"Frontmatter". In: Analysis of Financial Time Series

FACTOR-VOLATILITY MODELS 383considered there allows for time-varying correlations, but in a relatively restrictivemanner. Additional references of multivariate volatility model include Harvey, Ruiz,and Shephard (1995). We discuss MCMC methods to volatility modeling in Chapter10.9.4 FACTOR-VOLATILITY MODELSAnother approach to simplifying the dynamic structure of a multivariate volatilityprocess is to use factor models. In practice, the “common factors” can be determineda priori by substantive matter or empirical methods. As an illustration, we use thefactor analysis of Chapter 8 to discuss factor-volatility models. Because volatilitymodels are concerned with the evolution over time of the conditional covariancematrix of a t ,wherea t = r t − µ t , a simple way to identify the “common factors” involatility is to perform a principal component analysis (PCA) on a t ; see the PCA ofChapter 8. Building a factor volatility model thus involves a three-step procedure:• select the first few principal components that explain a high percentage of variabilityin a t ,• build a volatility model for the selected principal components, and• relate the volatility of each a it series to the volatilities of the selected principalcomponents.The objective of such a procedure is to reduce the dimension, but maintain an accurateapproximation of the multivariate volatility.Example 9.4. Consider again the monthly log returns, in percentages, ofIBM stock and the S&P 500 index of Example 9.2. Using the bivariate AR(3) modelof Example 8.4, we obtain an innovational series a t . Performing a PCA on a t basedon its covariance matrix, we obtained eigenvalues 63.373 and 13.489. The first eigenvalueexplains 82.2% of the generalized variance of a t . Therefore, we may choosethe first principal component x t = 0.797a 1t + 0.604a 2t as the common factor. Alternatively,as shown by the model in Example 8.4, the serial dependence in r t is weakand hence, one can perform the PCA on r t directly. For this particular instance, thetwo eigenvalues of the sample covariance matrix of r t are 63.625 and 13.513, whichare essentially the same as those based on a t .Thefirst principal component explainsapproximately 82.5% of the generalized variance of r t , and the corresponding commonfactor is x t = 0.796r 1t + 0.605r 2t . Consequently, for the two monthly logreturn series considered, the effect of the conditional mean equations on PCA is negligible.Based on the prior discussion and for simplicity, we use x t = 0.796r 1t + 0.605r 2tas a common factor for the two monthly return series. Figure 9.11(a) shows the timeplot of this common factor. If univariate Gaussian GARCH models are entertained,we obtain the following model for x t :