PROC AUTOREG produces two types of predicted values for the response series — predicted values based on only the structural part of the model, x'β, and predicted values based on both the structural part of the model and the error process, x'β + v_t|t-1. For an AR(m) error process, when m previous values of the structural residuals are available, v_t|t-1 is computed using the estimated AR parameters and the m previous structural residuals:

v_t|t-1 = -φ₁v_t-1 - φ₂v_t-2 - ... - φ_mv_t-m

The negative signs in front of the AR parameters in this equation result from the specification of the error process in PROC AUTOREG which has negative signs in front of the AR parameters. See "Regression with Autocorrelated Errors" in the Details section of the PROC AUTOREG documentation.

When the structural residuals from the previous period are not available, then a minimum variance linear predictor of the error term given past values of the structural residuals is used together with the estimated AR parameters to compute v_t|t-1. The following example shows how the predicted values can be computed in a DATA step. Note that the predicted values cannot be easily replicated for the first NLAG=n observations in the data set, or for the first NLAG=n observations following a missing dependent value. For these observations computation of predicted values use a Kalman filtering technique.

Sometimes you might want to compute predicted values for a data set when you do not have a sufficient number of observations to obtain reliable parameter estimates for the AR(m) model, or you just want to use predetermined parameters to compute predicted values for your data. In these situations, you can specify the GARCH=(q=1) option in the MODEL statement, and use the RESTRICT statement to restrict the _AH_1 parameter to zero and all the AR model parameters to fixed values. You can use this approach to compute predicted values in the AR model provided that you have at least k+1 observations on both the dependent and independent variables in the beginning of the data set, where k is the total number of parameters in the model in which GARCH=(q=1) is specified. An example is given in this note.

Example:

The following DATA step creates the data to be analyzed. The variable X contains the annual real GNP series from 1901 to 1973. The dependent variable in this analysis is the natural log of real GNP, Y. The independent variable in the model is the time trend variable, T. For the years from 1974 through 1983, no real GNP is observed. To have PROC AUTOREG produce predictions for these years, values of T are provided and the dependent variable, Y, is set to missing.

      data gnp; 
         date = intnx( 'year', '01jan1901'd, _n_-1 ); 
         format date year4.; 
         input x @@; 
         y = log(x); 
         t = _n_; 
         label y = 'Real GNP' 
               t = 'Time Trend'; 
         datalines; 
       137.87  139.13  146.10  144.21  155.04  172.97  175.61  161.22 
       180.93  185.98  191.90  201.51  203.38  195.96  193.63  208.19 
       211.43  244.33  228.99  214.22  199.94  229.54  254.09  256.39 
       276.03  291.81  293.27  296.22  315.69  285.52  263.46  227.04 
       222.13  239.09  260.04  295.54  310.16  296.75  319.83  344.16 
       400.40  461.80  531.66  569.13  560.17  478.28  470.39  489.74 
       492.24  534.57  579.17  600.63  623.46  615.64  657.37  671.72 
       683.57  680.88  721.72  737.01  756.59  800.29  832.20  875.96 
       929.00  984.62 1011.38 1058.06 1087.58 1085.58 1122.42 1185.92 
      1254.20     .       .       .       .       .       .       .
          .       .       .
      ; 

These statements display the data set.

      proc print data=gnp noobs; 
         run;

