# Regressing a ratio on a component of the ratio

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.

• thanks for your reply, if i did regress log(debt) on log(income), but was interested in the debt to income ratio, how would i interpret my coefficients. could i do to a regression of debt/income to income in levels? May 7, 2014 at 23:48