The PDLREG Procedure

RESTRICT Statement

RESTRICT equation , …, equation;

The RESTRICT statement places restrictions on the parameter estimates for covariates in the preceding MODEL statement. A parameter produced by a distributed lag cannot be restricted with the RESTRICT statement.

Each restriction is written as a linear equation. If you specify more than one restriction in a RESTRICT statement, the restrictions are separated by commas.

You can refer to parameters by the name of the corresponding regressor variable. Each name used in the equation must be a regressor in the preceding MODEL statement. Use the keyword INTERCEPT to refer to the intercept parameter in the model.

RESTRICT statements can be given labels. You can use labels to distinguish results for different restrictions in the printed output. Labels are specified as follows:

label : RESTRICT …

The following is an example of the use of the RESTRICT statement, in which the coefficients of the regressors X1 and X2 are required to sum to 1:

proc pdlreg data=a;
   model y = x1 x2;
   restrict x1 + x2 = 1;
run;

Parameter names can be multiplied by constants. When no equal sign appears, the linear combination is set equal to 0. Note that the parameters associated with the variables are restricted, not the variables themselves. Here are some examples of valid RESTRICT statements:

restrict x1 + x2 = 1;
restrict x1 + x2 - 1;
restrict 2 * x1 = x2 + x3 , intercept + x4 = 0;
restrict x1 = x2 = x3 = 1;
restrict 2 * x1 - x2;

Restricted parameter estimates are computed by introducing a Lagrangian parameter ${\lambda }$ for each restriction (Pringle and Rayner, 1971). The estimates of these Lagrangian parameters are printed in the parameter estimates table. If a restriction cannot be applied, its parameter value and degrees of freedom are listed as 0.

The Lagrangian parameter, ${\lambda }$ , measures the sensitivity of the SSE to the restriction. If the restriction is changed by a small amount ${\epsilon }$ , the SSE is changed by 2 ${\lambda }$ ${\epsilon }$ .

The t ratio tests the significance of the restrictions. If ${\lambda }$ is zero, the restricted estimates are the same as the unrestricted ones.

You can specify any number of restrictions in a RESTRICT statement, and you can use any number of RESTRICT statements. The estimates are computed subject to all restrictions specified. However, restrictions should be consistent and not redundant.