# Plim of ols estimator

Suppose you are given 2 models. Model of 1 has $$y$$ as the dependent variable and $$x$$ as the independent. Model 2 has $$w$$ as the dependent variable and $$x$$ as the independent variable. Both models have their own disturbance term.

Then you are given a fitted model in which $$y$$ is dependent on $$w$$. Now the question is to find plim of $$\hat{b}$$. I tired making $$x$$ the subject from model 1, plugged the $$x$$ in model 2 and then plugged $$w$$ (model 2) in fitted. Now when I took $$Cov(xy)$$ after all this substitution my fitted model $$y$$ doesn't have and $$x$$. I can't further solve for $$plim$$.

• Please write explicitly the three model equations, so that the relations are clear. For example, we cannot tell whether a constant term(s) exist, or from which model does $\hat b$ is estimated – Alecos Papadopoulos Apr 29 at 18:13

If your question is how to express the plim ($$b$$, say) of the slope estimator from the regression of $$y$$ on $$w$$ in terms of the plim's ($$a$$ and $$c$$, say) of the two slope estimators for your two models (one $$y$$ on $$x$$, and the other $$w$$ on $$x$$), my answer is no that's not possible. You did not give us enough information.
If $$y$$, $$x$$ and $$w$$ have unit variance (for simplification), what you want is $$b=cor(y,w)$$ expressed in terms of $$a=cor(y,x)$$ and $$c=cor(w,x)$$. But that's not possible. You cannot find $$b$$ from $$a$$ and $$c$$. $$Cor(y,w)$$ is not a function of $$cor(y,x)$$ and $$cor(w,x)$$. It is like we cannot find the joint distribution of $$(A,B)$$ from the individual marginal distributions of $$A$$ and $$B$$.
Here are two examples, with the same $$a$$ and $$c$$, but different $$b$$. (i) $$x$$ and $$y$$ are mutually independent ($$a=0$$), $$x$$ and $$w$$ are mutually independent ($$c=0$$), and $$y$$ and $$w$$ are mutually independent ($$b=0$$). (ii) $$x$$ and $$y$$ are mutually independent ($$a=0$$), $$x$$ and $$w$$ are mutually independent ($$c=0$$), and $$y=w$$ ($$b=1$$). You cannot find $$b$$ from $$a$$ and $$c$$.