Lee's Lecture Note_ARCH

Lee's Lecture Note_ARCH - Lecture 6 Volatility and ARCH...

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Lecture 6 Volatility and ARCH Models Dr. Junsoo Lee EC 413
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Why Volatility clustering? 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
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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: This is the same as exponential smoothing method (except that y t is replaced with r t-1 2 ). Recall from Lecture 3: t = α y t-1 + (1 - α ) t-1 t-1 = α y t-2 + (1 - α ) t-2 t-2 = α y t-3 + (1 - α ) t-3 = α y t-1 + (1- α ) α y t-2 + (1- α ) 2 α y t-3 + ….. + (1- α ) k α y t-k+1 Similarly, 2 = (1- λ29 λ i-1 r t-i 2 (with λ = 1-α29 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 (no error term!) (1) Mean Equation : Usual ARMA models or others The Model Explaining "Y" (the "mean" equation): y t = c + φ y t-1 + u t .. AR(1) model, for instance Var(u t | t-1 ) = h t .. conditional variance of u t where t-1 is the information set available at t-1. 3
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(2) Variance Equation: ARCH(Q) equation The Model Explaining the Conditional Variance: h t = ϖ + α 1 u t-1 2 + … + α q u t-q 2 These two equations consist of an AR(1)-ARCH(Q) model! 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): h t = ϖ + α 1 u t-1 2 + … + α q u t-q 2 + β 1 h t-1 + …. + β p h t-p Note: People often use GARCH (1,1) models, without having to search for optimal models. GARCH(1, 1): h t = ϖ + α 1 u t-1 2 + β 1 h t-1 A complete specification of the AR(1)-GARCH(1,1) model , for example: y t = c + φ y t-1 + u t .. AR(1) model, for instance Var(u t | t-1 ) = h t .. conditional variance of u t h t = ϖ + α 1 u t-1 2 + β 1 h t-1 ... GARCH(1,1), for instance More examples: 4
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Lee's Lecture Note_ARCH - Lecture 6 Volatility and ARCH...

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