date x y t 1901 137.87 4.92631 1 1902 139.13 4.93541 2 1903 146.10 4.98429 3 1904 144.21 4.97127 4 1905 155.04 5.04368 5 1906 172.97 5.15312 6 1907 175.61 5.16827 7 1908 161.22 5.08277 8 1909 180.93 5.19811 9 1910 185.98 5.22564 10 1911 191.90 5.25697 11 1912 201.51 5.30584 12 1913 203.38 5.31508 13 1914 195.96 5.27791 14 1915 193.63 5.26595 15 1916 208.19 5.33845 16 1917 211.43 5.35389 17 1918 244.33 5.49852 18 1919 228.99 5.43368 19 1920 214.22 5.36700 20 1921 199.94 5.29802 21 1922 229.54 5.43608 22 1923 254.09 5.53769 23 1924 256.39 5.54670 24 1925 276.03 5.62051 25 1926 291.81 5.67610 26 1927 293.27 5.68109 27 1928 296.22 5.69110 28 1929 315.69 5.75476 29 1930 285.52 5.65431 30 1931 263.46 5.57390 31 1932 227.04 5.42513 32 1933 222.13 5.40326 33 1934 239.09 5.47684 34 1935 260.04 5.56084 35 1936 295.54 5.68880 36 1937 310.16 5.73709 37 1938 296.75 5.69289 38 1939 319.83 5.76779 39 1940 344.16 5.84111 40 1941 400.40 5.99246 41 1942 461.80 6.13513 42 1943 531.66 6.27600 43 1944 569.13 6.34411 44 1945 560.17 6.32824 45 1946 478.28 6.17020 46 1947 470.39 6.15356 47 1948 489.74 6.19387 48 1949 492.24 6.19897 49 1950 534.57 6.28146 50 1951 579.17 6.36160 51 1952 600.63 6.39798 52 1953 623.46 6.43528 53 1954 615.64 6.42266 54 1955 657.37 6.48825 55 1956 671.72 6.50984 56 1957 683.57 6.52733 57 1958 680.88 6.52339 58 1959 721.72 6.58164 59 1960 737.01 6.60260 60 1961 756.59 6.62882 61 1962 800.29 6.68497 62 1963 832.20 6.72407 63 1964 875.96 6.77532 64 1965 929.00 6.83411 65 1966 984.62 6.89226 66 1967 1011.38 6.91907 67 1968 1058.06 6.96419 68 1969 1087.58 6.99171 69 1970 1085.58 6.98987 70 1971 1122.42 7.02324 71 1972 1185.92 7.07827 72 1973 1254.20 7.13425 73 1974 . . 74 1975 . . 75 1976 . . 76 1977 . . 77 1978 . . 78 1979 . . 79 1980 . . 80 1981 . . 81 1982 . . 82 1983 . . 83

The following PROC AUTOREG code specifies a regression model with AR(2) error process. The NLAG=2 option specifies the order of AR error process. The METHOD=ML option specifies that the maximum likelihood estimation method is to be used. The OUTEST=EST option saves the parameter estimates to a data set named EST which will be used in a subsequent DATA step to replicate the predicted values produced by PROC AUTOREG. The OUTPUT statement saves the predicted values and structural residuals to a data set named B. The PM= option names the variable containing the structural predictions. The P= option names the variable containing the predicted values that incorporate both the structural model and the AR error process. The RM= option names the variable containing the structural residuals.

      proc autoreg data=gnp outest=est; 
         model y = t / nlag=2 method=ml; 
         output out=b p=py rm=r pm=pm;
         run; 

These statements display the data set containing the parameter estimates.

      proc print data=est label noobs; 
         run;

Label of Model Type of Estimation Convergence Status Name of Dependent Variable Method of Estimation Variable Names for Rows of COV Matrix Estimate of Variance Sum of Squares Error Intercept Parameter Parameter Estimate for y Parameter Estimate for t Parameter Estimate for _A_1 Parameter Estimate for _A_2   PARM 0 Converged y ML   .003369253 0.23248 4.82694 -1 0.029974 -1.21087 0.38254

The following DATA step computes the predicted values. The RETAIN statement initializes the lagged residuals to be zero. The structural predictions, YHAT, are computed using the estimated intercept and the estimated parameter on the independent variable, T. In order to compute v_t|t-1, the lagged residuals, v_t-1, v_t-2,..., v_t-m, are obtained as the observed Y minus the structural predictions, YHAT. When V at a specific lag cannot be obtained due to missing Y, a minimum variance predictor, VHAT, is used which is obtained using the estimated AR parameters and the available lagged residuals. Note the negative signs in front of the AR parameters, _A_1, and _A_2, in the equation for computing VHAT. The final predictions, PREDY, are computed using the structural predictions plus the predictions on the error term v_t|t-1. The variable DIFF is created to compare the predicted values computed in the DATA step, PREDY, with those obtained from the OUTPUT statement in PROC AUTOREG, PY.

