Lectures for Part II: Time Series Models in Finance

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Plot of M t (4)/S t (4) for IBM M(4)/S(4) 0.0 0.2 0.4 0.6 0.8 1.0 MaPhySto Workshop 9/04 0 2000 4000 6000 8000 10000 Remark: For IID data, M t (k)/S t (k) → 0 as t →∞iff E|X , | k < ∞, where t k k M t = maxs= 1,..., t | X s | and St = ∑| X s | t s= 1 22
Hill’s estimator of tail index The marginal distribution F for heavy-tailed data is often modeled using Pareto-like tails, 1-F(x) = x -α L(x), for x large, where L(x) is a slowly varying function (L(xt)/ L(x)→1, as x →∞). Now if X~ F, then P(log X > x) = P(X > exp(x))=exp(-αx)L(exp(x)), and hence log X has an approximate exponential distribution for large x. The spacings, log( X (j) ) − log(X (j+1) ) , j=1,2,. . . ,m, from a sample of size n from an exponential distr are approximately independent and Exp(αj) distributed. This suggests estimating α −1 by MaPhySto Workshop 9/04 αˆ −1 = = 1 m 1 m m ∑( log X ( j) − log X ( j+ 1) ) j= 1 m ∑( log X ( j) − log X ( m+ 1) ) j= 1 j 23
Page 1 and 2: Nonlinear Time Series Modeling Part
Page 3 and 4: 1. Classification of White Noise As
Page 5 and 6: 2. Examples (1) All-pass processs.
Page 7 and 8: 2. Examples (cont) MaPhySto Worksho
Page 13 and 14: 2. Examples (cont) Likelihood funct
Page 15 and 16: 2. Examples (cont) The sample ACF.
Page 17 and 18: MaPhySto Workshop 9/04 17
Page 19 and 20: Log returns for IBM 1/3/62-11/3/00
Page 21: ACF of squares for IBM (a) 1961-198
Page 25 and 26: Hill’s plot of tail index for IBM
Page 27 and 28: 4. ARCH and GARCH Models (cont) MaP
Page 37 and 38: 4. ARCH and GARCH Models (cont) i M
Page 41 and 42: 4. ARCH and GARCH Models (cont) SS
Page 49 and 50: Parameter Estimation for Finite-Var
Page 67 and 68: 5. Forecasting with GARCH (cont) Ma
Page 69 and 70: 7. Stochastic Volatility Models MaP
Page 71 and 72: 7. Stochastic Volatility Models (co
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7. Stochastic Volatility Models (co
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MaPhySto Workshop 9/04 81
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8. Regular variation and applicatio
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8.2 Regular variation — multivari
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2. If X 1 = X 2 > 0, then X= (X 1 ,
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8.3 Applications of multivariate re
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8.4 Applications of theorem 1. Kest
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8.4 Examples (cont) Example of ARCH
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8.6 Extremes for GARCH and SV proce
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8.6 Extremes for GARCH and SV proce
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8.7 Summary of results for ACF of G
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Sample ACF for Squares of GARCH (10
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Example: Amazon returns May 16, 199
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Lectures for Part II: Time Series Models in Finance

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