The NLIN Procedure

Covariance Matrix of Parameter Estimates

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

\[  \mr {mse} \times (\mb {X}’\mb {X})^{-1}  \]

for the gradient, Marquardt, and Gauss methods and

\[  \mr {mse} \times \mb {H}^{-1}  \]

for the Newton method. Recall that $\mb {X}$ is the matrix of the first partial derivatives of the nonlinear model with respect to the parameters. The matrices are evaluated at the final parameter estimates. The mean squared error, the estimate of the residual variance $\sigma ^2$, is computed as

\[  \mr {mse} = \mb {r}’\mb {r} / (n - p)  \]

where n is the number of nonmissing (used) observations and p is the number of estimable parameters. The standard error reported for the parameter estimates is the square root of the corresponding diagonal element of this matrix. If you specify a value for the residual variance with the SIGSQ= option, then that value replaces $\mr {mse}$ in the preceding expressions.

Now suppose that restrictions or bounds are active. Equality restrictions can be written as a vector function, $ h({\btheta }) =\mb {0}$. Inequality restrictions are either active or inactive. When an inequality restriction is active, it is treated as an equality restriction.

Assume that the vector ${h({\btheta })}$ contains the current active restrictions. The constraint matrix $\mb {A}$ is then

\[  \mb {A}(\widehat{\btheta }) = \frac{{\partial } h(\widehat{\btheta })}{{\partial } \widehat{\btheta }}  \]

The covariance matrix for the restricted parameter estimates is computed as

\[  \mb {Z} ( \mb {Z}’ \mb {HZ} )^{-1} \mb {Z}’  \]

where $\mb {H}$ is the Hessian (or approximation to the Hessian) and $\mb {Z}$ collects the last $(p - n_ c)$ columns of $\mb {Q}$ from an LQ factorization of the constraint matrix. Further, $n_ c$ is the number of active constraints, and p denotes the number of parameters. See Gill, Murray, and Wright (1981) for more details about the LQ factorization. The covariance matrix for the Lagrange multipliers is computed as

\[  \left(\mb {A} \mb {H}^{-1} \mb {A}’\right)^{-1}  \]