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Econometrics Procedures
 

CSSM Procedure

Covariance Parameterization

The covariance matrices specified by the COV and COV1 options in the STATE statement must be positive semidefinite. When these matrices are of general form and are not user-specified, they are internally parameterized by their Cholesky root. Suppose that normal upper Sigma, an m times m positive semidefinite matrix of rank normal r, is such a covariance matrix. Then, normal upper Sigma can always be written as

normal upper Sigma equals upper R upper R Superscript prime

where the (generalized) Cholesky root, normal upper R, is an m times r lower triangular matrix with nonnegative diagonal elements (that is, upper R left-bracket i comma j right-bracket equals 0 normal i normal f j greater-than i and upper R left-bracket i comma i right-bracket greater-than-or-equal-to 0 comma 1 less-than-or-equal-to i less-than-or-equal-to r). The CSSM procedure parameterizes normal upper Sigma by the elements of its Cholesky root, which adds r asterisk left-parenthesis r plus 1 right-parenthesis slash 2 plus r asterisk left-parenthesis m minus r right-parenthesis elements to the parameter vector theta theta.

Last updated: July 09, 2026