Applications of state space models in finance

5.1 Autoregressive conditional heteroskedasticity 61
(1986) refer to the class of models with γ1+δ1 = 1 as integrated GARCH (IGARCH). For
IGARCH models the finite unconditional variance is not defined. Although the IGARCH
model is not covariance-stationary, it can be shown that it is strictly stationary. Standard
inference procedures remain asymptotically valid given that the sample size is large; cf.
Bollerslev et al. (1992) where further references can be found.
To understand why the GARCH(1,1) process is qualified to model the stylized facts
of a small first-order autocorrelation and a slow decay simultaneously, we have a look
at the autocorrelations of ɛ 2 t :
γ
ρ1 = γ1 +
2 1δ1
1 − 2γ1δ1 − δ2 1
, (5.9)
ρτ = (γ1 + δ1) τ−1 ρ1, for τ = 2, 3, . . . . (5.10)
With the decay factor of the exponentially declining autocorrelations being equal to
γ1 + δ1, the autocorrelations will decrease the more gradually the closer this sum is to
unity (cf. Franses and van Dijk 2000, ¢ 4.1.1).
5.1.1.2 Forecasting
With respect to forecasts of the conditional variance, it follows from (5.5) that the
optimal l-step ahead forecast can be calculated directly from ht+1, which is part of the
information set Ωt:
ˆh t+l|t = ω + γ1ˆɛ 2 t+l−1|t + δ1 ˆ h 2 t+l−1|t . (5.11)
For a covariance-stationary GARCH(1,1) process, it can be shown that this is equivalent
to
ˆh t+l|t = σ 2 + (γ1 + δ1) l−1 � ht+1 − σ 2� , (5.12)
where the forecast reverts to σ 2 at an exponential rate (cf. Andersen et al. 2005).
5.1.2 Nonlinear extensions
The basic GARCH model is based on the assumption that positive and negative past
shocks have the same effect on the conditional variance. However, many financial time
series are asymmetric: negative shocks tend to have a bigger influence on future volatility
than equally sized positive shocks. Asymmetric effects are often observed for aggregate
equity indices (cf. Andersen et al. 2005), which will be in the focus of this thesis. This
asymmetry, first observed by Black (1976), is commonly referred to as leverage effect.
According to its definition, the fall of the value of equity amounts in an increased debtto-equity
ratio, the leverage. This implies an increased level of riskiness, which results in
increased future volatility. 12 In the standard GARCH model, the conditional variance
does not depend on the sign of the shocks such that asymmetries cannot be accommodated.
Over the last fifteen years various nonlinear extensions of the GARCH model
12 The volatility-feedback hypothesis by Campbell and Hentschel (1992) represents an alternative
but less regarded explanation for volatility asymmetries, according to which positive
volatility shocks lead to lower future returns.