I'm cross-posting this question on both Economics and Cross Validated to get answers from a different perspective on each field. It is generally accepted to cross-post if the question is tailored to fit better in each community. See: https://meta.stackexchange.com/a/64069/510233
Cross-posted in Cross Validated: https://stats.stackexchange.com/q/403213/243829
I'll try to be as detailed as possible to best deliver what I want to know in this question.
In economic data, there are some variables that are composed of many other variables. Mostly indices or multiples. I can think of some examples like CPI, Consumer Sentiment Index, or even EV/EBITDA, PER.
Better and more general example would be something like Wind chill Temperature Index which is
WCI= wind chill index, kcal/m2/h
v= wind velocity, m/s
Ta= air temperature, °C
What I want to know is whether a variable like WCI can have more meanings(effects) than v and Ta seperately combined.
Based on what I learned in regression analysis from Ecnonometrics class, a simple interaction term like would be interpreted as the difference in marginal effect between a control group (D=0) and a treatment group (D=1), when expressed like:
Also, more general interaction term like VariableA * VariableB would be interpreted as an impact of VariableA on the coefficient of VariableB, in some examples like
PollutionLevel = B0 + B1*Population + B2*NumberOfCars + B3*Population*NumberOfCars = B0 + B1*Population + (B2 + B3*Population)*NumberOfCars
(originally from a question posted on Cross Validated)
But I think these interpretations are too simple to capture the whole meaning(effect) of a new variable derived from other variables, which leads to other questions like:
What would be the interpretation of a super complex interaction term like
WCIin regression perspective? (and other data science methods like Random Forest or Deep Neural Network, etc if possible.)
Would it still be valid and preferable to include an explanatory variable like
WCIin the model when I already have
Ta? Would it make the model more accurate?
Intuitively, I feel like people would care more about important indices but not necessarily the variables it is composed of. Can a complex interaction term derived from other variables have more effect than a simple combination of the other variables?
The third one is related to my original question of whether one variable that is composed of other variables in a dataset can have more effect on the dependent variable.
One example I can think of is the steepness of the yield curve that is considered very important in the bond market, which is a mere slope of the interest rates of bonds with different YTM.
People seem to care about the slope of the yield curve more than individual rates of each bond so I think it is a legit explanatory variable but it is unclear whether it is justified to introduce this new variable from existing variables or how it should be interpreted.