GARCH models. Erik Lindström. FMS161/MASM18 Financial Statistics

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1 FMS161/MASM18 Financial Statistics

2 Time series models Let r t be a stochastic process. µ t = E[r t F t 1 ] is the conditional mean modeled by an AR, ARMA, SETAR, STAR etc. model. Having a correctly specified model for the conditional mean allows us to model the conditional variance. I will for the rest of the lecture assume that r t is the zero mean returns. = V [r t F t 1 ] is modeled using a dynamic variance model. σt 2

3 Dependence structures Dependence on the OMXS Autocorrelation, returns Autocorrelation, abs returns lag lag

4 The ARCH family ARCH (1982), Bank of Sweden... (2003) GARCH (1986) FIGARCH (1996) Special cases (IGARCH, A-GARCH, GJR-GARCH, EWMA) EGARCH (1991) SW-GARCH GARCH in Mean (1987)

5 ARCH The ARCH (Auto Regressive Conditional Heteroscedasticity) model The (mean free) model is given by r t = σ t z t, The conditional variance is given by σ 2 t = ω + p i=1 α i r 2 t i Easy to estimate as σt 2 F t 1! Q : Compute cov(r t,r t h ) and cov(rt 2,r2 t h ) for this model.

6 ARCH, solution We have that E[r t ] = E[E[σ t z t F t 1 ]] = E[σ t E[z t F t 1 ]] = 0. Next, we compute Cov(r t,r t h )) as E[σ t z t σ t h z t h ] = E[E[σ t z t σ t h z t h F t 1 ]] = E[σ t σ t h z t h E[z t F t 1 ]] = 0. Computing Cov(rt 2,r2 t h ) is harder. Introduce ν t = rt 2 σt 2 (white noise!). It then follows that rt 2 = σt 2 + ν t = ω + p i=1 α i r 2 t i + ν t. The rt 2 is thus a process (with heteroscedastic noise).

7 ARCH, limitations Large number of lags are needed to fit data. The model is rather restrictive, as the parameters must be bounded if moments should be finite (Exercise: Compute the restrictions for the ARCH(1) model to have finite variance and kurtosis.)

8 GARCH (Generalized ARCH) Is the most common dynamics variance model. The conditional variance is given by σ 2 t = ω + p i=1 A GARCH(1,1) is often sufficent! Conditions on the parameters. α i rt i 2 q + j=1 β j σ 2 t j

9 GARCH Cov(r t,r t h )= 0 as in the ARCH model. Computing Cov(rt 2,r2 t h ) is similar to the ARCH model. Reintroducing ν t = rt 2 σt 2 gives (assume p = q) rt 2 = σt 2 + ν t = ω + = ω + = ω + p i=1 p i=1 p i=1 α i rt i 2 p + j=1 α i rt i 2 p + j=1 β j σ 2 t j + ν t β j (r 2 t j ν t j) + ν t (α i + β i )rt i 2 p + β j ν t j + ν t The rt 2 is thus a process (with heteroscedastic noise). j=1

10 Estimation of GARCH(1,1) on OMXS30 logreturns ω = , α 1 = β 1 = OMXS30 logreturns Extimated GARCH(1,1) vol OMXS30 normalised logreturns Probability NORMPLOT OMXS30 normalised logreturns Data

11 GARCH, special cases An IGARCH (integrated GARCH) is a GARCH where α i + β i = 1 and ω > 0. The EWMA(exponentially weighted moving average) process is a process where α + β = 1 and ω = 0, i.e. the volatility is given by σt 2 = αrt (1 α)σt 1

12 Fractionally Integrated GARCH Recall the ARMA representation of the GARCH model (1 ψ(b))εt 2 = ω + (1 β(b))ν t (1) and the IGARCH representation is given by Φ(B)(1 B)εt 2 = ω + (1 β(b))ν t. (2) Can we have something in-between?

13 Fractionally Integrated GARCH Recall the ARMA representation of the GARCH model (1 ψ(b))ε 2 t = ω + (1 β(b))ν t (1) and the IGARCH representation is given by Φ(B)(1 B)ε 2 t = ω + (1 β(b))ν t. (2) Can we have something in-between? Yes, that is the FIGARCH model Φ(B)(1 B) d ε 2 t = ω + (1 β(b))ν t (3) with the process having finite variance if.5 < d < 0.5.

14 Fractionally Integrated GARCH Recall the ARMA representation of the GARCH model (1 ψ(b))ε 2 t = ω + (1 β(b))ν t (1) and the IGARCH representation is given by Φ(B)(1 B)ε 2 t = ω + (1 β(b))ν t. (2) Can we have something in-between? Yes, that is the FIGARCH model Φ(B)(1 B) d ε 2 t = ω + (1 β(b))ν t (3) with the process having finite variance if.5 < d < 0.5. The fractional differentiation can be computed as (1 B) d = k=0 Γ(k d) Γ(k + 1)Γ( d) Bk. (4)

15 EGARCH (Exponential GARCH) The conditional variance is given by logσt 2 = ω + logσ 2 may be negative! p i=1 α i f (r t i ) + q j=1 β j logσ 2 t j Thus no (fewer) restrictions on the parameters.

16 SW-?ARCH An advanced extension is the switching ARCH model. The conditional variance is given by a standard ARCH, GARCH or EGARCH (the later two are non-trivial, due to their non-markovian structure) The model is given by r t = g (S t ) σt 2z t, where g (1) = 1 and (g (n),n 2) are free parameters.

17 GARCH in Mean Asset pricing models may include variance terms as explanatory factors (think CAPM). This can be captured by GARCH in Mean models. r t = µ t + δf (σt 2 ) + σt 2z t.

18 Variations Several improvements can be applied to any of the models. Bad news tend to increase the variance more than good news. We can replace rt i 2 by (rt i + γ) 2 (Type I) ( rt i + crt i 2 ) (Type II) Replace αi with (α i + α i 1 {rt i <0}) (GJR, Glosten-Jagannathan-Runkle). Distributions Stationarity problems.

19 Multivariate models What about multivariate models? Huge number of models. VEC-MVGARCH (1988) BEKK-MVGARCH (1995) CCC-MVGARCH (1990) DCC-MVGARCH (2002) STCC-MVGARCH(2005)

20 Multivariate models What about multivariate models? Huge number of models. VEC-MVGARCH (1988) BEKK-MVGARCH (1995) CCC-MVGARCH (1990) DCC-MVGARCH (2002) STCC-MVGARCH(2005) Most are overparametrized. I recommend starting with the CCC-MVGARCH Returns: R t = H 1/2 t H t = t P c t where = diag(σ t,k ) Pc is a constant correlation matrix. Z t

21 log-likelihood The log-likelihood for a general Multivariate GARCH model is given by l t (θ) = 1 2 T t=1 ln det(2πh t ) 1 2 T t=1 r T t H 1 t r t. (5) Easy to optimize for CCC-MVGARCH, not so easy for other models. [Proof on the blackboard]

22 Some wellknown Swedish assets On the second computer exercise you will try to fit a CCC-MVGARCH model to this ABB Astrazeneca B Boliden Investor B Lundin MTG B Nordea Tele2 B