## Covariance Matrix

The estimated covariance matrix of the parameter estimates is computed
as the inverse Hessian matrix, and for unconstrained problems it
should be positive definite. If the final parameter estimates are
subjected to *n*_{act} > 0 active linear inequality constraints, the
formulas of the covariance matrices are modified similar to Gallant
(1987) and Cramer (1986, p. 38) and additionally generalized for
applications with singular matrices.
There are several steps available that enable you to tune the rank
calculations of the covariance matrix.

- You can use the ASINGULAR=, MSINGULAR=, and VSINGULAR= options
to set three singularity criteria for the inversion
of the Hessian matrix
*H*.
The singularity criterion used for the inversion is

where *d*_{j,j} is the diagonal pivot of the matrix *H*,
and ASING, VSING, and MSING are the specified values of
the ASINGULAR=, VSINGULAR=, and MSINGULAR= options. The
default values are
- ASING: the square root of the smallest positive double
precision value
- MSING: 1E-12 if you do not specify the SINGHESS= option
and
otherwise, where is the machine precision
- VSING: 1E-8 if you do not specify the SINGHESS= option
and the value of SINGHESS otherwise

Note that, in many cases, a normalized matrix *D*^{-1}*AD*^{-1}
is decomposed, and the singularity criteria are modified
correspondingly.
- If the matrix
*H* is found to be singular in the first step,
a generalized inverse is computed. Depending on the G4=
option, either a generalized inverse satisfying all four
Moore-Penrose conditions is computed or a generalized
inverse satisfying only two Moore-Penrose conditions is
computed. If the number of parameters *n* of the
application is less than or equal to G4=*i*, a G4 inverse
is computed; otherwise, only a G2 inverse is computed.
The G4 inverse is computed by the (computationally very
expensive but numerically stable) eigenvalue decomposition, and
the G2 inverse is computed by Gauss transformation.
The G4 inverse is computed using the eigenvalue
decomposition , where *Z* is the
orthogonal matrix of eigenvectors and is
the diagonal matrix of eigenvalues,
. The G4 inverse of *H* is set to

where the diagonal matrix
is defined using the COVSING= option.

If you do not specify the COVSING= option, the *nr*
smallest eigenvalues are set to zero, where *nr* is
the number of rank deficiencies found in the first step.

For optimization techniques that do not use second-order derivatives,
the covariance matrix is computed using finite difference
approximations of the derivatives.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.