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SUR, 3SLS, and FIML Estimation

In a multivariate regression model, the errors in different equations may be correlated. In this case the efficiency of the estimation may be improved by taking these cross-equation correlations into account.

Seemingly Unrelated Regression

The SUR option on the PROC SYSLIN statement specifies seemingly unrelated regression, as shown in the following statements:

   proc syslin data=in sur;
      demand: model q = p y s;
      supply: model q = p u;
   run;

INSTRUMENTS and ENDOGENOUS statements are not needed for SUR, since the SUR method assumes there are no endogenous regressors. For SUR to be effective, the models must use different regressors. SUR produces the same results as OLS unless the model contains at least one regressor not used in the other equations.

Three-Stage Least Squares

The 3SLS option on the PROC SYSLIN statement specifies the three-stage least-squares method, as shown in the following statements.

   proc syslin data=in 3sls;
      endogenous  p;
      instruments y u s;
      demand: model q = p y s;
      supply: model q = p u;
   run;

The 3SLS output begins with a two-stage least-squares regression to estimate the cross-model correlation matrix. This output is the same as the 2SLS results shown in Figure 19.3 and Figure 19.4, and is not repeated here. The next part of the 3SLS output prints the cross-model correlation matrix computed from the 2SLS residuals. This output is shown in Figure 19.5 and includes the cross-model covariances, correlations, the inverse of the correlation matrix, and the inverse covariance matrix.

The final 3SLS estimates are shown in Figure 19.6.

This output first prints the system weighted mean square error and system weighted R² statistics. The system weighted MSE and system weighted R² measure the fit of the joint model obtained by stacking all the models together and performing a single regression with the stacked observations weighted by the inverse of the model error variances. See the section "The R² Statistics" for details.

Next, the table of 3SLS parameter estimates for each model is printed. This output has the same form as for the other estimation methods.

Note that the 3SLS and 2SLS results may be the same in some cases. This results from the same principle that causes OLS and SUR results to be identical unless an equation includes a regressor not used in the other equations of the system. However, the application of this principle is more complex when instrumental variables are used. When all the exogenous variables are used as instruments, linear combinations of all the exogenous variables appear in the third-stage regressions through substitution of first-stage predicted values.

In this example, 3SLS produces different (and, it is hoped, more efficient) estimates for the demand equation. However, the 3SLS and 2SLS results for the supply equation are the same. This is because the supply equation has one endogenous regressor and one exogenous regressor not used in other equations. In contrast, the demand equation has fewer endogenous regressors than exogenous regressors not used in other equations in the system.

Full Information Maximum Likelihood

   proc syslin data=in fiml;
      endogenous  p q;
      instruments y u s;
      demand: model q = p y s;
      supply: model q = p u;
   run;

The FIML results are shown in Figure 19.7.

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