"Frontmatter". In: Analysis of Financial Time Series

SIMPLE AUTOREGRESSIVE MODELS 37Table 2.1. Sample Partial Autocorrelation Function and Akaike Information Criterionfor the Monthly Simple Returns of CRSP Value-Weighted Index from January 1926 toDecember 1997.p 1 2 3 4 5PACF 0.11 -0.02 −0.12 0.04 0.07AIC −5.807 −5.805 −5.817 −5.816 −5.819p 6 7 8 9 10PACF −0.06 0.02 0.06 0.06 −0.01AIC −5.821 −5.819 −5.820 −5.821 −5.818As an example, consider the monthly simple returns of CRSP value-weightedindex from January 1926 to December 1997. Table 2.1 gives the first 10 lags ofsample PACF of the series. With T = 864, the asymptotic standard error of thesample PACF is approximately 0.03. Therefore, using the 5% significant level, weidentify an AR(3) or AR(5) model for the data (i.e., p = 3or5).Information CriteriaThere are several information criteria available to determine the order p of an ARprocess. All of them are likelihood based. For example, the well-known Akaike InformationCriterion (Akaike, 1973) is defined asAIC = −2T ln(likelihood) + 2 T× (number of parameters), (2.13)where the likelihood function is evaluated at the maximum likelihood estimates andT is the sample size. For a Gaussian AR(l) model, AIC reduces toAIC(l) = ln( ˆσ 2 l ) + 2lT ,where ˆσ 2 l is the maximum likelihood estimate of σ 2 a , which is the variance of a t,and T is the sample size; see Eq. (1.18). In practice, one computes AIC(l) forl = 0,...,P, whereP is a prespecified positive integer and selects the order kthat has the minimum AIC value. The second term of the AIC in Eq. (2.13) is calledthe penalty function of the criterion because it penalizes a candidate model by thenumber of parameters used. Different penalty functions result in different informationcriteria.Table 2.1 also gives the AIC for p = 1,...,10. The AIC values are close to eachother with minimum −5.821 occurring at p = 6 and 9, suggesting that an AR(6)model is preferred by the criterion. This example shows that different approachesfor order determination may result in different choices of p. There is no evidence