      data predict;
         retain vl1 vl2 0;              
         if _n_ = 1 then set est(rename=(t=beta));  
         set b;
         yhat = intercept + beta * t;   
         v  = y - yhat;  
         vhat = -_a_1 * vl1 - _a_2 * vl2;  
         if v = . then v = vhat;
         predy = intercept + beta * t - _a_1 * vl1 - _a_2 * vl2; 
         diff = py - predy;
         diff = round(diff , .00000001);
         output;
         vl2 = vl1;                    
         vl1 = v;
         run;

The following statements compare the predicted values computed in the DATA step with those computed by PROC AUTOREG. The values of DIFF show that the DATA step produced predicted values identical to those from PROC AUTOREG except for the first NLAG=2 observations as explained above.

      proc print data=predict noobs;
         var y py predy diff pm yhat v r vhat;
         run;

y py predy diff pm yhat v r vhat 4.92631 4.85691 4.85691 0.000000 4.85691 4.85691 0.06940 0.06940 0.00000 4.93541 4.94767 4.97092 -0.023252 4.88688 4.88688 0.04853 0.04853 0.08404 4.98429 4.94907 4.94907 0.000000 4.91686 4.91686 0.06743 0.06743 0.03221 4.97127 5.00992 5.00992 0.000000 4.94683 4.94683 0.02444 0.02444 0.06309 5.04368 4.98060 4.98060 0.000000 4.97680 4.97680 0.06688 0.06688 0.00380 5.15312 5.07841 5.07841 0.000000 5.00678 5.00678 0.14634 0.14634 0.07163 5.16827 5.18837 5.18837 0.000000 5.03675 5.03675 0.13151 0.13151 0.15162 5.08277 5.16999 5.16999 0.000000 5.06673 5.06673 0.01604 0.01604 0.10327 5.19811 5.06582 5.06582 0.000000 5.09670 5.09670 0.10141 0.10141 -0.03088 5.22564 5.24333 5.24333 0.000000 5.12667 5.12667 0.09897 0.09897 0.11666 5.25697 5.23769 5.23769 0.000000 5.15665 5.15665 0.10033 0.10033 0.08104 5.30584 5.27025 5.27025 0.000000 5.18662 5.18662 0.11922 0.11922 0.08363 5.31508 5.32257 5.32257 0.000000 5.21659 5.21659 0.09848 0.09848 0.10598 5.27791 5.32021 5.32021 0.000000 5.24657 5.24657 0.03134 0.03134 0.07364 5.26595 5.27682 5.27682 0.000000 5.27654 5.27654 -0.01059 -0.01059 0.00028 5.33845 5.28170 5.28170 0.000000 5.30652 5.30652 0.03194 0.03194 -0.02482 5.35389 5.37921 5.37921 0.000000 5.33649 5.33649 0.01741 0.01741 0.04272 5.49852 5.37532 5.37532 0.000000 5.36646 5.36646 0.13206 0.13206 0.00886 5.43368 5.54968 5.54968 0.000000 5.39644 5.39644 0.03724 0.03724 0.15325 5.36700 5.42099 5.42099 0.000000 5.42641 5.42641 -0.05941 -0.05941 -0.00542 5.29802 5.37020 5.37020 0.000000 5.45638 5.45638 -0.15837 -0.15837 -0.08618 5.43608 5.31732 5.31732 0.000000 5.48636 5.48636 -0.05028 -0.05028 -0.16904 5.53769 5.51603 5.51603 0.000000 5.51633 5.51633 0.02136 0.02136 -0.00030 5.54670 5.59140 5.59140 0.000000 5.54631 5.54631 0.00039 0.00039 0.04510 5.62051 5.56859 5.56859 0.000000 5.57628 5.57628 0.04423 0.04423 -0.00769 5.67610 5.65966 5.65966 0.000000 5.60625 5.60625 0.06985 0.06985 0.05341 5.68109 5.70389 5.70389 0.000000 5.63623 5.63623 0.04487 0.04487 0.06766 5.69110 5.69381 5.69381 0.000000 5.66620 5.66620 0.02490 0.02490 0.02761 5.75476 5.70916 5.70916 0.000000 5.69617 5.69617 0.05859 0.05859 0.01299 5.65431 5.78756 5.78756 0.000000 5.72615 5.72615 -0.07184 -0.07184 0.06142 5.57390 5.64673 5.64673 0.000000 5.75612 5.75612 -0.18222 -0.18222 -0.10940 5.42513 5.59293 5.59293 0.000000 5.78609 5.78609 -0.36097 -0.36097 -0.19316 5.40326 5.44869 5.44869 0.000000 5.81607 5.81607 -0.41281 -0.41281 -0.36738 5.47684 5.48427 5.48427 0.000000 5.84604 5.84604 -0.36920 -0.36920 -0.36177 5.56084 5.58688 5.58688 0.000000 5.87602 