Stability of random coefficient ARCH models and aggregation schemes

Recent Advances in ARCH Modelling

“…in view of general result by Kazakevičius et al (2004) on stability of random coefficient ARCH models. The covariance function of the limit processr 2 t is absolutely summable, i.e.r 2 t has short memory, but the aggregated process r 2 N,t and the limit processr 2 t are no longer ARCH(∞) processes.…”

“…We also note an alternative approach in Kazakevičius et al (2004) to the problem of the existence of fourth order stationary solution of ARCH(∞), which leads to equivalent necessary and sufficient conditions as (10).…”

“…Further investigation of this model was conducted by Kazakevičius et al (2004), who assumed that σ 2 > 0 is a constant, α k are i.i.d. random variables on [0, 1] with the mean Eα k = µ > 0 and β k are i.i.d.…”

“…Beta(p, q) random variables independent of α k . For p + q = 1 and 0 < q < 1/2, they showed that Kazakevičius et al (2004), where the limit r 2 t is ARCH(∞) model with…”

“…The long memory property was rigorously established for some of these models including the Gaussian subordinated stochastic volatility model (Robinson, 2001), with general form of nonlinearity, the FIEGARCH and related exponential volatility models (Harvey, 1998;Surgailis and Viano, 2002), the LARCH model (Giraitis et al, 2000c), the stochastic volatility model of Zaffaroni (1997, 1998). The long memory property (and even the existence of stationary regime) of some other models (FIGARCH, LM-ARCH) has not been theoretically established; see Giraitis et al (2000a) Mikosch andStȃricȃ (2000, 2003), Kazakevičius et al (2004). Covariance long memory was also proved for some regime switching SV models (Liu, 2000; Leipus et al, 2005).…”