# Two step Generalized Method of Moments (Newey 1994). $\hat{W}$ matrix depending on the nuisance parameter

Suppose that I am working in a model containing a nuisance parameter $$h$$ and a finite dimensional parameter of interest $$\theta$$, whose true values are $$h_0$$ and $$\theta_0$$, respectively. Newey (1994) proves the asymptotic properties of a two-step GMM with a non parametric first step, and studies the properties of $$\hat{\theta}= \arg \max_{\theta} m_{n}(\theta, \hat{h})^{T} \hat{W} m_{n}(\theta,\hat{h})$$ where $$m_{n}(\theta,h)=\frac{1}{n} \sum\limits_{i=1}^n m(W_i,\theta,h)$$ and $$\mathbb{E}[m(W,\theta_0,h_0)]=0$$.

In this setup, can one choose $$\hat{W}$$ (positive semi-definite) to depend on $$\hat{h}$$ and still expect the asymptotic results to hold, if all the regularity conditions are met and $$\hat{W} \to W$$ in probability ? There are no indications in the papers that $$\hat{W}$$ cannot depend on the nuisance parameter.

As long as $$\hat{W}\rightarrow W$$ where $$W$$ is the inverse of the variance-covariance matrix of moments, then all of the asymptotic properties hold. You seem to be asking if this applies in your case.
I think you would need to justify that $$\hat{h}\rightarrow h$$ and that $$\hat{W}(h)$$ is "continuous" in $$h$$. Given this, $$\hat{W}(\hat{h})\rightarrow W$$ and all is well.
• I am afraid that nuisance parameters are not satisfying the requirement $\hat{h}\rightarrow h_0$ when sample size goes to infinity. Sep 14, 2023 at 11:24