"Frontmatter". In: Analysis of Financial Time Series

VECTOR MA MODELS 321turn used to compute a t recursively using Eq. (8.25) and starting with a 1 =˜r 1 +Θâ 0 .The resulting {a t }t=1 T are then used to evaluate the exact likelihood function of thedata to update the estimates of θ 0 , Θ, andΣ. The whole process is then repeateduntil the estimates converge. This iterative method to evaluate the exact likelihoodfunction applies to the general VMA(q) models.From the previous discussion, the exact likelihood method requires more intensivecomputation than the conditional likelihood approach does. But it provides moreaccurate parameter estimates, especially when some eigenvalues of Θ is close to 1in modulus. Hillmer and Tiao (1979) provide some comparison between the conditionaland exact likelihood estimations of VMA models. In multivariate time seriesanalysis, the exact maximum likelihood method becomes important if one suspectsthat the data might have been overdifferenced. Overdifferencing may occur in manysituations (e.g., differencing individual components of a co-integrated system; seediscussion later on co-integration).In summary, building a VMA model involves three steps: (a) use the sample crosscorrelationmatrixes to specify the order q—for a VMA(q) model, ρ l = 0 for l>q;(b) estimate the specified model by using either the conditional or exact likelihoodmethod—the exact method is preferred when the sample size is not large; and (c) thefitted model should be checked for adequacy (e.g., applying the Q k (m) statistics tothe residual series). Finally, forecasts of a VMA model can be obtained by using thesame procedure as a univariate MA model.Example 8.5. Consider again the bivariate series of monthly log returns inpercentages of IBM stock and the S&P 500 index from January 1926 to December1999. Since significant cross-correlations occur mainly at lags 1 and 3, we employthe VMA(3) modelr t = θ 0 + a t − Θ 1 a t−1 − Θ 3 a t−3 (8.27)for the data. Table 8.5 shows the estimation results of the model. The Q k (m) statisticsfor the residuals of the simplified model give Q 2 (4) = 17.25 and Q 2 (8) = 39.30.Compared with chi-squared distributions with 12 and 28 degrees of freedom, thep values of these statistics are 0.1404 and 0.0762, respectively. Thus, the model isadequate at the 5% significance level.From Table 8.5, we make the following observations:1. The difference between conditional and exact likelihood estimates is small forthis particular example. This is not surprising because the sample size is notsmall and, more important, the dynamic structure of the data is weak.2. The VMA(3) model provides essentially the same dynamic relationship for theseries as that of the VAR(3) model in Example 8.4. The monthly log return ofIBM stock depends on the previous returns of the S&P 500 index. The marketreturn, in contrast, does not depend on lagged returns of IBM stock. In otherwords, the dynamic structure of the data is driven by the market return, notby IBM return. The concurrent correlation between the two returns remainsstrong, however.