Lecture_6_ARCH_Falll_2009

Lecture_6_ARCH_Falll_2009 - Lecture 6 Volatility and GARCH...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 6 Volatility and GARCH Models Read (Handbook Ch 3) EC 413/513 Economic Forecast and Analysis (Professor Lee)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 = growth rates of price = rate of returns 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
Background image of page 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: 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. A uto- R egressive C onditional H eteroskedasticity (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 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! 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 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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: GARCH(1, 2): h t = ϖ + α 1 u t-1 2 + α 2 u t-2 2 + β 1 h t-1 GARCH(2, 1): h t = ϖ + α 1 u t-1 2 + β 1 h t-1 + β 2 h t-2 Case Study: WPI Inflation (handout) Questions: 1. Why conditional? 2.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 29

Lecture_6_ARCH_Falll_2009 - Lecture 6 Volatility and GARCH...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online