ARCH.pdf.pdf (EC3400)

# ARCH.pdf.pdf (EC3400) - ARCH Ser-Huang Poon Financial...

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ARCH Ser-Huang Poon August 11, 2008 Financial market volatility is known to cluster. A volatile period is knowntopers istforsomet imebeforethemarketreturnstonorma l ity .The ARCH (AutoRegressive Conditional Heteroskedasticity) model proposed by Engle (1982) was to capture volatility persistence in in f ation. The ARCH model was later found to F tmany F nancial times and the wide spread impact on many F nance applications has led to the Nobel Committee’s recognition of Rob Engle’s work in 2003. The ARCH e f ect has been shown to lead to high kurtosis which F ts in well with the empirically observed tail thickness of many asset return distributions. The leverage e f ect, a phenomenon relates to high volatility being induced by negative return, is often modelled with a sign based return variable in the conditional volatility equation. 1 Engle (1982) The ARCH (Autoregressive Conditional Heteroskedasticity) model, F rst in- troduced by Engle (1982) has been extended by many researchers and ex- tensively surveyed in Bera and Higgins (1993), Bollerslev, Chou and Kroner (1992), Bollerslev, Engle and Nelson (194) and Diebold and Lopez (1995). In contrast to the Historical Volatility models described in the previous chap- ter, ARCH models do not make use of the sample standard deviations, but formulate conditional variance, h t , of asset returns via maximum likelihood 1

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procedures. (We follow the ARCH literature here by writing σ 2 t = h t .) To illustrate this, f rst write returns, r t ,as r t = μ + ε t , ε t = p h t z t , (1) where z t is a white noise, z t D (0 , 1) . The distribution D is often taken as Normal. The process z t is scaled by h t , the conditional variance, which in turn is a function of past squared residual returns. In the ARCH ( q ) process proposed by Engle (1982), h t = ω + q X j =1 α j ε 2 t j (2) the condition ω> 0 and α j 0 are set to ensure strictly positive variance. Typically, q is of high order because of the phenomenon of volatility persis- tence in f nancial markets. From the way in which volatility is constructed in (2), h t is known at time t 1 . So the one-step ahead forecast is readily available. The multi-step ahead forecasts can be formulated by assuming E £ ε 2 t + τ ¤ = h t + τ . The unconditional variance of r t is σ 2 = ω 1 P q j =1 α j . The process is covariance stationary if and only if the sum of the autore- gressive parameters is less than one P q j =1 α j < 1 . 2 Generalised ARCH For high order ARCH ( q ) process, it is more parsimonous to model volatility as a GARCH ( p, q ) (Generalised ARCH due to Bollerslev (1986) and Taylor (1986)), where additional dependencies are permitted on p lags of past h t as shown below h t = ω + p X i =1 β i h t i + q X j =1 α j ε 2 t j 2
and ω> 0 .F o rG A R C H (1 , 1) , the constriants α 1 0 and β 1 0 are needed to ensure h t is strictly positive. For higher orders of GARCH, the constraints on β i and α j are more complex (see Nelson and Cao (1992) for details). The unconditional variance equals σ 2 = ω 1 P p i =1 β i P q j =1 α j The GARCH ( p, q ) model is covariance stationary if and only if P p i =1 β i +

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## This note was uploaded on 02/28/2010 for the course ECO 211 taught by Professor Gilo during the Spring '10 term at Young Harris.

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ARCH.pdf.pdf (EC3400) - ARCH Ser-Huang Poon Financial...

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