"Frontmatter". In: Analysis of Financial Time Series

THE ARCH MODEL 83corrected asset return a t is serially uncorrelated, but dependent, and (b) the dependenceof a t can be described by a simple quadratic function of its lagged values.Specifically, an ARCH(m) model assumes thata t = σ t ɛ t , σ 2t = α 0 + α 1 a 2 t−1 +···+α ma 2 t−m , (3.5)where {ɛ t } is a sequence of independent and identically distributed (iid) random variableswith mean zero and variance 1, α 0 > 0, and α i ≥ 0fori > 0. The coefficientsα i must satisfy some regularity conditions to ensure that the unconditional varianceof a t is finite. In practice, ɛ t is often assumed to follow the standard normal or astandardized Student-t distribution.From the structure of the model, it is seen that large past squared shocks {at−i 2 }m i=1imply a large conditional variance σt 2 for the mean-corrected return a t . Consequently,a t tends to assume a large value (in modulus). This means that, under the ARCHframework, large shocks tend to be followed by another large shock. Here I use theword tend because a large variance does not necessarily produce a large variate. Itonly says that the probability of obtaining a large variate is greater than that of asmaller variance. This feature is similar to the volatility clusterings observed in assetreturns.The ARCH effect also occurs in other financial time series. Figure 3.2 shows thetime plots of (a) the percentage changes in Deutsche Mark/U.S. Dollar exchange ratemeasured in 10-minute intervals from June 5, 1989 to June 19, 1989 for 2488 observations,and (b) the squared series of the percentage changes. Big percentage changesoccurred occasionally, but there exist certain stable periods. Figure 3.3(a) shows thesample ACF of the percentage change series. Clearly, the series has no serial correlation.Figure 3.3(b) shows the sample PACF of the squared series of percentagechanges. It is seen that there are some big spikes in the PACF. Such spikes suggestthat the percentage changes are not independent and have some ARCH effects.Remark: Some authors use h t to denote the conditional variance in Eq. (3.5). Inthis case, the shock becomes a t = √ h t ɛ t .3.3.1 Properties of ARCH ModelsTo understand the ARCH models, it pays to carefully study the ARCH(1) modela t = σ t ɛ t , σ 2t = α 0 + α 1 a 2 t−1 ,where α 0 > 0andα 1 ≥ 0. First, the unconditional mean of a t remains zero becauseE(a t ) = E[E(a t | F t−1 )]=E[σ t E(ɛ t )]=0.Second, the unconditional variance of a t can be obtained asVar(a t ) = E(a 2 t ) = E[E(a2 t | F t−1 )]=E(α 0 + α 1 a 2 t−1 ) = α 0 + α 1 E(a 2 t−1 ).