BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

56 ACTA WASAENSIAarriving at the following definition of FIGARCH:φ(L)(1 − L) d r 2 t = α 0 + {1 − β(L)}ν t , ν t = r 2 t − σ2 t (3.25)with α 0 > 0andzerosofbothφ(L) ={1 − α(L) − β(L)}(1 − L) −1 and 1 − β(L)outside the unit circle. The aim of this model is to replace the exponential decay inthe autocorrelation function of conventional GARCH models with hyperbolic decay asempirically oberved in financial markets 97 .The practical relevance of FIGARCH for financial modelling is however still unclear,as the model has been aspersed for not being properly specified 98 . A possible correctionhas been suggested by Chung (1999). Baillie et al. (1996) claim furthermorethat returns following a FIGARCH process have infinite variance and can thus not becovariance stationary. This would however invalidate classical estimation and inferencetechniques for the same reasons as for the IGARCH model 99 .Similarly, Ding & Granger (1996) aim to introduce long memory into GARCH bymodelling the conditional variance as a weighted sum of infinitely many GARCH typevariances, which they call Long Memory (LM) ARCH. The authors claim a hyperbolicdecline of the autocorrelation function, provided that the returns have a finite4th moment. This statement has however been disproved by Giraitis, Kokoszka &Leipus (2000) for parameter values ensuring stationarity of the model, while for otherparameter values it is not yet known whether LM-ARCH has a stationary solution atall 100 .Other recent attempts to include long memory into the GARCH framework includeMaheu (2005) and Zumbach (2004). Overall it appears from the discussion abovethat the GARCH subclass of stochastic volatility models might be less well suited formodeling the stylized facts of financial returns, than what their popularity suggests.97 see section 2.6.98 see e.g. Chung (1999) and Mikosch (2003a).99 see Mikosch (2003a).100 see Mikosch (2003a).