# Consistent estimator - consistent with what exactly?

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?

• Yes, $$\hat\gamma_2$$ is consistent for $$\beta_2$$
• No, $$\hat\gamma_2$$ is not consistent for $$\gamma_2$$ (or for $$\beta_0$$ or lots of other things).
In this case, the causal assumptions suggest you'd be more interested in whether it was consistent for $$\gamma_2$$, but you still need to say "consistent for $$\gamma_2$$", not just "consistent". The same is true for 'biased' and 'unbiased': an estimator is biased or unbiased for a parameter