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."