Lets assume, that the real DGP (real world data) is generated from the model:
$$y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + \varepsilon_i$$
Lets further assume, that $x_1$ and $x_2$ are correlated. Precisely, $x_1$ is a confounder variable, that causes $x_2$:
$$x_{2i} = \alpha_0 + \alpha_1 x_{1i} + u_i$$
Researcher does not know above information, he is sure, that the true model has only one variable, and assumes following functional form:
$$ y_i = \gamma_0 + \gamma_2x_{2i} + v_i $$
What can we, who know everything, tell about the consistency of the estimator $\hat \gamma_2$?
- It is inconsistent, because consistent estimator has limit in the 'real world parameter', which in this case is $\beta_2$.
- It is consistent, because consistent estimator has limit in the parameters of 'assumed model'. In this case $\gamma_2$. It is the model, which does not fit real world, not the estimator.
I see these two possibilities. Which one is (more) true, and what is most important - why?