The R2 Statistics

The SYSLIN Procedure

The R² Statistics

As explained in the section "ANOVA Table for Instrumental Variables Methods," when instrumental variables are used, the regression sum of squares (RSS) and the error sum of squares (ESS) do not sum to the total corrected sum of squares. In this case, there are several ways that the R² statistic can be defined.

The definition of R² used by the SYSLIN procedure is

R² = [RSS/(RSS + ESS )]

This definition is consistent with the F-test of the null hypothesis that the true coefficients of all regressors are zero. However, this R² may not be a good measure of the goodness of fit of the model.

System Weighted R² and System Weighted Mean Square Error

The system weighted R², printed for the 3SLS, IT3SLS, SUR, ITSUR, and FIML methods, is computed as follows:

R² = Y' W R (X'X)^-1 R' W Y / Y' W Y

In this equation the matrix X'X is R'W R, and W is the projection matrix of the instruments:

$W = S^{-1} {\otimes} Z(Z'Z)^{-1} Z'$

The matrix Z is the instrument set, R is the the regressor set, and S is the estimated cross-model covariance matrix.

The system weighted MSE, printed for the 3SLS, IT3SLS, SUR, ITSUR, and FIML methods, is computed as follows:

MSE = [1/tdf] ( Y' W Y - Y' W R (X'X)^-1 R' W Y )

In this equation tdf is the sum of the error degrees of freedom for the equations in the system.

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