TAILIEUCHUNG - SAS/ETS 9.22 User's Guide 109

SAS/Ets User's Guide 109. 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 | 1072 F Chapter 18 The MODEL Procedure HESSIAN GLS has better convergence properties in general but COVBEST CROSS produces the most pessimistic standard error bounds. When the HESSIAN option is used the default estimator of the variance-covariance of O is the inverse of the Hessian selected. Multivariate t Distribution Estimation The multivariate t distribution is specified by using the ERRORMODEL statement with the T option. Other method specifications FIML and OLS for example are ignored when the ERRORMODEL statement is used for a distribution other than normal. The probability density function for the multivariate t distribution is p __r df bm C q0 yt xt Q S d 1q y xt Q 2 q n df r df X d 1 df where m is the number of equations and df is the degrees of freedom. The maximum likelihood estimators of O and a are the O and a that minimize the negative loglikelihood function ln 0 a n r df m z _ y-1_ Vf X X ln 1 2 __ A C dt q @ n df 2 r df v f A C2 ln ï - Xln t 1 @qt @y t The ERRORMODEL statement is used to request the t distribution maximum likelihood estimation. An OLS estimation is done to obtain initial parameter estimates and estimates. Use NOOLS to turn off this initial estimation. If the errors are distributed normally t distribution estimation produces results similar to FIML. The multivariate model has a single shared degrees-of-freedom parameter which is estimated. The degrees-of-freedom parameter can also be set to a fixed value. The log-likelihood value and the l2 norm of the gradient of the negative log-likelihood function are shown in the estimation summary. t Distribution Details Since a variance term is explicitly specified by using the ERRORMODEL statement X O is estimated as a correlation matrix and q yt xt Q is normalized by the variance. The gradient of the negative log-likelihood function with respect to the degrees of freedom is @ln @df nm n 0 dfCm 2df 2 1 df m 0 5 log 1 C Tp df C nlf C 2 r f df C m q0X-1q

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