5.87602 -0.31518 -0.31518 -0.28914 5.68880 5.66558 5.66558 0.000000 5.90599 5.90599 -0.21719 -0.21719 -0.24041 5.73709 5.79355 5.79355 0.000000 5.93596 5.93596 -0.19888 -0.19888 -0.14241 5.69289 5.80821 5.80821 0.000000 5.96594 5.96594 -0.27305 -0.27305 -0.15773 5.76779 5.74136 5.74136 0.000000 5.99591 5.99591 -0.22812 -0.22812 -0.25455 5.84111 5.85411 5.85411 0.000000 6.02588 6.02588 -0.18478 -0.18478 -0.17177 5.99246 5.91938 5.91938 0.000000 6.05586 6.05586 -0.06339 -0.06339 -0.13648 6.13513 6.07975 6.07975 0.000000 6.08583 6.08583 0.04930 0.04930 -0.00608 6.27600 6.19975 6.19975 0.000000 6.11581 6.11581 0.16020 0.16020 0.08395 6.34411 6.32090 6.32090 0.000000 6.14578 6.14578 0.19833 0.19833 0.17512 6.32824 6.35462 6.35462 0.000000 6.17575 6.17575 0.15249 0.15249 0.17887 6.17020 6.31450 6.31450 0.000000 6.20573 6.20573 -0.03553 -0.03553 0.10877 6.15356 6.13435 6.13435 0.000000 6.23570 6.23570 -0.08214 -0.08214 -0.10136 6.19387 6.17981 6.17981 0.000000 6.26567 6.26567 -0.07180 -0.07180 -0.08587 6.19897 6.24013 6.24013 0.000000 6.29565 6.29565 -0.09668 -0.09668 -0.05552 6.28146 6.23602 6.23602 0.000000 6.32562 6.32562 -0.04416 -0.04416 -0.08960 6.36160 6.33911 6.33911 0.000000 6.35560 6.35560 0.00600 0.00600 -0.01649 6.39798 6.40973 6.40973 0.000000 6.38557 6.38557 0.01241 0.01241 0.02416 6.43528 6.42827 6.42827 0.000000 6.41554 6.41554 0.01974 0.01974 0.01273 6.42266 6.46467 6.46467 0.000000 6.44552 6.44552 -0.02285 -0.02285 0.01916 6.48825 6.44026 6.44026 0.000000 6.47549 6.47549 0.01276 0.01276 -0.03523 6.50984 6.52965 6.52965 0.000000 6.50546 6.50546 0.00438 0.00438 0.02419 6.52733 6.53586 6.53586 0.000000 6.53544 6.53544 -0.00811 -0.00811 0.00042 6.52339 6.55392 6.55392 0.000000 6.56541 6.56541 -0.04203 -0.04203 -0.01149 6.58164 6.54760 6.54760 0.000000 6.59539 6.59539 -0.01375 -0.01375 -0.04779 6.60260 6.62479 6.62479 0.000000 6.62536 6.62536 -0.02276 -0.02276 -0.00057 6.62882 6.63304 6.63304 0.000000 6.65533 6.65533 -0.02651 -0.02651 -0.02230 6.68497 6.66191 6.66191 0.000000 6.68531 6.68531 -0.00033 -0.00033 -0.02340 6.72407 6.72502 6.72502 0.000000 6.71528 6.71528 0.00879 0.00879 0.00974 6.77532 6.75603 6.75603 0.000000 6.74525 6.74525 0.03007 0.03007 0.01077 6.83411 6.80827 6.80827 0.000000 6.77523 6.77523 0.05888 0.05888 0.03304 6.89226 6.86500 6.86500 0.000000 6.80520 6.80520 0.08705 0.08705 0.05980 6.91907 6.91806 6.91806 0.000000 6.83518 6.83518 0.08390 0.08390 0.08289 6.96419 6.93343 6.93343 0.000000 6.86515 6.86515 0.09904 0.09904 0.06828 6.99171 6.98296 6.98296 0.000000 6.89512 6.89512 0.09659 0.09659 0.08783 6.98987 7.00416 7.00416 0.000000 6.92510 6.92510 0.06477 0.06477 0.07907 7.02324 6.99655 6.99655 0.000000 6.95507 6.95507 0.06817 0.06817 0.04148 7.07827 7.04281 7.04281 0.000000 6.98504 6.98504 0.09323 0.09323 0.05777 7.13425 7.10183 7.10183 0.000000 7.01502 7.01502 0.11924 0.11924 0.08681 . 7.15371 7.15371 0.000000 7.04499 7.04499 0.10871 . 0.10871 . 7.16099 7.16099 0.000000 7.07497 7.07497 0.08603 . 0.08603 . 7.16752 7.16752 0.000000 7.10494 7.10494 0.06258 . 0.06258 . 7.17778 7.17778 0.000000 7.13491 7.13491 0.04287 . 0.04287 . 7.19285 7.19285 0.000000 7.16489 7.16489 0.02797 . 0.02797 . 7.21233 7.21233 0.000000 7.19486 7.19486 0.01747 . 0.01747 . 7.23528 7.23528 0.000000 7.22483 7.22483 0.01045 . 0.01045 . 7.26078 7.26078 0.000000 7.25481 7.25481 0.00597 . 0.00597 . 7.28802 7.28802 0.000000 7.28478 7.28478 0.00323 . 0.00323 . 7.31639 7.31639 0.000000 7.31476 7.31476 0.00163 . 0.00163

