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

$Y_{2i} = \beta_0+\delta_1X_i+\delta_2Y_{1i}+\epsilon_i$

?

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

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

?