Economics Stack Exchange is a question and answer site for those who study, teach, research and apply economics and econometrics. It only takes a minute to sign up.
Sign up to join this community
Suppose we are using GMM to estimate the model $y=m(x;\theta_0)+u$ (for some function $m$ and $\theta_0 \in\mathbb{R^p}$) and we are using the moment condition: $\mathbb{E}[g(\theta_0)]=0$ where $g:\mathbb{R^p}\rightarrow\mathbb{R^p}$ so the model is just identified. My question is how do we know that our estimator $\hat\theta$ is not dependent on the weight matrix?
Thanks!
In the exactly identified case, it is natural to assume that a unique $\theta$ satisfies $\bar{g}(\theta) = 0$, because the number of parameters is equal to the number of equations. Let $\hat\theta$ denote such $\theta$ value.
When $W$ is positive definite, $\bar{g}(\theta)' W \bar{g}(\theta) \ge 0$ (because $W$ is positive definite), and $\bar{g}(\theta)' W \bar{g}(\theta)$ attains zero if and only if $\bar{g}(\theta)=0$. That is, the global minimizer of $\bar{g}(\theta)' W \bar{g}(\theta)$ equals the solution to $\bar{g}(\theta)=0$, which is $\hat\theta$, for whatever positive definite $W$ matrix. Thus, $W$ is irrelevant.
Note that this argument does not hold if $W$ is not positive definite. For example, if $W$ is positive semi-definite but not positive definite, then the GMM estimator may not be unique.
