) e− = x− , x ≈ → ∞ 2 Neither (log σt )t Z nor (Yt)t Z show cluster behaviour. • ∈ ∈ (Lindner, Meyer (2001)) Similar results for stochastic volatility models by Breidt and Davis (1998)

19 Other GARCH type models Many many other GARCH type models exist, like

MGARCH (multiplicative GARCH) • NGARCH (non-linear asymmetric GARCH) • TGARCH (threshold GARCH) • FIGARCH (fractional integrated GARCH) • •··· •··· See Gouri´eroux(1997) or Duan (1997) for some of them.

20 References Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307-327.

Bougerol, P. and Picard, N. (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115-127.

Brandt, A. (1986) The stochastic equation Yn+1 = AnYn +Bn with stationary coeﬃcients. Adv. Appl. Probab. 18, 211-220.

Breidt, F. and Davis, R (1998) Extremes of stochastic volatility models. Ann. Appl. Probab. 8, 664-675. de Haan, L., Resnick, S., Rootz´en,H. and de Vries, C. (1989) Extremal behaviour of solutions to a stochastic diﬀernce equation with applications to ARCH processes. Stoch. Proc. Appl. 32, 213-224.

Duan, J.-C. (1997) Augmented GARCH(p, q) process and its diﬀusion limit. Journal of Econometrics 79, 97-127.

Engle, R. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inﬂation. Econometrica 50, 987-1007.

Goldie, C. (1991) Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126-166.

Gouri´eroux,C. (1997) ARCH Models and Financial Applications. Springer, New York.

Kesten, H. (1973) Random diﬀerence equations and renewal theory for prod- ucts of random matrices. Acta Math. 131, 207-248.

21 Lindner, A. and Meyer, K. (2001) Extremal behavior of ﬁnite EGARCH processes. Preprint.

Mikosch, T. and St˘aric˘a,C (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Statist. 28, 1427-1451.

Vervaat, W. (1979) On a stochastic diﬀerence equation and a representation of non-negative inﬁnitely divisible random variables. Adv. Appl. Probab. 11, 750-783.