GARCH processes probabilistic properties (Part 1)

Transcription

1 GARCH processes probabilistic properties (Part 1) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D Garching Germany lindner@ma.tum.de Maphysto Workshop Non-Linear Time Series Modeling Copenhagen September 27 30,

2 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

3 GARCH(p,q) on N 0 (ε t ) t N0 i.i.d., P (ε 0 = 0) = 0 (1) α 0 > 0, α 1,..., α p 0, β 1,..., β q 0. (σ 2 0,..., σ 2 max(p,q) 1 ) independent of {ε t : t max(p, q) 1} Y t = σ t ε t, σ 2 t = α 0 + p α i Y 2 t i i=1 }{{} + } ARCH {{} GARCH (Y t ) t N0 GARCH(p,q) process (2) q β j σt j 2, t max(p, q). (3) j=1 3

4 GARCH(p,q) on Z (1), (2) and (3) for t Z, and ε t independent of Y t 1 := {Y s : s t 1}. ARCH(p): Engle (1981) GARCH(p,q): Bollerslev (1986) Generalized AutoRegressive Conditional Heteroscedasticity 4

5 Why conditional volatility? Suppose σt 2 Y t 1, or equivalently σt 2 ε t 1, t Z (the process is causal ) Suppose Eε t = 0, Eε 2 t = 1, EY t <. Then E(Y t Y t 1 ) = E(σ t ε t Y t 1 ) = σ t E(ε t Y t 1 ) = 0, V (Y t Y t 1 ) = E(σ 2 t ε 2 t Y t 1 ) = σ 2 t E(ε 2 t Y t 1 ) = σ 2 t Hence σ 2 t is the conditional variance 5

6 Markov property (a) GARCH(1,1): (σt 2 ) t Z is a Markov process, since σt 2 = α 0 + (α 1 ε 2 t 1 + β 1 )σt 1 2 (σt 2, Y t ) t Z is Markov process, but (Y t ) t Z is not. (b) GARCH(p,q), max(p, q) > 1 (σ 2 t ) t Z is not Markov process (σ 2 t,..., σ 2 t max(p,q)+1 ) t Z is Markov process. 6

7 Strict stationarity - GARCH(1,1) σ 2 t = α 0 + α 1 σ 2 t 1ε 2 t 1 + β 1 σ 2 t 1 = (α 1 ε 2 t 1 + β 1 )σ 2 t 1 + α 0 =: A t σ 2 t 1 + B t (4) (A t, B t ) t Z is i.i.d. sequence. (4) is called a random recurrence equation. σ 2 t = A t σt B t = A t A t 1 σt A t B t 1 + B t. = A t A t k σ 2 t k 1 + k i=0 A t A t i+1 B t i }{{} =α 0. (5) Let k and hope that (5) converges. 7

8 Strict stationarity - continued Suppose γ := E log A 1 < 0 1 k (log A t + log A t log A t k+1 ) }{{} random walk! k E log A 1 a.s. = a.s. ω n 0 (ω) : 1 k log (A t A t k+1 ) γ/2 < 0 k n 0 (ω) = A t A t k+1 e kγ/2 k n 0 (ω) Hence (5) converges almost surely. 8

9 Theorem: (Nelson, 1990) If E log(α 1 ε β 1 ) < 0, (6) then GARCH(1,1) has a strictly stationary solution. Its marginal distribution is given by α 0 i=0 (α 1 ε β 1 ) (α 1 ε 2 i + β 1 ) (7) If β 1 > 0 (i.e. GARCH but not ARCH) and a strictly stationary solution exists, then (6) holds. Remark: If E log + A 1 <, then { ( ) 1 γ = inf E n + 1 log A 0A 1 A n } : n N is called the Lyapunov exponent of the sequence (A n ) n Z. 9

10 When is EY 2 t <? σ0 2 = α 0 (α 1 ε β 1 ) (α 1 ε 2 i + β 1 ) i=0 ( Eσ0 2 = α 0 E(α1 ε β 1 ) ) i < i=0 α 1 Eε β 1 < 1 EY 2 t = Eσ 2 t Eε 2 t So stationary solution (Y t ) t Z has finite variance iff α 1 Eε β 1 < 1 10

11 Tail behaviour of stochastic recurrence equations Theorem (Goldie 1991, Kesten 1973, Vervaat 1979) Suppose (Z t ) t N satisfies the stochastic recurrence equation Z t = A t Z t 1 + B t, t N, where ((A t, B t )) 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 (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 lim x xκ P (Z > x) = E[((AZ + B)+ ) κ ((AZ) + ) κ ] κe A κ log + =: C 0 A (8) 11

12 Tail behaviour of GARCH(1,1) Corollary: Suppose ε 1 is continuous, P (ε 1 > 0) > 0 and all of its moments exist. Further, suppose that E log(α 1 ε β 1 ) < 0. Then κ > 0 and C 1 > 0, C 2 > 0 such that for the stationary solutions of (σ 2 t ) and (Y t ): lim x xκ P (σ 2 0 > x) = C 1 lim x x2κ P (Y 0 > x) = C 2 Mikosch, Stărică (2000): GARCH(1,1) de Haan, Resnick, Rootzén, de Vries: ARCH(1) 12

