<p>As long as <span class="math-container">$\hat{W}\rightarrow W$</span> where <span class="math-container">$W$</span> 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.</p>
<p>I think you would need to justify that <span class="math-container">$\hat{h}\rightarrow h$</span> and that <span class="math-container">$\hat{W}(h)$</span> is &quot;continuous&quot; in <span class="math-container">$h$</span>. Given this, <span class="math-container">$\hat{W}(\hat{h})\rightarrow W$</span> and all is well.</p>

CC BY-SA 4.0
2023-09-12T11:02:00.337Z
Does it have to be the inverse variance-covariance matrix of moments? Or W hat could be a positive definite matrix, that converges to a positive definite matrix W. Not looking for efficiejxy.
mich95
CC BY-SA 4.0
2023-09-13T07:30:31.183Z
If you aren't looking for efficiency, then you are correct, you just need convergence to a positive definite matrix.
Michael Gmeiner
CC BY-SA 4.0
2023-09-14T11:24:22.493Z
I am afraid that nuisance parameters are not satisfying the requirement $\hat{h}\rightarrow h_0$ when sample size goes to infinity.
Bertrand
CC BY-SA 4.0
2023-09-12T08:46:20.360Z
2023-09-12T08:46:20.360Z
Michael Gmeiner
<p>Suppose that I am working in a model containing a nuisance parameter <span class="math-container">$h$</span> and a finite dimensional parameter of interest <span class="math-container">$\theta$</span>, whose true values are <span class="math-container">$h_0$</span> and <span class="math-container">$\theta_0$</span>, respectively.
Newey (1994) proves the asymptotic properties of a two-step GMM with a non parametric first step, and studies the properties of
<span class="math-container">$\hat{\theta}= \arg \max_{\theta} m_{n}(\theta, \hat{h})^{T} \hat{W} m_{n}(\theta,\hat{h}) $</span>
where
<span class="math-container">$m_{n}(\theta,h)=\frac{1}{n} \sum\limits_{i=1}^n m(W_i,\theta,h)$</span>
and <span class="math-container">$\mathbb{E}[m(W,\theta_0,h_0)]=0$</span>.</p>
<p>In this setup, can one choose <span class="math-container">$\hat{W}$</span> (positive semi-definite) to depend on <span class="math-container">$\hat{h}$</span> and still expect the asymptotic results to hold, if all the regularity conditions are met and <span class="math-container">$\hat{W} \to W$</span> in probability ?
There are no indications in the papers that <span class="math-container">$\hat{W}$</span> cannot depend on the nuisance parameter.</p>

<econometrics><theory><gmm>
Two step Generalized Method of Moments (Newey 1994). $\hat{W}$ matrix depending on the nuisance parameter
mich95
# Two step Generalized Method of Moments (Newey 1994). $\hat{W}$ matrix depending on the nuisance parameter

(Asked by: username_0 on 2023-09-08)

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.

**Tags:** econometrics, theory, gmm

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## 1 answers

### Answer by username_1 on 2023-09-12. Score: 2

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.

> **Comments:**
> **username_0**: Does it have to be the inverse variance-covariance matrix of moments? Or W hat could be a positive definite matrix, that converges to a positive definite matrix W. Not looking for efficiejxy.  
> **username_1**: If you aren't looking for efficiency, then you are correct, you just need convergence to a positive definite matrix.  
> **username_2**: I am afraid that nuisance parameters are not satisfying the requirement $\hat{h}\rightarrow h_0$ when sample size goes to infinity.  

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