Numerical Solution Methods
If the SINGLE option is not used,
PROC MODEL computes values that simultaneously satisfy the
model equations for the variables named in the SOLVE statement.
PROC MODEL provides three iterative methods, Newton,
Jacobi, and Seidel, for computing a simultaneous solution of the system
of nonlinear equations.
SingleEquation Solution
For normalizedform equation systems,
the solution can either simultaneously satisfy all the equations or can
be computed for each equation separately, using the actual values of the solution
variables in the current period to compute each predicted value.
By default, PROC MODEL computes a simultaneous solution.
The SINGLE option on the SOLVE statement selects
singleequation solutions.
Singleequation simulations are often made to produce residuals (which
estimate the random terms of the stochastic equations) rather than the
predicted values themselves.
If the input data and range are the same as that used for parameter estimation,
a static singleequation simulation will reproduce the residuals of the estimation.
Newton's Method
The NEWTON option on the SOLVE statement
requests Newton's method to simultaneously solve the equations for
each observation. Newton's method is the default solution method.
Newton's method is an iterative scheme that uses the
derivatives of the equations with respect
to the solution variables, J, to compute a change vector as
PROC MODEL builds and solves J using efficient sparse matrix techniques.
The solution variables y^{i} at the ith iteration
are then updated as
d is a damping factor between 0 and 1 chosen iteratively so that
The number of subiterations allowed for finding a suitable d
is controlled by the MAXSUBITER= option. The number of iterations
of Newton's method allowed for each observation is controlled by
MAXITER= option.
Refer to Ortega and Rheinbolt (1970) for more details.
Jacobi Method
The JACOBI option on the
SOLVE statement selects a matrixfree
alternative to Newton's method. This method is the traditional nonlinear
Jacobi method found in the literature.
The Jacobi method as implemented in PROC MODEL substitutes predicted values for
the endogenous variables and iterates until a fixed point is reached.
Then necessary derivatives are computed only for the diagonal elements
of the jacobian, J.
If the normalizedform equation is
the Jacobi iteration has the form
Seidel Method
The Seidel method is an orderdependent alternative to the Jacobi method.
The Seidel method is selected by the SEIDEL option on the SOLVE statement
and is applicable only to normalizedform equations.
The Seidel method is like the Jacobi method except that in the
Seidel method the model is
further edited to substitute the predicted values into the solution variables
immediately after they are computed.
Seidel thus differs from the other methods in that the values
of the solution variables are not fixed within an iteration.
With the other methods, the order of the equations in the model program makes no
difference, but the Seidel method may work much differently when the
equations are specified in a different sequence. Note that this
fixed point method is the traditional nonlinear Seidel method found
in the literature.
The iteration has the form
where y^{i+1}_{j} is the jth
equation variable at the ith iteration and
If the model is recursive, and if the equations are in recursive order,
the Seidel method will converge at once.
If the model is blockrecursive, the Seidel method may converge faster
if the equations are grouped by block and the blocks are placed
in blockrecursive order.
The BLOCK option can be used to determine the blockrecursive form.
Comparison of Methods
Newton's method is the default and should work better than the others
for most small to mediumsized models. The Seidel method is always faster
than the Jacobi for recursive models with equations in recursive order.
For very large models and some highly nonlinear smaller models,
the Jacobi or Seidel methods can sometimes be faster. Newton's method
uses more memory than the Jacobi or Seidel methods.
Both the Newton's method and the Jacobi method are orderinvariant in
the sense that the order in which equations are specified in the model
program has no effect on the operation of the iterative solution
process. In orderinvariant methods, the values of the solution
variables are fixed for the entire execution of the model program.
Assignments to model variables are automatically
changed to assignments to corresponding equation variables.
Only after the model program has completed execution are the results
used to compute the new solution values for the next iteration.
Troubleshooting Problems
In solving a simultaneous nonlinear dynamic model you may encounter
some of the following problems.
[cmodelfsolvemi]Missing Values
For SOLVE tasks, there can be no missing parameter values.
If there are missing righthandside variables,
this will result in a missing lefthandside variable for that observation.
Unstable Solutions
A solution may exist but be unstable.
An unstable system can cause the Jacobi and Seidel methods to diverge.
Explosive Dynamic Systems
A model may have wellbehaved solutions at
each observation but be dynamically unstable. The solution may
oscillate wildly or grow rapidly with time.
Propagation of Errors
During the solution process, solution
variables can take on values that cause computational errors. For
example, a solution variable that appears in a LOG function may be
positive at the solution but may be given a negative value during one of
the iterations. When computational errors occur, missing values are
generated and propagated, and the solution process may collapse.
Convergence Problems
The following items can cause convergence problems:
 illegal function values ( that is )
 local minima in the model equation
 no solution exists
 multiple solutions exist
 initial values too far from the solution
 the CONVERGE= value too small.
When PROC MODEL fails to find a solution to the system, the current iteration
information and the program data vector are printed. The simulation
halts if actual values are not available for the simulation to proceed.
Consider the following program:
data test1;
do t=1 to 50;
x1 = sqrt(t) ;
y = .;
output;
end;
proc model data=test1;
exogenous x1 ;
control a1 1 b1 29 c1 4 ;
y = a1 * sqrt(y) + b1 * x1 * x1 + c1 * lag(x1);
solve y / out=sim forecast dynamic ;
run;
which produces the output shown in Figure 14.70.
The MODEL Procedure 
Dynamic SingleEquation Forecast 
ERROR: 
Could not reduce norm of residuals in 10 subiterations. 

