"Frontmatter". In: Analysis of Financial Time Series

APPLICATION 385[ ]σ11,t=σ 22,t⎡ ⎤ ⎡ ⎤ ⎡ ⎤19.08 0.098 ·[ ⎢(3.70)⎥⎣−5.62⎦ + ⎢(0.044)⎥ a2]0.3331,t−1⎣ · · ⎦ a2,t−12 + ⎢(0.076)⎥⎣ 0.596 ⎦ σ t 2 , (9.35)(2.36)(0.050)where, as before, standard errors are in parentheses, and σt2 is obtained from model(9.34). The conditional correlation equation isρ t = exp(q t)1 + exp(q t ) , q t =−2.098 + 4.120ρ t−1 + 0.078 a 1,t−1a 2,t−1√ σ11,t−1 σ 22,t−1, (9.36)where standard errors of the three parameters are 0.025, 0.038, and 0.015, respectively.Defining the standardized residuals as before, we obtain Q(4) = 15.37(0.29)and Q(8) = 34.24(0.23), where the number in parentheses denotes p value.Therefore, the standardized residuals have no serial correlations. Yet we haveQ(4) = 20.25(0.09) and Q(8) = 61.95(0.0004) for the squared standardizedresiduals. The volatility model in Eq. (9.35) does not adequately handle the conditionalheteroscedasticity of the data especially at higher lags. This is not surprisingas the single common factor only explains about 82.5% of the generalized varianceof the data.Comparing the factor model in Eqs. (9.35) and (9.36) with the time-varying correlationmodel in Eqs. (9.24) and (9.25), we see that (a) the correlation equations ofthe two models are essential the same, (b) as expected the factor model uses fewerparameters in the volatility equation, and (c) the common-factor model provides areasonable approximation to the volatility process of the data.Remark: In Example 9.4, we used a two-step estimation procedure. In the firststep, a volatility model is built for the common factor. The estimated volatility istreated as given in the second step to estimate the multivariate volatility model. Suchan estimation procedure is simple, but may not be efficient. A more efficient estimationprocedure is to perform a joint estimation. This can be done relatively easilyprovided that the common factors are known. For example, for the monthly logreturns of Example 9.4, a joint estimation of Eqs. (9.34)–(9.36) can be performed ifthe common factor x t = 0.769r 1t + 0.605r 2t is treated as given.9.5 APPLICATIONWe illustrate the application of multivariate volatility models by considering theValue at Risk (VaR) of a financial position with multiple assets. Suppose that aninvestor holds a long position in the stocks of Cisco Systems and Intel Corporationeach worth $1 million. We use the daily log returns for the two stocks from January 2,1991 to December 31, 1999 to build volatility models. The VaR is computed usingthe 1-step ahead forecasts at the end of data span and 5% critical values.