Composite variable as DV. Can I include part of it as independent variable?

Question

I am struggling to form a regression model as I have to explain financial performance of a company.

The problem is that I have a variable like Return on assets $\frac{ebit}{total assets}$ as a dependent variable and I would like to control for total assets. I understand that if I include "total assets" as Independent Variable, I will end up with a probably significant coefficient as there is obvious correlation between dependent Variable and Independent Variable. My questions is regarding the interpretation of other variable coefficients.

Are they ok or do they suffer from the inclusion of "total assets" into the model?

model example:

$\frac{Ebit}{total assets}=constant+b_1*total assets+b_2*AGE+e$

I see this kind of situation a lot in the paper, but it is not considered a problematic. I personally think that its important to control for "total assets" in this situation but does someone see a problem here?

Answer 1

I'm not certain—so don't take me at 100% here—but it doesn't feel right to me. Predicting y/x by x itself is strange. Of course you are going to get a significant prediction, because x is used to calculate the DV. I generated some random data (assuming no relationship between any of the variables)...

set.seed(1839) # setting seed for replicability
ebit <- sample(1:7, 200, T) # creating ebit scores
assets <- sample(1:7, 200, T) # creating assets scores
age <- sample(1:7, 200, T) # creating age scores
data <- data.frame(roa=ebit/assets, ebit, assets, age) # making dataset

and ran the models:

summary(lm(roa~assets+age, data)) # running roa as DV

Call:
lm(formula = roa ~ assets + age, data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.8400 -0.6483 -0.0154  0.4882  4.3598 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.32694    0.22118  15.042   <2e-16 ***
assets      -0.44517    0.03770 -11.807   <2e-16 ***
age         -0.04832    0.03768  -1.282    0.201    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.025 on 197 degrees of freedom
Multiple R-squared:  0.4189,    Adjusted R-squared:  0.413 
F-statistic:    71 on 2 and 197 DF,  p-value: < 2.2e-16

And

summary(lm(ebit~assets+age, data)) # running ebit as DV

Call:
lm(formula = ebit ~ assets + age, data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.0695 -1.7712  0.1377  1.4421  3.5400 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  3.35110    0.43397   7.722 5.66e-13 ***
assets       0.04386    0.07398   0.593    0.554    
age          0.06503    0.07393   0.880    0.380    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.011 on 197 degrees of freedom
Multiple R-squared:  0.005828,  Adjusted R-squared:  -0.004265 
F-statistic: 0.5774 on 2 and 197 DF,  p-value: 0.5623

The effect of assets on roa makes sense in the first model. All it says is: As the denominator gets higher, the value of the quotient gets lower. How does that help you? The second model is more straightforward: It is the effect of assets and age on ebit simultaneously.