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

<p>Assuming a linear setup, we look at</p>

<p>$$\mathbf \Pi = \mathbf X\mathbf a + \mathbf u \tag{1}$$</p>

<p>and 
$$\mathbf R = \mathbf X \mathbf b + \mathbf v \tag{2}$$</p>

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

<p>$$\mathbf \Pi = \gamma \mathbf R  + \mathbf \varepsilon \tag{3}$$</p>

<p>Inserting $(2)$ into $(3)$ we get</p>

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

<p>Looking at $(1)$ and $(4)$, we have the equivalence </p>

<p>$$\mathbf a = \gamma \mathbf b \tag{5}$$</p>

<p>Will we obtain it through estimation? Not that easily.</p>

<p>Least -squares estimation will give us, for $(1)$</p>

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

<p>and for $(2)$</p>

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

<p>Insert $(3)$ into $(6)$ to get</p>

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

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

<p>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 $(3)$.  </p>

<p>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 <em>not</em> recover $\gamma$, since $\left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf \varepsilon$ won't converge to zero.</p>

CC BY-SA 3.0
2015-09-06T00:02:25.897Z
2015-09-07T13:10:50.200Z
2015-09-07T13:10:50.200Z
Alecos Papadopoulos
<p>I was wondering if it is possible to make any economic implications by regressing the same set of independent variables with different outcome variables.</p>

<p>For instance, regressing [Industry, Years, Production Level] on [Revenue, Profits].</p>

<p>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].</p>

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

CC BY-SA 3.0
2015-06-03T06:03:35.290Z
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
<econometrics>
What are the economic implications of regressing the same set of independent variables with different outcome variables
valarmorghulis
# What are the economic implications of regressing the same set of independent variables with different outcome variables

(Asked by: username_0 on 2015-06-03)

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?

**Tags:** econometrics

> **Comments:**
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2015-09-06. Score: 4

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.


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