MODEL Procedure

Solution Mode Output

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 79. 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 79: Solve Step Summary Output

The MODEL Procedure

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 80, gives more information about the failed observation.

Figure 80: Solve Step Error Message

The MODEL Procedure
Dynamic Single-Equation Forecast


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    


Iteration Errors - Missing.

                              The MODEL Procedure                               
                        Dynamic Single-Equation 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 81.

Figure 81: Solution Summary Report

The MODEL Procedure
Dynamic Single-Equation Forecast

Data Set Options
DATA= SASHELP.CITIMON
OUT= SIM

Solution Summary
Variables Solved 1
Forecast Lag Length 1
Solution Method NEWTON
CONVERGE= 1E-8
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.

Summary Statistics

The STATS and THEIL options are used to select goodness-of-fit 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.

STATS Option Output

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 82 and the THEIL output in Figure 83 are generated.

Figure 82: STATS Output

The MODEL Procedure
Dynamic Single-Equation Simulation
 
Solution Range DATE = FEB1980 To NOV1991

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 R-Square 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 StartFraction 1 Over upper N EndFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts left-parenthesis ModifyingAbove y With caret Subscript j Baseline minus y Subscript j Baseline right-parenthesis
Mean % Error StartFraction 100 Over upper N EndFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts left-parenthesis ModifyingAbove y With caret Subscript j Baseline minus y Subscript j Baseline right-parenthesis slash y Subscript j
Mean Abs Error StartFraction 1 Over upper N EndFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts StartAbsoluteValue ModifyingAbove y With caret Subscript j Baseline minus y Subscript j Baseline EndAbsoluteValue
Mean Abs % Error StartFraction 100 Over upper N EndFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts StartAbsoluteValue left-parenthesis ModifyingAbove y With caret Subscript j Baseline minus y Subscript j Baseline right-parenthesis slash y Subscript j Baseline EndAbsoluteValue
RMS Error StartRoot StartFraction 1 Over upper N EndFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts left-parenthesis ModifyingAbove y With caret Subscript j Baseline minus y Subscript j Baseline right-parenthesis squared EndRoot
RMS % Error 100 StartRoot StartFraction 1 Over upper N EndFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts left-parenthesis left-parenthesis ModifyingAbove y With caret Subscript j Baseline minus y Subscript j Baseline right-parenthesis slash y Subscript j Baseline right-parenthesis squared EndRoot
R-square 1 minus normal upper S normal upper S normal upper E slash normal upper C normal upper S normal upper S normal upper A
SSE sigma-summation Underscript j equals 1 Overscript upper N Endscripts left-parenthesis ModifyingAbove y With caret Subscript j Baseline minus y Subscript j Baseline right-parenthesis squared
SSA sigma-summation Underscript j equals 1 Overscript upper N Endscripts left-parenthesis y Subscript j Baseline right-parenthesis squared
CSSA normal upper S normal upper S normal upper A minus left-parenthesis sigma-summation Underscript j equals 1 Overscript upper N Endscripts y Subscript j Baseline right-parenthesis squared
ModifyingAbove y With caret predicted value
y actual value

When the RANDOM= option is specified, the statistics do not include the unperturbed (_REP_=0) solution.

THEIL Option Output

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

ModifyingAbove y c With caret equals left-parenthesis ModifyingAbove y With caret minus normal l normal a normal g left-parenthesis y right-parenthesis right-parenthesis slash normal l normal a normal g left-parenthesis y right-parenthesis
y c equals left-parenthesis y minus normal l normal a normal g left-parenthesis y right-parenthesis right-parenthesis slash normal l normal a normal g left-parenthesis y right-parenthesis

where ModifyingAbove y c With caret is the relative change for the predicted value and y c is the relative change for the actual value.

Figure 83: 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:

Corr (R)

is the correlation coefficient, rho, between the actual and predicted values.

rho equals StartFraction normal c normal o normal v left-parenthesis y comma ModifyingAbove y With caret right-parenthesis Over sigma Subscript a Baseline sigma Subscript p Baseline EndFraction

where sigma Subscript p and sigma Subscript a are the standard deviations of the predicted and actual values.

Bias (UM)

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

StartStartFraction left-parenthesis normal upper E left-parenthesis y right-parenthesis minus normal upper E left-parenthesis ModifyingAbove y With caret right-parenthesis right-parenthesis squared OverOver StartFraction 1 Over upper N EndFraction sigma-summation Underscript t equals 1 Overscript upper N Endscripts left-parenthesis y Subscript t Baseline minus ModifyingAbove y With caret Subscript t Baseline right-parenthesis squared EndEndFraction
Reg (UR)

is defined as left-parenthesis sigma Subscript p Baseline minus rho asterisk sigma Subscript a Baseline right-parenthesis squared slash normal upper M normal upper S normal upper E. Consider the regression

y equals alpha plus beta ModifyingAbove y With caret

If ModifyingAbove beta With caret equals 1, UR will equal zero.

Dist (UD)

is defined as left-parenthesis 1 minus rho squared right-parenthesis sigma Subscript a Baseline sigma Subscript a slash normal upper M normal upper S normal upper E and represents the variance of the residuals obtained by regressing y c on ModifyingAbove y c With caret.

Var (US)

is the variance proportion. US indicates the ability of the model to replicate the degree of variability in the endogenous variable.

upper U upper S equals StartFraction left-parenthesis sigma Subscript p Baseline minus sigma Subscript a Baseline right-parenthesis squared Over normal upper M normal upper S normal upper E EndFraction
Covar (UC)

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

upper U upper C equals StartFraction 2 left-parenthesis 1 minus rho right-parenthesis sigma Subscript p Baseline sigma Subscript a Baseline Over normal upper M normal upper S normal upper E EndFraction
U1

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

upper U Baseline italic 1 equals StartFraction StartRoot normal upper M normal upper S normal upper E EndRoot Over StartRoot StartFraction 1 Over upper N EndFraction sigma-summation Underscript t equals 1 Overscript upper N Endscripts left-parenthesis y Subscript t Baseline right-parenthesis squared EndRoot EndFraction
U

is the Theil’s inequality coefficient defined as follows:

upper U equals StartFraction StartRoot normal upper M normal upper S normal upper E EndRoot Over StartRoot StartFraction 1 Over upper N EndFraction sigma-summation Underscript t equals 1 Overscript upper N Endscripts left-parenthesis y Subscript t Baseline right-parenthesis squared EndRoot plus StartRoot StartFraction 1 Over upper N EndFraction sigma-summation Underscript t equals 1 Overscript upper N Endscripts left-parenthesis ModifyingAbove y With caret Subscript t Baseline right-parenthesis squared EndRoot EndFraction
MSE

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

normal upper M normal upper S normal upper E equals StartFraction 1 Over upper N EndFraction sigma-summation Underscript t equals 1 Overscript upper N Endscripts left-parenthesis ModifyingAbove y c With caret Subscript t Baseline minus y c Subscript t Baseline right-parenthesis squared

For more information about these statistics, see Maddala (1977, pp. 344–347) and Pindyck and Rubinfeld (1981, pp. 364–365).

Last updated: June 19, 2025