# Can every identifiable model be estimated by GMM?

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:

$$E(g(x_i,\theta))=0$$

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?