The SYSLIN Procedure

SRESTRICT Statement

SRESTRICT equation , …, equation ;

The SRESTRICT statement imposes linear restrictions that involve parameters in two or more MODEL statements. The SRESTRICT statement is like the RESTRICT statement but is used to impose restrictions across equations, whereas the RESTRICT statement applies only to parameters in the immediately preceding MODEL statement.

Each restriction is written as a linear equation. Parameters are referred to as label.variable, where label is the model label and variable is the name of the regressor to which the parameter is attached. (If the MODEL statement does not have a label, you can use the dependent variable name as the label for the model, provided the dependent variable uniquely labels the model.) Each variable name used must be a regressor in the indicated MODEL statement. The keyword INTERCEPT is used to refer to intercept parameters.

SRESTRICT statements can be given labels. The labels are used in the printed output to distinguish results for different restrictions. Labels are specified as follows:

label : SRESTRICT …;

The following is an example of the use of the SRESTRICT statement, in which the coefficient for the regressor X2 is constrained to be the same in both models.

   proc syslin data=a 3sls;
      endogenous y1 y2;
      instruments x1 x2;
      model y1 = y2 x1 x2;
      model y2 = y1 x2;
      srestrict y1.x2 = y2.x2;
   run;

When no equal sign is used, the linear combination is set equal to 0. Thus, the restriction in the preceding example can also be specified as

      srestrict y1.x2 - y2.x2;

Any number of restrictions can be specified on an SRESTRICT statement, and any number of SRESTRICT statements can be used. The estimates are computed subject to all restrictions specified. However, restrictions should be consistent and not redundant.

When a system restriction is requested for a single equation estimation method (such as OLS or 2SLS), PROC SYSLIN produces the restricted estimates by actually using a corresponding system method. For example, when SRESTRICT is specified along with OLS, PROC SYSLIN produces the restricted OLS estimates via a two-step process equivalent to using SUR estimation with the SDIAG option. First, the unrestricted OLS results are produced. Then, the GLS (SUR) estimation with the system restriction is performed, using the diagonal of the covariance matrix of the residuals. When SRESTRICT is specified along with 2SLS, PROC SYSLIN produces the restricted 2SLS estimates via a multistep process equivalent to using 3SLS estimation with the SDIAG option. First, the unrestricted 2SLS results are produced. Then, the GLS (3SLS) estimation with the system restriction is performed, using the diagonal of the covariance matrix of the residuals.

The results of the SRESTRICT statements are printed after the parameter estimates for all the models in the system. The format of the SRESTRICT statement output is the same as the “Parameter Estimates” table. In this output the parameter estimate is the Lagrangian parameter ${\lambda }$ used to impose the restriction.

The Lagrangian parameter ${\lambda }$ measures the sensitivity of the system sum of square errors to the restriction. The system SSE is the system MSE shown in the printed output multiplied by the degrees of freedom. If the restriction is changed by a small amount ${\epsilon }$ , the system SSE is changed by 2 ${\lambda }$ ${\epsilon }$ .

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

The model degrees of freedom are not adjusted for the cross-model restrictions imposed by SRESTRICT statements.

Note: The SRESTRICT statement is not supported for the LIML and the FIML estimation methods.