Lee's Lecture Note_GARCH

# Lee's Lecture Note_GARCH - Lecture 5 Volatility and ARCH...

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Lecture 5 Volatility and ARCH Models Dr. Junsoo Lee EC 413 Read RATS Handout Chapter 5

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Why Volatility? What are the issues? Time varying risk premia Heteroskedastic variance o not constant variance News arrivals are serially (auto) correlated . o News tends to cluster in time Asymmetric reactions (leverage effects): o “People react more when prices fall.” Non-linearity in the model o Time deformation (economic activity does not match calendar time) Leptokurtic distribution o Fat-tails and excess peakness at the mean Volatility Models 1. Moving Average Models m-day historic volatility estimate 2 = r t-i 2 where r t-i is the m most recent returns r t = log(P t ) = log(P t ) - log(P t-1 ) : first difference of the price data = growth rates of price Questions: o How to determine m ? o Equal weights for each term, r t-i 2 ? o Could we lose valuable information by smoothing out the series? 2
2. Exponentially Weighted Moving Averages (EWMA) 2 = α r t-1 2 + (1 - α ) 2 = α (1 - α ) i-1 r t-i 2 Note: Brooks (p. 443) 2 = (1- λ29 λ i-1 r t-i 2 Note: This is the same as exponential smoothing method (except that y t is replaced with r t-1 2 ). Remarks: o If α = 1, it’s a naive model for volatility with r t-1 2 . o If α is close to 1, recent values of r t-i 2 are heavily weighted. o The predicted volatility remains constant as the estimate at T. 3. Auto-Regressive Conditional Heteroskedasticity (ARCH) Models “Express u t 2 in terms of past values of u t 2 (lagged squared residuals ).” u t 2 = ϖ + α 1 u t-1 2 + … + α q u t-q 2 Formally, (1) Mean Equation : Usual ARMA models or others y t = c + φ y t-1 + u t .. AR(1) model, for instance Let Var(u t | t-1 ) = h t .. conditional variance of u t where t-1 is the information set available at t-1. (2) Variance Equation: ARCH(Q) equation h t = ϖ + α 1 u t-1 2 + … + α q u t-q 2 This is an AR(1)-ARCH(Q) model! 3

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More examples: ARCH(2): h t = ϖ + α 1 u t-1 2 + α 2 u t-2 2 ARCH(3): h t = ϖ + α 1 u t-1 2 + α 2 u t-2 2 + α 3 u t-3 2 4. GARCH (Generalized ARCH) Models Additionally include lagged variance terms , h t-j , j=1,. .,P. GARCH(P, Q):
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## This note was uploaded on 02/21/2012 for the course ECON 500 taught by Professor Professor during the Fall '10 term at SUNY Buffalo.

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Lee's Lecture Note_GARCH - Lecture 5 Volatility and ARCH...

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