# Regressing a function of independent variables on another function of independent variables

For a particular economic sector, I want to assess whether expenditures on technology, equipment, and construction (which I will call capital expenditures) are cost-increasing or cost-decreasing relative to labour costs. That is, it could be the case that capital serves as a strong substitute for labour and leads to decreased overall costs. Alternatively, it could be the case that capital investments require significant additional labour for upkeep and do not offer much substitutive value and thus are relatively cost-increasing.

My supervisor recommended regressing total expenditures for firm $$i$$ in year $$t$$ on the capital-to-labour ratio for that firm in that year, while controlling for that firm's production in that year.

Thus I am using the following econometric model:

$$TotalExpenditures_{it} = \alpha_i + \beta_1(K/L)_{it}+\beta_2Production_{it}+\beta_3Controls_{it}+\epsilon_{it}$$

However, total expenditures are obviously a function of capital and labour expenditures, since in our data,

$$TotalExpenditures_{it} = K_{it} + L_{it} +Other_{it}$$

Are there issues with regressing a function of independent variables on another function of independent variables? It feels wrong to me, but I can't pinpoint what the issue is. I'm not sure if our model is answering the question of interest.

One issue I do see is that $$Other_{it}$$ is clearly correlated with $$K_{it}$$, since $$K_{it} = TotalExpenditures_{it} - L_{it} - Other_{it}$$ so we would have to control for this, right? Beyond that, are there major issues here?

Note: I have read Dependent variable is a function of independent variables; can I sensibly include them in a regression? , but am not sure that it totally applies to my particular example.

## 1 Answer

One issue I do see is that $$Other_{it}$$ is clearly correlated with $$K_{it}$$, since $$K_{it}=TotalExpenditures_{it}−L_{it}−Other_{it}$$ so we would have to control for this, right? Beyond that, are there major issues here?

The mathematical derivation of $$TotalExpenditures_{it}$$ as a sum of three factors does not necessarily mean that those factors are correlated with each other. Think of your own personal budget. Does the fact that your monthly budget consists of spending on housing, food, and say transportation, imply that expenditures on housing and transportation are correlated? For some people, there could be a positive correlation (if income allows it and personal preferences/lifestyle choices guide those spending decisions), but for other people, there could be a negative correlation. Overall, there could be no correlation.

Are there issues with regressing a function of independent variables on another function of independent variables?

From this perspective, there is no issue with your model specification here. In that sense, the responses to the question you linked to are relevant to your situation. The model specification should follow your research question. If you know that some variables are related to the dependent variable by construction, that is not a problem in itself. It would only have been a problem if there was a deterministic relationship between the dependent variable and a single explanatory variable (e.g., total expenditures were equal to the number of employees a firm hired multiplied by the employee wage), as that would eliminate the need for a regression in the first place.

• Ok, thank you for your response! Jun 10, 2019 at 12:59
• Regarding the first question you highlighted, if I think there is a conceptual correlation between either capital expenditures and other expenditures or labour expenditures and other expenditures, then I would have to control for other expenditures, right? Jun 10, 2019 at 13:26
• And then if I include other expenditures, is there any need for other controls? If other controls, like customer demographics impact total expenditures, wouldn't this impact have to be through either the K, L, or other expenditures channels? Jun 10, 2019 at 13:33
• @leecarvallo Your variable of interest appears to be the ratio of capital to labor. You should control for variables that are correlated with that ratio and with total expenditures. Think about your own budget again. Do your food expenditures affect the ratio between housing and transportation expenditures? It does not seem like that is necessarily true, but if it is, controlling for other expenditures makes sense. Same for customer demographics: it is a confounding variable if it affects both the dependent variable (DV) and the ratio. Jun 10, 2019 at 18:03
• @leecarvallo You can also include variables that affect the DV only, as that can make standard errors smaller. Jun 10, 2019 at 18:03