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Lecture 5a: ARCH Models - Miami University

Lecture 5a: ARCH Models1 Big use ARMA model for the conditional use ARCH model for the conditional and ARCH model can be used together to describe bothconditional mean and conditional variance2 Price and ReturnLetptdenote the price of a financial asset (such as a stock). Thenthe return of buying yesterday and selling today (assuming nodividend) isrt=pt pt 1pt 1 log(pt) log(pt 1).The approximation works well whenrtis close to Compounded ReturnAlternatively,rtmeasures the continuously compounded ratert= log(pt) log(pt 1)(1) ert=ptpt 1(2) pt=ertpt 1(3) pt= limn!1(1 +rtn)npt 1(4)4 Why conditional variance? asset is risky if its returnrtis volatile (changing a lot overtime) statistics we use variance to measure volatility (dispersion),and so the are more interested in conditional variance, denoted byvar(rt|rt 1,rt 2,..) =E(r2t|rt 1,rt 2,..),because we want to use the past history to forecast the last equality holds ifE(rt|rt 1,rt 2,..) = 0,which is truein most stylized fact about financial market is volatility clustering.

Consider the first order autoregressive conditional heteroskedasticity (ARCH) process rt = σtet (5) et ∼ white noise(0, 1) (6) σt = √ ω + α1r2 t 1 (7) where rt is the return, and is assumed here to be an ARCH(1) process. et is a white noise with zero mean and variance of one. et may or may not follow normal distribution. 7

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  Conditional, Autoregressive, Autoregressive conditional

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Transcription of Lecture 5a: ARCH Models - Miami University

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