In a multivariate regression model, the errors in different equations might be correlated. In this case, the efficiency of the estimation can be improved by taking these crossequation correlations into account. Seemingly unrelated regression (SUR), also called joint generalized least squares (JGLS) or Zellner estimation, is a generalization of OLS for multiequation systems.
Like SUR in the least squares setting, the generalized maximum entropy SUR (GMESUR) method assumes that all the regressors are independent variables and uses the correlations among the errors in different equations to improve the regression estimates. The GMESUR method requires an initial entropy regression to compute residuals. The entropy residuals are used to estimate the crossequation covariance matrix.
In the iterative GMESUR (ITGMESUR) case, the preceding process is repeated by using the residuals from the GMESUR estimation to estimate a new crossequation covariance matrix. ITGMESUR method alternates between estimating the system coefficients and estimating the crossequation covariance matrix until the estimated coefficients and covariance matrix converge.
The estimation problem becomes the generalized maximum entropy system adapted for multiequations as follows:








where






y denotes the MT column vector of observations of the dependent variables; denotes the (MT x KM ) matrix of observations for the independent variables; p denotes the LKM column vector of weights associated with the points in Z; w denotes the LMT column vector of weights associated with the points in V; , , and are L, KM, and MTdimensional column vectors, respectively, of ones; and and are (KM x KM) and (MT x MT) dimensional identity matrices. The subscript l denotes the support point , k denotes the parameter , m denotes the equation , and t denotes the observation .
Using this notation, the maximum entropy problem that is analogous to the OLS problem used as the initial step of the traditional SUR approach is








The results are GMESUR estimates with independent errors, the analog of OLS. The covariance matrix is computed based on the residual of the equations, . An factorization of the is used to compute the square root of the matrix.
After solving this problem, these entropybased estimates are analogous to the Aitken twostep estimator. For iterative GMESUR, the covariance matrix of the errors is recomputed, and a new is computed and factored. As in traditional ITSUR, this process repeats until the covariance matrix and the parameter estimates converge.
The estimation of the parameters for the normedmoment version of SUR (GMESURNM) uses an identical process. The constraints for GMESURNM is defined as:

The estimation of the parameters for GMESURNM uses an identical process as outlined previously for GMESUR.