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Suppose I want to see the impact of an explanatory variable $X$ on two different dependent variables: $Y_1$ and $Y_2$. Suppose also that I find that $Y_1$ and $Y_2$ are correlated.

Assuming that all other necessary assumptions for OLS regression are satisfied, can I assess the two impacts on individuals $i$ by just running OLS on

$Y_{1i} = \beta_0+\beta_1X_i+\beta_2Y_{2i}+\epsilon_i$

and

$Y_{2i} = \delta_0+\delta_1X_i+\delta_2Y_{1i}+v_i$

?

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  • $\begingroup$ Did you intend to have $\delta_0$ as the intercept in the $Y_2$ equation? What is the relationship between the epsilons in the two equations? $\endgroup$
    – dimitriy
    Commented May 16, 2019 at 0:26
  • $\begingroup$ My bad - yes it was meant to be a delta0 and a vi. $\endgroup$
    – leecarvallo
    Commented May 16, 2019 at 18:35
  • $\begingroup$ What do you know about how the $\epsilon_i$ and $\upsilon_i$ might be correlated? That determines how to proceed. For instance, do you know their correlations independently of these data? Can you assume their correlations are unvarying? $\endgroup$
    – whuber ♦
    Commented May 16, 2019 at 18:36
  • $\begingroup$ For my purposes, they would likely be correlated. $\endgroup$
    – leecarvallo
    Commented May 16, 2019 at 18:37
  • $\begingroup$ Right--that's not what I'm asking. I'm asking for more specific, quantitative information or assumptions about that correlation. Moreover, do you really intend to be regressing the $Y_{1i}$ and $Y_{2i}$ against each other? What sense does that make? Isn't this some kind of typographical error in the formulas? $\endgroup$
    – whuber ♦
    Commented May 16, 2019 at 18:39

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