The SPATIALREG Procedure

The SPATIALREG procedure enables you to specify the estimation method for the covariance matrix. The COVEST=HESSIAN option estimates the covariance matrix based on the inverse of the Hessian matrix, COVEST=OP uses the outer product of gradients, and COVEST=QML produces the covariance matrix based on both the Hessian and outer product matrices. By default, COVEST=HESSIAN.

Although all three methods produce asymptotically equivalent results, they differ in computational intensity and produce results that might differ in finite samples. The COVEST=OP option provides the covariance matrix that is usually the easiest to compute. In some cases, the OP approximation is considered more efficient than the Hessian or QML approximation because it contains fewer random elements. The QML approximation is computationally the most complex, because both the outer product of gradients and the Hessian matrix are required. In most cases, the OP or Hessian approximation is preferred to QML. The need to use QML approximation arises in some cases when the model is misspecified and the information matrix equality does not hold.

When Taylor approximation or Chebyshev approximation is used for the SAR and SDM models, only COVEST=HESSIAN is supported.