Is there anything wrong with regressing a ratio on itself? For example can we regress the log(savings ratio) on log(income) or the log(debt to income) ratio on log(income)? If not, should we use the log transformation to take income to the RHS and regress log(savings) on log(income)? If using a log transformation is the way to go, how do I interpret my results, given that the savings ratio is what I am interest in, not savings?
The log of a ratio is the difference of the logs. So log(debt/income)~log(income) = log(debt)-log(income)~log(income). That can't be good. With random data you get a p value with 16 0's:
set.seed(123) income <- 10^(rnorm(100))*10000 debt <- 10^(rnorm(100))*1000 debtincome <- debt/income m1 <- lm(log(debtincome)~log(income)) summary(m1)
log(savings)~log(income) is at least statistically reasonable; whether it is what you want, I can't say.