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ch10 - ARCH(1 process(autore-gressive conditional...

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10 Time series models in Finance and Econometrics 10.1 Background Recall that the AR(1) model is given by X t = αX t - 1 + Z t (10.1) where { Z t } is white noise, ie E ( Z t ) = 0 (10.2) and E ( Z t Z s ) = b σ 2 , t = s 0 , t n = s (10.3) Equation (10.3) says that the unconditional variance of { Z t } (and hence for { X t } as well) does not depend on time. As speciFed the conditional variance does not depend on time either since the Z t are independent. However, we could change the model for { Z t } such that (i) the unconditional variance is still constant (ii) the conditional variance changes with time. 10.2 A new model for { Z t } Instead of { Z t } being white noise we will specify the following model for { Z t } : We assume that Z t = r h t V t (10.4) where { V t } is white noise, ie iid N (0 , 1), and h t = ν + φ 1 Z 2 t - 1 (10.5) with ν > 0 and φ 1 0. A process { Z t } that can be written in this form is called an

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Unformatted text preview: ARCH(1) process (autore-gressive conditional heteroscedastic) and h t is called volatility . The conditional variance of an ARCH(1) process { Z t } depends on time V ( Z t | Z t-1 , Z t-2 , . . . ) = ν + φ 1 Z 2 t-1 , but the conditional variance is still constant V ( Z t ) = ν 1 − φ 1 . 10.3 Extensions The GARCH model (generalised ARCH) is h t = ν + φ 1 Z 2 t-1 + φ 2 Z 2 t-2 + · · · + φ m Z 2 t-m + δ 1 h t-1 + δ 2 h t-2 + · · · + δ r h t-r . The EGARCH model (exponential GARCH) is log( h t ) = ν t + ∞ s j =1 π j {| V t-j | − E ( | V t-j | ) + XV t-j } ....
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• Spring '09
• jsdkasj
• Probability theory, Autoregressive conditional heteroskedasticity, conditional variance, zt, Finance and Econometrics

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