GARCH models Erik Lindstrm FMS161/MASM18 Financial Statistics Erik - - PowerPoint PPT Presentation

GARCH models

Erik Lindström

FMS161/MASM18 Financial Statistics

Erik Lindström GARCH models

Time series models

Let rt be a stochastic process.

◮ µt = E[rt|Ft−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 rt is the zero

mean returns.

◮ σ2 t = V[rt|Ft−1] is modeled using a dynamic variance

model.

Erik Lindström GARCH models

Dependence structures

Dependence on the OMXS30.

5 10 15 20 25 30 −0.2 0.2 0.4 0.6 0.8 1 1.2 lag Autocorrelation, returns 50 100 150 200 250 300 −0.2 0.2 0.4 0.6 0.8 1 1.2 lag Autocorrelation, abs returns

Erik Lindström GARCH models

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)

Erik Lindström GARCH models

ARCH

The ARCH (Auto Regressive Conditional Heteroscedasticity) model

◮ The (mean free) model is given by

rt = σtzt,

◮ The conditional variance is given by

σ2

t = ω + p

∑

i=1

αir 2

t−i ◮ Easy to estimate as σ2 t ∈ Ft−1! ◮ Q : Compute cov(rt,rt−h) and cov(r 2 t ,r 2 t−h) for this model.

Erik Lindström GARCH models

ARCH, solution

◮ We have that E[rt] = E[E[σtzt|Ft−1]] = E[σtE[zt|Ft−1]] = 0. ◮ Next, we compute Cov(rt,rt−h)) as

E[σtztσt−hzt−h] = E[E[σtztσt−hzt−h|Ft−1]] = E[σtσt−hzt−hE[zt|Ft−1]] = 0.

◮ Computing Cov(r 2 t ,r 2 t−h) is harder. Introduce νt = r 2 t −σ2 t

(white noise!). It then follows that r 2

t = σ2 t +νt = ω + p

∑

i=1

αir 2

t−i +νt.

The r 2

t is thus a . . . . . . . . . process (with heteroscedastic

noise).

Erik Lindström GARCH models

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.)

Erik Lindström GARCH models

GARCH (Generalized ARCH)

◮ Is the most common dynamics variance model. ◮ The conditional variance is given by

σ2

t = ω + p

∑

i=1

αir 2

t−i + q

∑

j=1

βjσ2

t−j ◮ A GARCH(1,1) is often sufficent! ◮ Conditions on the parameters.

Erik Lindström GARCH models

GARCH

◮ Cov(rt,rt−h)= 0 as in the ARCH model. ◮ Computing Cov(r 2 t ,r 2 t−h) is similar to the ARCH model.

Reintroducing νt = r 2

t −σ2 t gives (assume p = q)

r 2

t

= σ2

t +νt = ω + p

∑

i=1

αir 2

t−i + p

∑

j=1

βjσ2

t−j +νt

= ω +

p

∑

i=1

αir 2

t−i + p

∑

j=1

βj(r 2

t−j −νt−j)+νt

= ω +

p

∑

i=1

(αi +βi)r 2

t−i + p

∑

j=1

−βjνt−j +νt The r 2

t is thus a . . . . . . . . . process (with heteroscedastic

noise).

Erik Lindström GARCH models

Estimation of GARCH(1,1) on OMXS30 logreturns

ω = 1.9·10−6, α1 = 0.0775 β1 = 0.9152

2000 2010 −0.05 0.05 0.1 OMXS30 logreturns 2000 2010 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Extimated GARCH(1,1) vol 2000 2010 −4 −2 2 4 OMXS30 normalised logreturns −4 −2 2 4 0.001 0.003 0.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997 0.999 Data Probability NORMPLOT OMXS30 normalised logreturns

Erik Lindström GARCH models

GARCH, special cases

◮ An IGARCH (integrated GARCH) is a GARCH where

∑αi +βi = 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 σ2

t = αr 2 t−1 +(1−α)σ2 t−1

Erik Lindström GARCH models

Fractionally Integrated GARCH

Recall the ARMA representation of the GARCH model (1−ψ(B))ε2

t = ω +(1−β(B))νt

(1) and the IGARCH representation is given by Φ(B)(1−B)ε2

t = ω +(1−β(B))νt.

(2) Can we have something in-between?

Erik Lindström GARCH models

Fractionally Integrated GARCH

Recall the ARMA representation of the GARCH model (1−ψ(B))ε2

t = ω +(1−β(B))νt

(1) and the IGARCH representation is given by Φ(B)(1−B)ε2

t = ω +(1−β(B))νt.

(2) Can we have something in-between? Yes, that is the FIGARCH model Φ(B)(1−B)dε2

t = ω +(1−β(B))νt

(3) with the process having finite variance if −.5 < d < 0.5.

Erik Lindström GARCH models

Fractionally Integrated GARCH

Recall the ARMA representation of the GARCH model (1−ψ(B))ε2

t = ω +(1−β(B))νt

(1) and the IGARCH representation is given by Φ(B)(1−B)ε2

t = ω +(1−β(B))νt.

(2) Can we have something in-between? Yes, that is the FIGARCH model Φ(B)(1−B)dε2

t = ω +(1−β(B))νt

(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)

Erik Lindström GARCH models

EGARCH (Exponential GARCH)

◮ The conditional variance is given by

logσ2

t = ω + p

∑

i=1

αif(rt−i)+

q

∑

j=1

βj logσ2

t−j ◮ logσ2 may be negative! ◮ Thus no (fewer) restrictions on the parameters.

Erik Lindström GARCH models

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

rt =

• g(St)σ2

t zt, ◮ where g(1) = 1 and (g(n),n ≥ 2) are free parameters.

Erik Lindström GARCH models

GARCH in Mean

Asset pricing models may include variance terms as explanatory factors (think CAPM). This can be captured by GARCH in Mean models. rt = µt +δf(σ2

t )+

• σ2

t zt.

Erik Lindström GARCH models

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 r 2

t−i by

◮ (rt−i +γ)2 (Type I) ◮ (|rt−i|+cr 2

t−i) (Type II)

◮ Replace αi with (αi + ˜

αi1{rt−i<0}) (GJR, Glosten-Jagannathan-Runkle).

◮ Distributions ◮ Stationarity problems.

Erik Lindström GARCH models

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) Erik Lindström GARCH models

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: Rt = H1/2 t

Zt

◮ Ht = ∆tPc∆t where

◮ ∆ = diag(σt,k) ◮ Pc is a constant correlation matrix. Erik Lindström GARCH models

log-Likelihood

The log-likelihood for a general Multivariate GARCH model is given by ℓt(θ) = −1 2

T

∑

t=1

ln|det(2πHt)|− 1 2

T

∑

t=1

r T

t H−1 t

rt. (5) Easy to optimize for CCC-MVGARCH, not so easy for other

• models. [Proof on the blackboard]

Erik Lindström GARCH models

Some wellknown Swedish assets

On the second computer exercise you will try to fit a CCC-MVGARCH model to this.

2005 2006 2007 2008 2009 2010 100 200 300 400 500 600 ABB AstrazenecaB Boliden InvestorB Lundin MTGB Nordea Tele2

B

Erik Lindström GARCH models