A model suffering from omitted variable bias can be said to be unidentified?

Question

If my regression model $$ y_i = \alpha + \beta x_i + \epsilon_i $$

suffers from OVB the error contains one variable which we assume correlated with $$ \epsilon_i = \gamma w_i + u_i $$

my estimate of $\beta$ will be biased by a component which depends on $\gamma Cov(x_i,w_i)$. I would colloquially say that OLS can't distinguish between variation that's due solely to $x_i$ from variation that's shared with the omitted variable. Can it then be said the parameter in the model is not uniquely identified in the original model? Are identification and endogeneity the same thing here? And if yes, is that always the case?

Answer 1

From the tag:

A model is identifiable if a single set of parameters can be found that will yield the best fit.

Unidentifiability is typically used to mean the model can't uniquely assign values to the coefficients of two parameters, but in your case, there is no parameter for the omitted variable, so there is no possibility for unidentifiability (otherwise it wouldn't be omitted in the first place).

Hence I would not call this unidentified, or unidentifiable.