"Frontmatter". In: Analysis of Financial Time Series

THE ARCH MODEL 87sample mean from the data if the sample mean is significantly different from zero.For some daily return series, a simple AR model might be needed. The squared seriesat 2 is used to check for conditional heteroscedasticity, where a t = r t −µ t is the residualof the ARMA model. Two tests are available here. The first test is to check theusual Ljung–Box statistics of at 2 ; see McLeod and Li (1983). The second test forconditional heteroscedasticity is the Lagrange multiplier test of Engle (1982). Thistest is equivalent to the usual F statistic for testing α i = 0(i = 1,...,m) in thelinear regressiona 2 t = α 0 + α 1 a 2 t−1 +···+α ma 2 t−m + e t,t = m + 1,...,T,where e t denotes the error term, m is a prespecified positive integer, and T is thesample size. Let SSR 0 = ∑ Tt=m+1 (a 2 t −¯ω) 2 ,where ¯ω is the sample mean of a 2 t ,and SSR 1 = ∑ Tt=m+1 ê 2 t ,whereê t is the least squares residual of the prior linearregression. Then we haveF = (SSR 0 − SSR 1 )/mSSR 1 /(T − 2m − 1) ,which is asymptotically distributed as a chi-squared distribution with m degrees offreedom under the null hypothesis.Order DeterminationIf the test statistic F is significant, then conditional heteroscedasticity of a t isdetected, and we use the PACF of at 2 to determine the ARCH order. Using PACF ofat 2 to select the ARCH order can be justified as follows. From the model in Eq. (3.5),we haveσ 2t = α 0 + α 1 a 2 t−1 +···+α ma 2 t−m .For a given sample, at 2 is an unbiased estimate of σt 2.Therefore, we expect that a2 tis linearly related to at−1 2 ,...,a2 t−m in a manner similar to that of an autoregressivemodel of order m. Note that a single at2 is generally not an efficient estimate ofσt 2 , but it can serve as an approximation that could be informative in specifying theorder m.Alternatively, define η t = at 2 − σt 2.It can be shown that {η t} is an un-correlatedseries with mean 0. The ARCH model then becomesa 2 t = α 0 + α 1 a 2 t−1 +···+α ma 2 t−m + η t,which is in the form of an AR(m) model for at 2, except that {η t} is not an iid series.From Chapter 2, PACF of at2 is a useful tool to determine the order m. Because{η t } are not identically distributed, the least squares estimates of the prior model areconsistent, but not efficient. The PACF of at 2 may not be effective when the samplesize is small.