Is it possible to have a valid regression with debt to income as the D.V and income as one of the I.V's?

Question

I have been assigned the following question

"On the basis of the Bank of England's 2013 survey of the Financial position of British households, given in the data file [households], examine the statistical and econometric relationships between household debt-to-income ratios and the respondents background characteristics, in particular: social grade; age-group; income; saving and housing tenure. Consider the importance of your results with respect to the impact on household debt-to-income ratios of: (a) an increase in interest rates; and (b) a fall in the rate of price inflation.

My question is is it valid to regress D/Y=f(Y,...) since Y appears on both sides? If not then what would be the way to deal with the problem. I thought perhaps loging the regression so that ln(D) can be regressed on ln(Y) so that an implicit relationship between D/Y and Y can be obtained after the regression.

Any help will be much appreciated.

Answer 1

I would say no because if you use random noise variables for debt and income and then run such a model, you get a very significant result. E.g. in R:

set.seed(1010381)
debt <- rnorm(1000, 1000, 5)
income <- 10^(rnorm(1000, 1, .3))*1000
di <- debt/income

m1 <- lm(di~income)
summary(m1)

gives $R^2$ of about .7 and a p value with 15 0's.