A parameter in a statistical model is identified if there is a bijection between values of that parameter, and the set of probability distributions (over observable variables) that are consistent with the model.
Here are a couple of questions relating to parameter identification:
Conjecture. The existence of an estimator $b$ such that $plim(b)=\beta$ does not necessarily imply that $\beta$ is identified.
My intuition says this must be true for two reasons: 1. It might still be that in finite samples, there are multiple distributions consistent with $\beta$, even if in the infinite sample case there is a bijection. 2. $plim(b)= \beta$ does not even imply that there is a bijection in infinite samples, since for what ever value $\beta$ takes (say $5$), we can simply take as an estimator $b=5$ regardless of the sample. In this case $plim b = \beta$ trivially, so whether a parameter is identified or not, we can have a consistent estimator of it.
Is my argument correct?