Assume a model with parameters $\theta$ is identifiable. Then that means that for every probability distribution over observable variables $p(x|\theta)$, there is a unique parameter value $\theta$.
The GMM estimator relies on the moment condition:
My question is as follows: The expectation of some random variable is based on the distribution of that variable, but it loses information relative to the distribution. Therefore I am not so sure whether:
Conjecture. If a model with paramters $\theta$ is identifiable, then there exists a function $g$ such that $$E(g(x_i,\theta))=0$$
Is this correct?