I employ a two-stage estimation routine: In the first stage, a large vector $\hat \theta_1$ (around 2000 elements) is estimated with maximum likelihood. In the second stage, I estimate $\hat \theta_2 ( \hat \theta_1)$ using GMM, and I'm interested in the standard errors of $\theta_2$.
If I were to use GMM for estimation of my parameters in one step only, I know that I would obtain consistent estimates of the standard errors using the outer product of the gradient estimator. Conversely, if I were to use ML also in the second stage, I could apply the Murphy-Topel correction for standard errors.
How can I correct for standard-errors given that I use GMM in the second stage?