TimeSeriesBook.pdf

In addition z 2 t is a causal and invertible armamax

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In addition, { Z 2 t } is a causal and invertible ARMA(max { p, q } , q ) process satisfying the following difference equation: Z 2 t = α 0 + p X j =1 α j Z 2 t - j + q X j =1 β j σ 2 t - j + e t = α 0 + max { p,q } X j =1 ( α j + β j ) Z 2 t - j + e t - q X j =1 β j e t - j , 5 A detailed exposition of the GARCH(1,1) model is given in Section 8.1.4. 6 Zadrozny (2005) derives a necessary and sufficient condition for the existence of the fourth moment.
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188 CHAPTER 8. MODELS OF VOLATILITY where α p + j = β q + j = 0 for j 1 and “error term” e t = Z 2 t - σ 2 t = ( ν 2 t - 1) α 0 + p X j =1 α j Z 2 t - j + q X j =1 β j σ 2 t - j ! . Note, however, there is a circularity here because the noise process { e t } is defined in terms of Z 2 t and is therefore not an exogenous process driving Z 2 t . Thus, one has to be precautious in the interpretation of { Z 2 t } as an ARMA process. Further generalizations of the GARCH(p,q) model can be obtained by allowing deviations from the normal distribution for ν t . In particular, distri- butions such as the t-distribution which put more weight on extreme values have become popular. This seems warranted as prices on financial markets exhibit large and sudden fluctuations. 7 Threshold GARCH Model Assuming a symmetric distribution for ν t and specifying a linear relationship between σ 2 t and Z 2 t - j bzw. σ 2 t - j , j > 0, leads to a symmetric distribution for Z t . It has, however, been observed that downward movements seem to be different from upward movements. This asymmetric behavior is accounted for by the asymmetric GARCH(1,1) model or threshold GARCH(1,1) model (TGARCH(1,1) model). This model was proposed by Glosten et al. (1993) and Zako¨ ıan (1994): asymmetric GARCH(1 , 1) : Z t = ν t σ t with σ 2 t = α 0 + α 1 Z 2 t - 1 + βσ 2 t - 1 + γ 1 { Z t - 1 < 0 } Z 2 t - 1 . 1 { Z t - 1 < 0 } denotes the indicator function which takes on the value one if Z t - 1 is negative and the value zero otherwise. Assuming, as before, that all pa- rameters α 0 , α 1 , β and γ are greater than zero, this specification postu- lates a leverage effect because negative realizations have a greater impact than positive ones. In order to obtain a stationary process the condition α 1 + β + γ/ 2 < 1 must hold. This model can be generalized in an obvious way by allowing additional lags Z 2 t - j and σ 2 t - j , j > 1 to enter the above specification. 7 A thorough treatment of the probabilistic properties of GARCH processes can be found in Nelson (1990), Bougerol and Picard (1992), Giraitis et al. (2000), and Kl¨uppelberg et al. (2004, theorem 2.1).
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8.1. SPECIFICATION AND INTERPRETATION 189 The Exponential GARCH model Another interesting and popular class of volatility models was introduced by Nelson (1991). The so-called exponential GARCH models or EGARCH models are defined as follows: log σ 2 t = α 0 + β log σ 2 t - 1 + γ Z t - 1 σ t - 1 + δ Z t - 1 σ t - 1 = α 0 + β log σ 2 t - 1 + γ | ν t - 1 | + δν t - 1 .
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