Parameter Covariance Matrix

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The NLIN Procedure

Parameter Covariance Matrix

For unconstrained estimates (no active bounds), the parameter covariance matrix is

(X'X)^-1 * mse

for the gradient, Marquardt, and Gauss methods and

H^-1 * mse

for Newton method. The mse is computed as

r'r / (nused - np)

where nused is the number of non-missing observations and np is the number of estimable parameters. The standard error reported for the parameters is the sqrt of the corresponding diagonal element of this matrix.

Equality restrictions can be written as a vector function

$h({\theta}) = 0$

Inequality restrictions are either active or inactive. When an inequality restriction is active, it is treated as an equality restriction.

For the following, assume the vector ${h({\theta})}$ contains all the current active restrictions. The constraint matrix A is

$A(\hat{{\theta}}) = \frac{{\partial} h(\hat{{\theta}})}{{\partial} \hat{{\theta}}}$

The covariance matrix for the restricted parameter estimates is computed as

Z ( Z' H Z )^-1 Z'

where H is Hessian or approximation to the Hessian, and Z is the last (np - nc) columns of Q. Q is from an LQ factorization of the constraint matrix, nc is the number of active constraints, and np is the number of parameters. Refer to Gill, Murray, and Wright (1981) for more details on LQ factorization.

The covariance matrix for the Lagrange multipliers is computed as

( A H^-1 A' )^-1

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