Lectures for Part II: Time Series Models in Finance

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8.6 Extremes for GARCH and SV processes (cont) (i) GARCH: P( b (ii) SV model: P( b M ≤ x) → exp{ −γx −1 −α n n M ≤ x) →exp{ −x −1 −α n n } } Remarks about extremal index. (i) (ii) γ < 1 implies clustering of exceedances Numerical example. Suppose c is a threshold such that Then, if γ = .5, P( b (iii) 1/γ is the mean cluster size of exceedances. (iv) Use γ to discriminate between GARCH and SV models. (v) P n −1 ( bn X1 −1 n M n ≤ c) ~ .95 ≤ c) ~ (.95) = .975 Even for the light-tailed SV model (i.e., {Z t } ~IID N(0,1), the extremal index is 1 (see Breidt and Davis `98 ) .5 MaPhySto Workshop 9/04 96
8.6 Extremes for GARCH and SV processes (cont) 0 10 20 30 ** * * *** *** 0 20 40 60 time MaPhySto Workshop 9/04 97
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Nonlinear Time Series Modeling Part
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1. Classification of White Noise As
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2. Examples (1) All-pass processs.
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2. Examples (cont) MaPhySto Worksho
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2. Examples (cont) Likelihood funct
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2. Examples (cont) The sample ACF.
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Log returns for IBM 1/3/62-11/3/00
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ACF of squares for IBM (a) 1961-198
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Hill’s estimator of tail index Th
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Hill’s plot of tail index for IBM
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4. ARCH and GARCH Models (cont) MaP
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4. ARCH and GARCH Models (cont) i M
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Lectures for Part II: Time Series Models in Finance

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