The ENTROPY Procedure (Experimental)

Maximum Entropy-Based Seemingly Unrelated Regression

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 cross-equation correlations into account. Seemingly unrelated regression (SUR), also called joint generalized least squares (JGLS) or Zellner estimation, is a generalization of OLS for multi-equation systems.

Like SUR in the least squares setting, the generalized maximum entropy SUR (GME-SUR) 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 GME-SUR method requires an initial entropy regression to compute residuals. The entropy residuals are used to estimate the cross-equation covariance matrix.

In the iterative GME-SUR (ITGME-SUR) case, the preceding process is repeated by using the residuals from the GME-SUR estimation to estimate a new cross-equation covariance matrix. ITGME-SUR method alternates between estimating the system coefficients and estimating the cross-equation covariance matrix until the estimated coefficients and covariance matrix converge.

The estimation problem becomes the generalized maximum entropy system adapted for multi-equations 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 MT-dimensional 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 GME-SUR 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 entropy-based estimates are analogous to the Aitken two-step estimator. For iterative GME-SUR, 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 normed-moment version of SUR (GME-SUR-NM) uses an identical process. The constraints for GME-SUR-NM is defined as:

The estimation of the parameters for GME-SUR-NM uses an identical process as outlined previously for GME-SUR.