Operating System and Release Information

Product Family	Product	System	SAS Release
			Reported	Fixed*
SAS System	SAS/ETS	z/OS
		OpenVMS VAX
		Microsoft® Windows® for 64-Bit Itanium-based Systems
		Microsoft Windows Server 2003 Datacenter 64-bit Edition
		Microsoft Windows Server 2003 Enterprise 64-bit Edition
		Microsoft Windows XP 64-bit Edition
		Microsoft® Windows® for x64
		OS/2
		Microsoft Windows 95/98
		Microsoft Windows 2000 Advanced Server
		Microsoft Windows 2000 Datacenter Server
		Microsoft Windows 2000 Server
		Microsoft Windows 2000 Professional
		Microsoft Windows NT Workstation
		Microsoft Windows Server 2003 Datacenter Edition
		Microsoft Windows Server 2003 Enterprise Edition
		Microsoft Windows Server 2003 Standard Edition
		Microsoft Windows Server 2003 for x64
		Microsoft Windows Server 2008
		Microsoft Windows Server 2008 for x64
		Microsoft Windows XP Professional
		Windows 7 Enterprise 32 bit
		Windows 7 Enterprise x64
		Windows 7 Home Premium 32 bit
		Windows 7 Home Premium x64
		Windows 7 Professional 32 bit
		Windows 7 Professional x64
		Windows 7 Ultimate 32 bit
		Windows 7 Ultimate x64
		Windows Millennium Edition (Me)
		Windows Vista
		Windows Vista for x64
		64-bit Enabled AIX
		64-bit Enabled HP-UX
		64-bit Enabled Solaris
		ABI+ for Intel Architecture
		AIX
		HP-UX
		HP-UX IPF
		IRIX
		Linux
		Linux for x64
		Linux on Itanium
		OpenVMS Alpha
		OpenVMS on HP Integrity
		Solaris
		Solaris for x64
		Tru64 UNIX

* For software releases that are not yet generally available, the Fixed Release is the software release in which the problem is planned to be fixed.

---

Type:	Usage Note
Priority:
Topic:	Analytics ==> Econometrics Analytics ==> Time Series Analysis SAS Reference ==> Procedures ==> AUTOREG

Date Modified:	2021-04-07 13:12:17
Date Created:	2010-06-28 15:14:52