Recent advances in ARCH modelling
نویسندگان
چکیده
where α0 > 0, αj ≥ 0, βi ≥ 0, p ≥ 0, q ≥ 0 are model parameters and {εj , j ∈ Z} are independent identically distributed (i.i.d.) zero mean random variables. The variables rt, σt, εt in (1) are usually interpreted as financial (log)returns (rt), their volatilities or conditional standard deviations (σt), and so-called innovations or shocks (εt), respectively; in (1) the innovations are supposed to follow a certain fixed distribution (e.g., standard normal). Later, a number of modifications of (1) were proposed, which account for asymmetry, leverage effect, heavy tails and other "stylized facts". For statistical and econometric aspects of ARCH modelling, see the surveys of Bollerslev et al. (1992), Shephard (1996), Bera and Higgins (1993), Bollerslev et al. (1994); for specific features of modelling the financial data, including ARCH, see Pagan (1996), Rydberg (2000), Mikosch (2003). Berkes et al. (2002b) review some recent results. One should mention here, besides the classical reference to Taylor (1986), the related monographs by Gouriéroux (1997), Fan and Yao (2002), Tsay (2002). Let us note that the GARCH model for returns is also related to the Autoregressive Conditional Duration (ACD) model proposed by Engle and Russell (1998) for modelling of durations between events.
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