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 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. ◮ σ 2 t = V [ r t | F t − 1 ] is modeled using a dynamic variance model. Erik Lindström GARCH models

Dependence structures Dependence on the OMXS30. 1.2 1.2 1 1 0.8 0.8 Autocorrelation, abs returns Autocorrelation, returns 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 0 5 10 15 20 25 30 0 50 100 150 200 250 300 lag lag 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 r t = σ t z t , ◮ The conditional variance is given by p σ 2 α i r 2 ∑ t = ω + t − i i = 1 ◮ Easy to estimate as σ 2 t ∈ F t − 1 ! ◮ Q : Compute cov( r t , r t − h ) and cov( r 2 t , r 2 t − h ) for this model. Erik Lindström GARCH models

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( r 2 t , r 2 t − h ) is harder. Introduce ν t = r 2 t − σ 2 t (white noise!). It then follows that p r 2 t = σ 2 α i r 2 ∑ t + ν t = ω + t − i + ν t . i = 1 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 p q σ 2 α i r 2 β j σ 2 ∑ ∑ t = ω + t − i + t − j i = 1 j = 1 ◮ A GARCH(1,1) is often sufficent! ◮ Conditions on the parameters. Erik Lindström GARCH models

GARCH ◮ Cov( r t , r t − h ) = 0 as in the ARCH model. ◮ Computing Cov( r 2 t , r 2 t − h ) is similar to the ARCH model. Reintroducing ν t = r 2 t − σ 2 t gives (assume p = q ) p p r 2 σ 2 ∑ α i r 2 ∑ β j σ 2 = t + ν t = ω + t − i + t − j + ν t t i = 1 j = 1 p p ∑ α i r 2 ∑ β j ( r 2 = ω + t − i + t − j − ν t − j )+ ν t i = 1 j = 1 p p ∑ ( α i + β i ) r 2 ∑ = ω + t − i + − β j ν t − j + ν t i = 1 j = 1 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 , α 1 = 0 . 0775 β 1 = 0 . 9152 OMXS30 logreturns Extimated GARCH(1,1) vol 0.1 0.04 0.035 0.05 0.03 0.025 0 0.02 0.015 −0.05 0.01 2000 2010 2000 2010 OMXS30 normalised logreturns NORMPLOT OMXS30 normalised logreturns 4 0.999 0.997 0.99 0.98 2 0.95 0.90 Probability 0.75 0 0.50 0.25 −2 0.10 0.05 0.02 0.01 0.003 −4 0.001 2000 2010 −4 −2 0 2 4 Data Erik Lindström GARCH models

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 σ 2 t = α r 2 t − 1 +( 1 − α ) σ 2 t − 1 Erik Lindström GARCH models

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? Erik Lindström GARCH models

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. Erik Lindström GARCH models

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 ∞ Γ( k − d ) ( 1 − B ) d = Γ( k + 1 )Γ( − d ) B k . ∑ (4) k = 0 Erik Lindström GARCH models

EGARCH (Exponential GARCH) ◮ The conditional variance is given by p q log σ 2 β j log σ 2 ∑ ∑ t = ω + α i f ( r t − i )+ t − j i = 1 j = 1 ◮ log σ 2 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 � g ( S t ) σ 2 r t = t z t , ◮ 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. � r t = µ t + δ f ( σ 2 σ 2 t )+ t z t . 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 ◮ ( r t − i + γ ) 2 (Type I) ◮ ( | r t − i | + cr 2 t − i ) (Type II) ◮ Replace α i with ( α i + ˜ α i 1 { r t − 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: R t = H 1 / 2 Z t t ◮ H t = ∆ t P c ∆ t where ◮ ∆ = diag ( σ t , k ) ◮ P c is a constant correlation matrix. Erik Lindström GARCH models

log-Likelihood The log-likelihood for a general Multivariate GARCH model is given by T T ℓ t ( θ ) = − 1 ln | det ( 2 π H t ) |− 1 r T t H − 1 ∑ ∑ r t . (5) t 2 2 t = 1 t = 1 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. 600 ABB Astrazeneca B Boliden Investor B 500 Lundin MTG B Nordea Tele2 B 400 300 200 100 0 2005 2006 2007 2008 2009 2010 Erik Lindström GARCH models