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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 $plim$.

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  • $\begingroup$ 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 $\endgroup$ – Alecos Papadopoulos Apr 29 at 18:13
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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 $c$.

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