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SAS/Ets 9.22 User's Guide 41. 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 | 392 F Chapter 8 The AUTOREG Procedure where o 2 is an estimator for o 2 1 -1 Evt vt -k. Note that the test rejects the null when Pt is small. The asymptotic power function for the point optimal test constructed with c under local alternatives with c is denoted by n c c . Then the power envelope is n c c because the test formed with c is the most powerful against the alternative c c. In other words the asymptotic function n c c is always below the power envelope n c except that at one point c c they are tangent. Elliott Rothenberg and Stock 1996 show that choosing some specific values for c can cause the asymptotic power function n c c of the point optimal test to be very close to the power envelope. The optimal c is 7 when zt 1 and 13.5 when zt 1 t . This choice of c corresponds to the tangent point where n 0.5. This is also true for the DF-GLS test. Elliott Rothenberg and Stock 1996 also propose the DF-GLS test given by the t statistic for testing o 0 in the regression p yf oyf-i C X Vi yd J C etp j 1 where yf is obtained in a first step detrending yf yt - a zt and 5 is least squares regression coefficient of ya on za. Regarding the lag length selection Elliott Rothenberg and Stock 1996 favor the Schwartz Bayesian information criterion. The optimal selection of the lag length p and the estimation of o 2 is further discussed in Ng and Perron 2001 . The lag length is selected from the interval 0 pmax for some fixed pmax by using the modified Akaike s information criterion 2. 2 tT p C P MAIC p log op C ---------- T Pmax where T rt tO iS2 1i r2 S T vf t2 inrl Pt2 T r 1 1 S T P2 Tor where Tt p Up Vo Lt pmax 1 yt-1 and p Sl pmax 2 t pmax 1 etp. For fixed lag length p an estimate of o 2 is given by .2 T - P -1 tT r 1 H ------------------2-- j1 - iV DF-GLS is indeed a superior unit root test according to Stock 1994 Schwert 1989 and Elliott Rothenberg and Stock 1996 . In terms of the size of the test DF-GLS is almost as good as the ADF t test DFr and better than the PP Zp
