MODEL Procedure
Restrictions and Bounds on Parameters
Using the BOUNDS and RESTRICT statements, PROC MODEL can compute optimal estimates subject to equality or inequality constraints on the parameter estimates.
Equality restrictions can be written as a vector function:
Inequality restrictions are either active or inactive. When an inequality restriction is active, it is treated as an equality restriction. All inactive inequality restrictions can be written as a vector function:
Strict inequalities, such as 
, are transformed into inequalities as 
, where the tolerance 
is controlled by the EPSILON= option in the FIT statement and defaults to 
. The ith inequality restriction becomes active if 
and remains active until its Lagrange multiplier becomes negative. Lagrange multipliers are computed for all the nonredundant equality restrictions and all the active inequality restrictions.
For the following, assume the vector 
contains all the current active restrictions. The constraint matrix 
is
The covariance matrix for the restricted parameter estimates is computed as
where 
is Hessian or approximation to the Hessian of the objective function (
for OLS), and 
is the last 
columns of 
. is from an LQ factorization of the constraint matrix, nc is the number of active constraints, and np is the number of parameters. For more information about LQ factorization, see Gill, Murray, and Wright (1981). The covariance column in Table 2 summarizes the Hessian approximation used for each estimation method.
The covariance matrix for the Lagrange multipliers is computed as
The p-value reported for a restriction is computed from a beta distribution rather than a t distribution because the numerator and the denominator of the t ratio for an estimated Lagrange multiplier are not independent.
The Lagrange multipliers for the active restrictions are printed with the parameter estimates. The Lagrange multiplier estimates are computed using the relationship
where the dimensions of the constraint matrix are the number of constraints by the number of parameters, 
is the vector of Lagrange multipliers, and g is the gradient of the objective function at the final estimates.
The final gradient includes the effects of the estimated 
matrix. For example, for OLS the final gradient would be
where r is the residual vector. Note that when nonlinear restrictions are imposed, the convergence measure R might have values greater than one for some iterations.