TAILIEUCHUNG - SAS/ETS 9.22 User's Guide 216

SAS/Ets User's Guide 216. 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 | 2142 F Chapter 32 The VARMAX Procedure Then the conditional approximate log-likelihood function can be written as follows Reinsel 1997 T -1 log S - 2 X s-1 t z t 1 log S w000-1 It 0 S-1 0-1w where w y - Pp 1 It 0 B1 y and 0 is such that e - Pq 1 IT 0 0 B1 e 0e. For the exact log-likelihood function of a VARMA p q model the Kalman filtering method is used transforming the VARMA process into the state-space form Reinsel 1997 . The state-space form of the VARMA p q model consists of a state equation zt F zt-1 G t and an observation equation yt H zt where for v max p q 1 Zt yt yt 1 t yt v-1 t F 0 0 Ik 0 0 Ik 0 0 G Ik 1 _ v-1 v-2 1 _ -1_ and H Ik 0 0 The Kalman filtering approach is used for evaluation of the likelihood function. The updating equation is zt t zt t-1 Kt t t-1 with Kt Pt t-1H 0 HPt t-1H V and the prediction equation is Zt t-1 Fzt_1 t-1 Pt t-1 FPt_1 t-1F0 GSG0 with Pt t I - Kt H Pt t-1 for t 1 2 . n. The log-likelihood function can be expressed as - 2 X log st t-1j- yt - yt t-1 0 r t1-1 yt - yt t-1 z t 1 VARMA and VARMAX Modeling F 2143 where yt t_1 and St t_1 are determined recursively from the Kalman filter procedure. To construct the likelihood function from Kalman filtering you obtain yt t_1 Hzt t_1 et t _1 yt yt t _1 and St t-i HPt t-1H0. Define the vector P P 01 . 0 01 . 0 vech S 0 where 0Z vec z and 0Z vec 0z . The log-likelihood equations are solved by iterative numerical procedures such as the quasi-Newton optimization. The starting values for the AR and MA parameters are obtained from the least squares estimates. Asymptotic Distribution of the Parameter Estimates Under the assumptions of stationarity and invertibility for the VARMA model and the assumption that t is a white noise process fi is a consistent estimator for f and VT f f converges in distribution to the multivariate normal N 0 V-1 as T 1 where V is the asymptotic information matrix of f. Asymptotic Distributions of Impulse Response Functions Defining the vector P p 0i . 0p 0i . 0q the

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