TAILIEUCHUNG - SAS/ETS 9.22 User's Guide 219

SAS/Ets User's Guide 219. Provides detailed reference material for using SAS/ETS software and guides you through the analysis and forecasting of features such as univariate and multivariate time series, cross-sectional time series, seasonal adjustments, multiequational nonlinear models, discrete choice models, limited dependent variable models, portfolio analysis, and generation of financial reports, with introductory and advanced examples for each procedure. You can also find complete information about two easy-to-use point-and-click applications: the Time Series Forecasting System, for automatic and interactive time series modeling and forecasting, and the Investment Analysis System, for time-value of money analysis of a variety of investments | 2172 F Chapter 32 The VARMAX Procedure proc varmax data simul2 model y1 y2 p 2 cointtest johansen iorder 2 run The last two columns in Figure explain the cointegration rank test with integrated order 1. The results indicate that there is the cointegrated relationship with the cointegration rank 1 with respect to the significance level because the test statistic of is smaller than the critical value of . Now look at the row associated with r 1. Compare the test statistic value to the critical value for the cointegrated order 2. There is no evidence that the series are integrated order 2 at the significance level. Figure Cointegrated I 2 Test IORDER Option The VARMAX Procedure Cointegration Rank Test for I 2 Trace 5 CV of r k-r-s 2 1 of I 1 I 1 0 1 5 CV I 2 Multivariate GARCH Modeling Stochastic volatility modeling is important in many areas particularly in finance. To study the volatility of time series GARCH models are widely used because they provide a good approach to conditional variance modeling. BEKK Representation Engle and Kroner 1995 propose a general multivariate GARCH model and call it a BEKK representation. Let F t 1 be the sigma field generated by the past values of et and let Ht be the conditional covariance matrix of the k-dimensional random vector et. Let Ht be measurable with respect to F t 1 then the multivariate GARCH model can be written as et F t - 1 N 0 Ht q p Ht C X A iet-ie t-i Ai X G Ht-i Gi i 1 i 1 where C Ai and Gi are k x k parameter matrices. Multivariate GARCH Modeling F 2173 Consider a bivariate GARCH 1 1 model as follows Ht c11 c12 a11 a12 0 e2 t-1 e1 t 1e2 t -1 2 a11 a12 c12 c22 a21 a22 e2 t -1 e1 t -1 e2 t-1 J a21 a22 g11 g12 0 Ht-1 g11 g12 g21 g22 g21 g22 or representing the univariate model 2 2 2 2 h i i t c i i a n e u _ j 2a 1 1 2 1 e 1 t - 1 62 t - 1 a2 1 e2 t - 1 g 21 h 1 1 t - 1 2g 1 1 g2 1 h 1 2 t - 1 g21

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