Hi Michael,
The coefficients of ridge regression are given by:
\beta^* = (X'X + k I)^{-1} X' y, ---------------- (1)
where k > 0 is the penalty parameter and I is the identity matrix.
The ridge estimates are related to OLS estimates \beta as follows:
\beta^* = Z \beta, ---------------------- (2)
where Z = [I + k(X'X)^{-1}]^{-1} , ------------------ (3)
Let \Sigma and \Sigma^* be the variance-covariance matrices of \beta and
\beta^*, resply. Therefore,
\Sigma^* = Z \Sigma Z' ----------------- (4)
In other words, for a fixed k, you can obtain the covariance matrix of \beta^*
by making use of (3) and (4).
You can read the original paper by Hoerl and Kennard (Technometrics 1970) for
more details.
Hope this is helpful,
Ravi.