How can I tell if a statistical model is "identified"?

Question

My econometrics professor used the term "identified" in class. We are considering data generating processes of the form $$Y = \beta_0 + \beta_1 X + U$$ where $X$ is a random variable and $U$ is a random error term. Our regression lines take the form of $$Y = \hat{\beta_0}+\hat{\beta_1}X$$

He gave the following definition of "identified":

$\beta_0$,$\beta_1$ are identified if a data set $\lbrace X_n\rbrace_{i=1}^{\infty}$ contains enough information to "pin down" unique values for $\beta_0$,$\beta_1$

I am dissatisfied with this definition because he neither specifies what "information" is nor what "pin down" means.

A Bit of Context

In one of our exercises, we were given $\Bbb E[UX] = \alpha \ne 0$. According to my professor, this violates an assumption called "Exogeneity" which is necessary for a model to be 'identifiable.'

Specifically, according to his lecture notes,

Exogeneity Assumption: The error term is uncorrelated with the regressors, or $\operatorname{Cov}(U_n,X_{nk}) = 0$ for all $k = 1,2,3...,K$. By assumption of $\Bbb E(U_n|X_{n1},X_{n2},...,X_{nK})$, this can be rewritten as $$\operatorname{Cov}(U_n,X_{nk}) = \Bbb E(U_nX_{nk}) =0$$ for all $k = 1,2,3...,K$

It seems in our problem, he is trying to get us to understand why, if this Exogeneity assumption fails, a model cannot be identified. So hopefully this can give answerers context for how he is using the term.

My Question

Can someone clarify what he means by "information" and "pin down"? Or give a better definition altogether.

EDIT:

Pulled from Wikipedia:

Observationally Equivalent --- two parameter values are considered observationally equivalent if they both result in the same probability distribution of observable data.

Identified --- any situation where a statistical model will invariably have more than one set of parameters that generate the same distribution of observations, meaning that multiple parametrizations are observationally equivalent.

This still doesn't really explain where "exogeneity" comes in and why it's related to being "identified."

Answer 1

Identifiability basically refers to whether or not consistent estimators exist for the parameters of the model. Put another way, if we are told the distribution of the data, can we recover the model parameters? If not then our model is unidentifiable.

Perhaps the simplest example of an unidentifiable model is the overparameterized ANOVA model. This model takes the form

$$ Y_{ij} = \mu + \alpha_i + \epsilon_{ij} $$

where $\mu$ and $\{ \alpha_i \}_{i=1}^{k}$ are arbitrary constants and $\epsilon_{ij} \sim$ normal$(0, \sigma^2)$. If we are given the information that $Y_{ij} \sim$ normal$(\mu_i, \sigma^2)$ for some sets of constants $\{ \mu_i \}_{i=1}^{k}$ and $\sigma^2$, and it is important to note that this is all we can ever hope to learn from the data, then there is no unique way to translate this back into constants $\mu$, $\{ \alpha_i \}_{i=1}^{k}$ and $\sigma^2$. This is because we can always take $\mu + c$ and $\alpha_i - c$ to arrive at the same mean parameter $\mu_i = \mu + \alpha_i$ for different values of the model parameters. Even if we had infinite data we could never hope to recover these values. For this reason we impose the constraint $\sum_{i=1}^{k} \alpha_i = 0$ which guarantees a one to one mapping between model and distribution parameters.