The following SAS statements dynamically forecast the solution to a nonlinear equation:
proc model data=sashelp.citimon; parameters a 0.010708 b 0.478849 c 0.929304; lhur = 1/(a * ip) + b + c * lag(lhur); solve lhur / out=sim forecast dynamic; run;
The first page of output produced by the SOLVE step is shown in Figure 19.77. This is the summary description of the model. The error message states that the simulation was aborted at observation 144 because of missing input values.
Figure 19.77: Solve Step Summary Output
Model Summary  

Model Variables  1 
Parameters  3 
Equations  1 
Number of Statements  1 
Program Lag Length  1 
Model Variables  LHUR 

Parameters(Value)  a(0.010708) b(0.478849) c(0.929304) 
Equations  LHUR 
The second page of output, shown in Figure 19.78, gives more information on the failed observation.
Figure 19.78: Solve Step Error Message
Solution values are missing because of missing input values for observation 144 at NEWTON iteration 0. 
Note:  Additional information on the values of the variables at this observation, which may be helpful in determining the cause of the failure of the solution process, is printed below. 
Observation  144  Iteration  0  CC  1.000000 

Missing  1 
The MODEL Procedure Dynamic SingleEquation Forecast  Listing of Program Data Vector  _N_: 144 ACTUAL.LHUR: . ERROR.LHUR: . IP: . LHUR: 7.10000 PRED.LHUR: . a: 0.01071 b: 0.47885 c: 0.92930 
Note:  Simulation aborted. 
From the program data vector, you can see the variable IP is missing for observation 144. LHUR could not be computed, so the simulation aborted.
The solution summary table is shown in Figure 19.79.
Figure 19.79: Solution Summary Report
Data Set Options  

DATA=  SASHELP.CITIMON 
OUT=  SIM 
Solution Summary  

Variables Solved  1 
Forecast Lag Length  1 
Solution Method  NEWTON 
CONVERGE=  1E8 
Maximum CC  0 
Maximum Iterations  1 
Total Iterations  143 
Average Iterations  1 
Observations Processed  

Read  145 
Lagged  1 
Solved  143 
First  2 
Last  145 
Failed  1 
Variables Solved For  LHUR 

This solution summary table includes the names of the input data set and the output data set followed by a description of the model. The table also indicates that the solution method defaulted to Newton’s method. The remaining output is defined as follows.
Maximum CC 
is the maximum convergence value accepted by the Newton 
procedure. This number is always less than the value 

for the CONVERGE= option. 

Maximum Iterations 
is the maximum number of Newton iterations performed 
at each observation and each replication of Monte 

Carlo simulations. 

Total Iterations 
is the sum of the number of iterations required for each 
observation and each Monte Carlo simulation. 

Average Iterations 
is the average number of Newton iterations required to 
solve the system at each step. 

Solved 
is the number of observations used times the number of 
random replications selected plus one, for Monte Carlo 

simulations. The one additional simulation is the original 

unperturbed solution. For simulations that do not involve Monte 

Carlo, this number is the number of observations used. 
The STATS and THEIL options are used to select goodnessoffit statistics. Actual values must be provided in the input data set for these statistics to be printed. When the RANDOM= option is specified, the statistics do not include the unperturbed (_REP_=0) solution.
The following statements show the addition of the STATS and THEIL options to the model in the previous section:
proc model data=sashelp.citimon; parameters a 0.010708 b 0.478849 c 0.929304; lhur= 1/(a * ip) + b + c * lag(lhur) ; solve lhur / out=sim dynamic stats theil; range date to '01nov91'd; run;
The STATS output in Figure 19.80 and the THEIL output in Figure 19.81 are generated.
Figure 19.80: STATS Output
Descriptive Statistics  

Variable  N Obs  N  Actual  Predicted  Label  
Mean  Std Dev  Mean  Std Dev  
LHUR  142  142  7.0887  1.4509  7.2473  1.1465  UNEMPLOYMENT RATE: ALL WORKERS, 16 YEARS 
Statistics of fit  

Variable  N  Mean Error  Mean % Error  Mean Abs Error  Mean Abs % Error  RMS Error  RMS % Error  RSquare  Label 
LHUR  142  0.1585  3.5289  0.6937  10.0001  0.7854  11.2452  0.7049  UNEMPLOYMENT RATE: ALL WORKERS, 16 YEARS 
The number of observations (Nobs), the number of observations with both predicted and actual values nonmissing (N), and the mean and standard deviation of the actual and predicted values of the determined variables are printed first. The next set of columns in the output are defined as follows:
Mean Error 

Mean % Error 

Mean Abs Error 

Mean Abs % Error 

RMS Error 

RMS % Error 

Rsquare 

SSE 

SSA 

CSSA 


predicted value 

actual value 
When the RANDOM= option is specified, the statistics do not include the unperturbed (_REP_=0) solution.
The THEIL option specifies that Theil forecast error statistics be computed for the actual and predicted values and for the relative changes from lagged values. Mathematically, the quantities are


where is the relative change for the predicted value and is the relative change for the actual value.
Figure 19.81: THEIL Output
Theil Forecast Error Statistics  

Variable  N  MSE  Corr (R)  MSE Decomposition Proportions  Inequality Coef  Label  
Bias (UM)  Reg (UR)  Dist (UD)  Var (US)  Covar (UC)  U1  U  
LHUR  142  0.6168  0.85  0.04  0.01  0.95  0.15  0.81  0.1086  0.0539  UNEMPLOYMENT RATE: ALL WORKERS, 16 YEARS 
Theil Relative Change Forecast Error Statistics  

Variable  Relative Change  MSE Decomposition Proportions  Inequality Coef  Label  
N  MSE  Corr (R)  Bias (UM)  Reg (UR)  Dist (UD)  Var (US)  Covar (UC)  U1  U  
LHUR  142  0.0126  0.08  0.09  0.85  0.06  0.43  0.47  4.1226  0.8348  UNEMPLOYMENT RATE: ALL WORKERS, 16 YEARS 
The columns have the following meaning:
is the correlation coefficient, , between the actual and predicted values.

where and are the standard deviations of the predicted and actual values.
is an indication of systematic error and measures the extent to which the average values of the actual and predicted deviate from each other.

is defined as . Consider the regression

If , UR will equal zero.
is defined as and represents the variance of the residuals obtained by regressing on .
is the variance proportion. US indicates the ability of the model to replicate the degree of variability in the endogenous variable.

represents the remaining error after deviations from average values and average variabilities have been accounted for.

is a statistic that measures the accuracy of a forecast defined as follows:

is the Theil’s inequality coefficient defined as follows:

is the mean square error. In the case of the relative change Theil statistics, the MSE is computed as follows:

More information about these statistics can be found in the references Maddala (1977, 344–347) and Pindyck and Rubinfeld (1981, 364–365).