# What are the economic implications of regressing the same set of independent variables with different outcome variables

I was wondering if it is possible to make any economic implications by regressing the same set of independent variables with different outcome variables.

For instance, regressing [Industry, Years, Production Level] on [Revenue, Profits].

Given the above variables, if both models are significant, what would be the implications? Does it imply that [Industry, Years, Production Level] can be used to explained both the change of [Revenue, Profits].

If it is, what are the appropriate regression models? Are there any concerned issues regarding the results?

• I think both models will be significant because given modest scalability Production Level will correlate with Revenue and given some nearly constant profit margin Profits will correlate Revenue. – Giskard Jun 3 '15 at 6:03

Using matrix notation, denote profits by $\Pi$, revenues by $R$, and the explanatory variables by $\mathbf X$.

Assuming a linear setup, we look at

$$\mathbf \Pi = \mathbf X\mathbf a + \mathbf u \tag{1}$$

and $$\mathbf R = \mathbf X \mathbf b + \mathbf v \tag{2}$$

Are profits correlated with Revenues? Experience say they are. So there is also a relationship

$$\mathbf \Pi = \gamma \mathbf R + \mathbf \varepsilon \tag{3}$$

Inserting $(2)$ into $(3)$ we get

$$\mathbf \Pi = \mathbf X(\gamma\mathbf b) + \gamma \mathbf v + \mathbf \varepsilon \tag{4}$$

Looking at $(1)$ and $(4)$, we have the equivalence

$$\mathbf a = \gamma \mathbf b \tag{5}$$

Will we obtain it through estimation? Not that easily.

Least -squares estimation will give us, for $(1)$

$$\mathbf {\hat a} = \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf \Pi \tag{6}$$

and for $(2)$

$$\mathbf {\hat b} = \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf R \tag{7}$$

Insert $(3)$ into $(6)$ to get

$$\mathbf {\hat a} = \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\left(\gamma \mathbf R + \mathbf \varepsilon\right)$$

$$\implies \mathbf {\hat a} = \gamma \mathbf {\hat b} + \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf \varepsilon \tag{8}$$

The interesting realization comes at this point: given the postulated relation $(1)$ it is unlikely that the variables in $\mathbf X$ will be uncorrelated with $\mathbf \varepsilon$, the error term in $(3)$.

This means that even with a large sample, calculating the ratios of the individual coefficients in the vectors $\mathbf {\hat a},\;\mathbf {\hat b}$, we will not recover $\gamma$, since $\left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf \varepsilon$ won't converge to zero.