Recent advances in ARCH modelling

Abstract

Econometric modelling of financial data received a broad interest in the last 20 years and the literature on ARCH and related models is vast. Starting with the path breaking works by Engle (1982) and Bollerslev (1986), one of the most popular models became the Generalized AutoRegressive Conditionally Heteroscedastic (GARCH) process. The classical GARCH(p, q) model is given by equations rt = σtεt, σ 2t = α 0 + Σpi=1 βiσ2t-i + Σqj=1 αjr2t-j, (1) where α0 > 0, αj ≥ 0, βi ≥ 0, p ≥ 0, q v 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 (65) are usually interpreted as financial (log)returns (rt), their volatilities or conditional standard deviations (σt), and so-called innovations or shocks (εt), respectively; in (65) the innovations are supposed to follow a certain fixed distribution (e.g., standard normal). Later, a number of modifications of (65) 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 Gouri-eroux (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. Under some additional conditions, similarly as in the case of ARMA models, the GARCH model can be written as ARCH(∞) model (see (3) below), i.e., σ2t can be represented as a moving average of the past squared returns r2s, s ≤ t, with exponentially decaying coefficients (see Bollerslev, 1988) and absolutely summable exponentially decaying autocovariance function. However, empirical studies of financial data show that sample autocorrelations of power series and volatilities (such as absolute values |rt| or squares r2t) remain non-zero for very large lags; see, e.g., Dacorogna et al. (1993), Ding et al. (1993), Baillie et al. (1996a), Ding and Granger (1996), Breidt et al. (1998), Mikosch and Sťariča (2003), Andersen et al. (2001). These studies provide a clearcut evidence in favor of models with autocovariances decaying slowly with the lag as k-γ, for some 0 ≤ γ ≤ 1. A number of such models (FIGARCH, LM-ARCH, FIEGARCH) were suggested in the ARCH literature. 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 Robinson and 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 and Sťariča (2000, 2003), Kazakevicius et al. (2004). Covariance long memory was also proved for some regime switching SV models (Liu, 2000; Leipus et al., 2005). One should also mention that some authors (Mikosch and Sťariča, 1999, 2004) argue that the observed long memory in sample autocorrelations can be explained by short memory GARCH models with structural breaks and/or slowly changing trends. The present paper reviews some recent theoretical findings on ARCH type models. We focus mainly on covariance stationary models which display empirically observed properties known as "stylized facts". One of the major issues to determine is whether the corresponding model r2t for squares has long memory or short memory, i.e. whether Σ∞ k=0 |Cov(r2k , r20)| = ∞ or ∞ k=0 |Cov(r2k , r20)| < ∞ holds. It is pointed out that for several ARCH-type models the behavior of Cov(r2k , r20) alone is sufficient to derive the limit distribution of ΣNj=1(r2j - Er2j) and statistical inferences, without imposing any additional (e.g. mixing) assumptions on the dependence structure. The review discusses ARCH(∞) processes and their modifications such as linear ARCH (LARCH), bilinear models, long memory EGARCH and stochastic volatility, regime switching SV models, random coefficient ARCH and aggregation. We give an overview of the theoretical results on the existence of a stationary solution, dependence structure, limit behavior of partial sums, leverage effect and long memory property of these models. Statistical estimation of ARCH parameters and testing for change-points are also discussed. © Springer Berlin Heidelberg 2007.