TAILIEUCHUNG - SAS/ETS 9.22 User's Guide 214

SAS/Ets User's Guide 214. 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 | 2122 F Chapter 32 The VARMAX Procedure Figure shows the orthogonalized responses of y1 and y2 to a forecast error impulse in y1 with two standard errors. Figure Plot of Orthogonalized Impulse Response Forecasting The optimal minimum MSE l-step-ahead forecast of yt i is P s q yt Cl It E j yt l-j t E xtcl-j t-E et i-j l q j 1 j 0 j l Ps ytcl i t Z j ytcl-ji t Z xt ci-ji t i q j 1 j o with yt cl _j t ytcl-j and x l-j t xt l _j for l j. For the forecasts x l _j t seethe section State-Space Representation on page 2105. Forecasting F 2123 Covariance Matrices of Prediction Errors without Exogenous Independent Variables Under the stationarity assumption the optimal minimum MSE l-step-ahead forecast of yt i has an infinite moving-average form yt i t 1 i j et i_j. The prediction error of the optimal l-step-ahead forecast is et i t yt i yt i t Pj Do jQ l-j with zero mean and covariance matrix i-1 i-1 S l i t X j S j X J 1 j 0 j 0 where o j P with a lower triangular matrix P such that S PP0. Under the assumption of normality of the et the l-step-ahead prediction error et i t is also normally distributed as multivariate N 0 S l . Hence it follows that the diagonal elements a l of S l can be used together with the point forecasts y t i t to construct l-step-ahead prediction intervals of the future values of the component series y t i. The following statements use the COVPE option to compute the covariance matrices of the prediction errors for a VAR 1 model. The parts of the VARMAX procedure output are shown in Figure and Figure . proc varmax data simul1 model y1 y2 p 1 noint lagmax 5 printform both print decompose 5 impulse all covpe 5 run Figure is the output in a matrix format associated with the COVPE option for the prediction error covariance matrices. Figure Covariances of Prediction Errors COVPE Option The VARMAX Procedure Prediction Error Covariances Lead Variable y1 y2 1 y1 y2 2 y1

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