What is the difference between Consistency and Identification? Dear experienced friends, I start to learn Econometrics recently and there is a question really confuses me. Suppose we have a simple linear model
$$
y = \beta*X. 
$$
From the definition, we know the consistency means the estimated parameters $\widehat{\beta}$ we get from data will converge to the true parameters $\beta$ when our sample size goes infinity.
Meanwhile, the definition of identification says: If we have enough data, the model is identified if $\widehat{\beta}$ converges to the true parameters $\beta$, and this true $\beta$ is unique.
From my point of view, both concepts look very similar: They all require the $\widehat{\beta}$ converge to $\beta$ as $n$ goes infinity. Can we say if the estimate of a model's parameter is consistent, then this model is identified?
Please feel free to point out any mistake I made. Thank you in advance!
 A: Revisiting the definition of identification.
Although your definition of consistency is fine, I think you're defining identification in a somewhat odd way, especially with regards to the usage of "if we have enough data."
Although we can loosely think about identification as having "infinite data," I'd suggest you instead consider a scenario where you know the true distribution of the observed data.
In this sense, let $P$ denote the true distribution of observed data where $P \in \mathcal{P} \equiv \{P_{\theta} : \theta \in \Theta\}$. We are interested in $\theta$ or some function $f(\theta)$.
Since $P \in \mathcal{P}$, we know that there exists some $\theta \in \Theta$ such that $P = P_{\theta}$. However, given $P$, we cannot distinguish $\theta$ from any other $\theta'$ such that $P = P_{\theta'}$. In words, this is saying that given $P$, we may not 'know' enough about $\theta$ to uniquely pin it down.
To illustrate when this can happen, suppose we observed $P = N(a+b,\sigma^2)$, so that $\theta = (a,b,\sigma^2)$, and we are interested in $f(\theta) = (a,b)$. Given $P$, I cannot uniquely pin down $f(\theta)$, because even though I know $a+b$, I can choose any $f(\theta) = (\theta_1,\theta_2)$ such that $\theta_1 + \theta_2 = a+b$. To make things super concrete, suppose $a = -1,b=1$ so that $a+b = 0$. Then both $(-1,1)$ and $(0,0)$ are consistent with $P$. Hence, even fully knowing the distribution $P$ does not give me enough to pin down the 'true' value of $(a,b)$.
In general, given $P$ and $\mathcal{P}$, the best we can say about $\theta$ is that $\theta \in \Theta^*(P)$ where
$$\Theta^*(P) \equiv \{\theta \in \Theta : P_\theta  = P\}.$$
This is simply defining $\Theta^*(P)$ to be the set of all $\theta$ that agree with the observed distribution $P$. We call this the identified set. We then say that $\theta$ is identified if $\Theta^*(P)$ is a singleton for all $P \in \mathcal{P}$. Here, by singleton, we are saying that given $P$, we can uniquely pin down $\theta$. We similarly define these terms for $f(\theta)$.
Relationship between identification and consistency.
It should hopefully be somewhat clear from the above that identification and consistency are closely related.
In particular, if $\theta$ is not identified, then it follows that consistent estimators cannot exist for $\theta$. Why? Well suppose that we had a consistent estimator $\hat{\theta}$. Because it is consistent, it should converge to all values in $\Theta^*(P)$. Since $\theta$ is not identified, then there are values $\tilde{\theta},\bar{\theta} \in \Theta^*(P)$ such that $\tilde{\theta} \neq \bar{\theta}$, and $\hat{\theta}$ cannot converge to two distinct values!
Conversely, if $\theta$ is identified, then consistent estimators may exist, though do not have to. Though most of the time, there will exist a consistent estimator (by appealing to law of large numbers, continuous mapping theorem, and so on), but exceptions exist (i.e., the mean for Pareto distributions with $\alpha < 1$).