13 Extremal behaviour of GARCH(1,1) Mikosch and Stărică (2000) and de Haan et al. (1989) gave extreme value theory for GARCH(1,1) and ARCH(1). Suppose a stationary solution (Y t ) t N0 exists. Let (Ỹt) t N0 be an i.i.d. sequence with Ỹ0 d = Y 0. Then under the previous assumptions, θ (0, 1) such that for a certain sequence c t > 0, t N: ( ) lim P max Y i c t x = exp( θx 2κ ), x R t i=1,...,t ( ) lim P max Ỹ i c t x = exp( x 2κ ), x R. t i=1,...,t The GARCH(1,1) process has an extremal index θ, i.e. exceedances over large thresholds occur in clusters, with an average cluster length of 1/θ 13

14 What can be done for GARCH(p,q)? τ t := (β 1 + α 1 ε 2 t, β 2,..., β p 1 ) R p 1 ξ t := (ε 2 t, 0,..., 0) R p 1 α := (α 2,..., α q 1 ) R q 2 I p 1 : (p 1) (p 1) identity matrix M t : (p + q 1) (p + q 1) matrix M t := τ t β p α α q I p ξ t I q 2 0 N := (α 0, 0,..., 0) R p+q 1 X t := (σ 2 t+1,..., σ 2 t p+2, Y 2 t,..., Y 2 t q+2) Then (Y t ) t Z solves the GARCH(p,q) equation if and only if X t+1 = M t+1 X t + N, t Z (9) 14

15 GARCH(p,q) - continued (9) is a random recurrence equation. Theory for existence of stationary solutions can be applied. For example, if Eε 1 = 0, Eε 2 1 = 1, then a necessary and sufficient condition for existence of a strictly stationary solution with finite second moments is p q α i + β j < 1. (10) i=1 j=1 (Bougerol and Picard (1992), Bollerslev (1986)) But stationary solutions can exist also if p q α 1 + β j = 1. 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

16 GARCH(p,q) is White Noise Suppose Eε t = 0, Eε 2 t = 1 and (10). Then for h 1, EY t = E(σ t ε t ) = E(σ t E(ε t Y t 1 )) = 0 E(Y t Y t+h ) = E(σ t σ t+h ε t E(ε t+h Y t+h 1 )) = 0 }{{} =0 But (Y 2 t ) t Z is not White Noise. u t := Y 2 t σ 2 t = σ 2 t (ε 2 t 1) Suppose Eu 2 t < and let h 1: Eu t = 0 E(u t u t+h ) = E(σ 2 t (ε 2 t 1)σ 2 t+h) E(ε 2 t+h 1) = 0 Hence (u t ) t Z is White Noise 16

17 The ARMA representation of Y 2 t Substitute σ 2 t = Y 2 t σ 2 t = α 0 + p i=1 u t. Then α i Y 2 t i + q β j σt j 2 j=1 Y 2 t p i=1 α i Y 2 t i q j=1 β j Y 2 t j = α 0 + u t q β j u t j j=1 Hence (Y 2 t ) t Z satisfies an ARMA(max(p, q), q) equation. Hence autocorrelation function and spectral density of (Y 2 t ) t Z is that of ARMA(max(p, q), q) process with the given parameters. 17

18 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 ensure positivity of σ 2 t. When estimating, however, sometimes best fits are achieved for negative parameters. 18

19 Exponential GARCH (EGARCH) (Nelson, 1991) log(σ 2 t ) = α 0 + (ε t ) t Z i.i.d.(0, 1) Y t = σ t ε t p α i g(ε t i ) + i=1 q β j log(σt j) 2 j=1 g(ε t ) = θε t + γ( ε t E ε t ), θ 2 + γ 2 0 Most often, ε t i.i.d. normal. EGARCH models often fit the data nicely, but log(σ 2 t ) has tails not much heavier than Gaussian tails (like x c e dx2, x, c, d > 0) tails of Y t are approximately like P (Y 0 > x) e (log x)2 = x log x, x Neither (log σ 2 t ) t Z nor (Y t ) t Z show cluster behaviour. (Lindner, Meyer (2001)) Similar results for stochastic volatility models by Breidt and Davis (1998) 19

20 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 Gouriéroux (1997) or Duan (1997) for some of them. 20

21 References Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, Bougerol, P. and Picard, N. (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, Brandt, A. (1986) The stochastic equation Y n+1 = A n Y n +B n with stationary coefficients. Adv. Appl. Probab. 18, Breidt, F. and Davis, R (1998) Extremes of stochastic volatility models. Ann. Appl. Probab. 8, de Haan, L., Resnick, S., Rootzén, H. and de Vries, C. (1989) Extremal behaviour of solutions to a stochastic differnce equation with applications to ARCH processes. Stoch. Proc. Appl. 32, Duan, J.-C. (1997) Augmented GARCH(p, q) process and its diffusion limit. Journal of Econometrics 79, Engle, R. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50, Goldie, C. (1991) Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, Gouriéroux, C. (1997) ARCH Models and Financial Applications. Springer, New York. Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Math. 131,

22 Lindner, A. and Meyer, K. (2001) Extremal behavior of finite EGARCH processes. Preprint. Mikosch, T. and Stărică, C (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Statist. 28, Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Probab. 11,