In ridge regression with design matrix $X$, outcomes $y$, fixed regularization parameter $\lambda$, and errors $\epsilon\sim\mathcal{N}(0, \sigma^2I)$, the computations for the ridge regression coefficients $\hat\beta$ (aka the solution to $\arg\min_b \big[(y-Xb)'(y-Xb) + \lambda b'b\big]$) and their variance-covariance matrix $var(\hat\beta)$ are:
$$ \begin{align*} M &:= (X'X + \lambda I)^{-1}X' \\ \hat\beta &= My \\ var(\hat\beta) &= \sigma^2MM' \end{align*} $$
The computation on $var(\hat\beta)$ relies on $\lambda$ being treated as a constant value, which makes $M$ constant. Thus, we can apply the identity $var(My) = Mvar(y)M'$ and the fact that $var(y) = \sigma^2I$ to derive $var(\hat\beta)$.
However when $\lambda$ is selected using $X$ and $y$ (for my application via cross-validation), $\lambda$ and $M$ become stochastic. How can $var(\hat\beta)$ be updated to deal with a stochastic $\lambda$ selected via cross-validation?
