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SAS/ETS 9.22 User's Guide 112

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SAS/Ets 9.22 User's Guide 112. 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 | 1102 F Chapter 18 The MODEL Procedure exp const inc income inc2 income income incsq income income fit exp white breusch 1 income incsq run Figure 18.39 Output for Heteroscedasticity Tests The MODEL Procedure Heteroscedasticity Test Equation Test Statistic DF Pr ChiSq Variables exp White s Test 21.16 4 0.0003 Cross of all vars Breusch-Pagan 15.83 2 0.0004 1 income incsq Correcting for Heteroscedasticity There are two methods for improving the efficiency of the parameter estimation in the presence of heteroscedastic errors. If the error variance relationships are known weighted regression can be used or an error model can be estimated. For details about error model estimation see the section Error Covariance Structure Specification on page 1112. If the error variance relationship is unknown GMM estimation can be used. Weighted Regression The WEIGHT statement can be used to correct for the heteroscedasticity. Consider the following model which has a heteroscedastic error term yt 250 e 2t - e 0 8t p9 t et The data for this model is generated with the following SAS statements. data test do t 1 to 25 y 250 exp -0.2 t - exp -0.8 t sqrt 9 t rannor 1 output end run If this model is estimated with OLS as shown in the following statements the estimates shown in Figure 18.40 are obtained for the parameters. proc model data test parms bl 0.1 b2 0.9 y 250 exp -b1 t - exp -b2 t fit y run Heteroscedasticity F 1103 Figure 18.40 Unweighted OLS Estimates The MODEL Procedure Nonlinear OLS Parameter Estimates Approx Approx Parameter Estimate Std Err t Value Pr t bl 0.200977 0.00l0l l98.60 .000l b2 0.826236 0.00853 96.82 .000l If both sides of the model equation are multiplied by the model has a homoscedastic error term. This multiplication or weighting is done through the WEIGHT statement. The WEIGHT statement variable operates on the squared residuals as e t et weight x q q so that the WEIGHT statement variable represents the square of the model multiplier. The following PROC MODEL .