ERROR: 
The solution failed because 1 equations are missing or have extreme values for observation 1 at NEWTON iteration 1. 

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 
1 
Iteration 
1 
CC 
1.000000 


Missing 
1 


Iteration Errors  Missing. 
_N_: 12 ACTUAL.x1: 1.41421 ACTUAL.y: .
ERROR.y: . PRED.y: . RESID.y: .
a1: 1 b1: 29 c1: 4
x1: 1.41421 y: 0.00109
@PRED.y/@y: . @ERROR.y/@y: .

Observation 
1 
Iteration 
1 
CC 
1.000000 


Missing 
1 


ERROR: 
1 execution errors for this observation 

NOTE: 
Check for missing input data or uninitialized lags. 

(Note that the LAG and DIF functions return missing values for the initial lag starting observations. This is a change from the 1982 and earlier versions of SAS/ETS which returned zero for uninitialized lags.) 
NOTE: 
Simulation aborted. 


Figure 14.70: SOLVE Convergence Problems
At the first observation the following equation is attempted to be
solved:
There is no solution to this problem.
The iterative
solution process got as close as it could to
making Y negative while
still being able to evaluate the model.
This problem can be avoided in this case by altering the equation.
In other models, the problem of missing values can be avoided
by either altering the data
set to provide better starting values for the solution variables or
by altering the equations.
You should be aware that, in general, a nonlinear
system can have any number of solutions, and the solution found may not
be the one that you want. When multiple solutions exist, the solution
that is found is usually determined by the starting values for the iterations.
If the value from the input data set for a solution variable is missing,
the starting value for it is taken from the solution of the last period (if
nonmissing) or else the solution estimate is started at 0.
Iteration Output
The iteration output, produced by the ITPRINT option,
is useful in determining the cause of a
convergence problem. The ITPRINT option forces the printing of
the solution approximation and equation errors at each iteration for
each observation. A portion of the ITPRINT output from
proc model data=test1;
exogenous x1 ;
control a1 1 b1 29 c1 4 ;
y = a1 * sqrt(abs(y)) + b1 * x1 * x1 + c1 * lag(x1);
solve y / out=sim forecast dynamic itprint;
run;
is shown in Figure 14.71.
The MODEL Procedure 
Dynamic SingleEquation Forecast 
Observation 
1 
Iteration 
0 
CC 
613961.39 
ERROR.y 
62.01010 
Predicted Values 
y 
0.0001000 
Iteration Errors 
y 
62.01010 
Observation 
1 
Iteration 
1 
CC 
50.902771 
ERROR.y 
61.88684 
Predicted Values 
y 
1.215784 
Iteration Errors 
y 
61.88684 
Observation 
1 
Iteration 
2 
CC 
0.364806 
ERROR.y 
41.752112 
Predicted Values 
y 
114.4503 
Iteration Errors 
y 
41.75211 

Figure 14.71: SOLVE, ITPRINT Output
For each iteration, the equation with the largest error is
listed in parentheses after
the Newton convergence criteria measure. From this output
you can determine which equation or equations in the system
are not converging well.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.