Ratio Variables in Regressions

I was reading Gibbs and Firebaugh (Criminology, 1990) on ratio variables in regressions. Suppose you regress Arrests/Crime on Crimes/population using city-by-city data, and in fact there is no causal connection. Will they be negative correlated anyway, since CRIMES is in both variables?

No, so long as all relevant control variables are in the regression. Here is a way to see it. Suppose we regress 1/Crime on Crimes/Population. Suppose too, that Crime and Crimes/Population are uncorrelated--- that bigger cities do not have a higher crime rate. Then 1/Crime and Crimes/Population will be uncorrelated.

If, of course, bigger cities do have higher crime rates, then 1/Crime and Crimes/Population will be correlated, but if we suspect that to be true, then in our original regression we should have regressed Arrests/Crime on not only Crimes/Population but on the control variable Crimes.

There is some issue of measurement error-- of false correlation arising if Crime has measurement error. Then we are regressing Arrests/(Crime+Error) on (Crime+Error)/Population. I think if we use (Crime +Error) as a control variable that will fix the problem, though.