# Murphy-Topel standard errors correction for GMM?

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?

• Is the weighting matrix for your second stage, GMM estimation coming from the first stage that you estimate through maximum likelihood? Is that how the second stage, GMM estimation depends on your first stage, MLE estimates? (Or are you doing something else?) – Matthew Gunn Sep 4 '18 at 18:55
• No, I just use the identity matrix as weighting matrix (yes, I know two-step GMM is efficient, but currently leave this aside). $\hat \theta_1$ enters in the moment conditions in a non-linear way. – bonifaz Sep 4 '18 at 19:16