# Using coefficient of variation as the DV in OLS regression

I would like to analyze income mixing in the neighbourhoods of a group of cities. One way to think of neighbourhood income mix is as the spread of incomes found in a neighbourhood. I have census variables for average income, median income, standard error of average income, and other sociodemographic characteristics. The data are aggregated at the neighbourhood level (I don't have access to microdata with information about individuals at this small scale). I would like to do OLS regression using the coefficient of variation (std.error of income divided by average income) as the dependent variable. I have three questions about this:

1. If I wanted to see whether wealthier neighbourhoods, controlling for other characteristics, have more or less income mixing than less wealthy ones, does it make sense to use the CV of income as the dependent variable and median income as an independent variable?
2. To use it as a dependent variable, I would be transforming the CV by taking its square root. How should I interpret the regression coefficients? For example, if a coefficient is significant and equal to 0.3, is it correct to say that 0.3 to the power of two, or 0.09, is the expected change in income CV for a one unit increase in X? What if the coefficient is negative?
3. Are you aware of published studies that use CV as the dependent variable?

1. It might make a little more sense, bringing things into more intuitive units, if instead of standard error you used standard deviation in the CV numerator. [S = SE*sqrt(N)]

2. I'm not seeing much reason to take the square root of the CV, regardless of whether you use SE or S in its numerator. But the answer to your question might be found at this excellent thread.

3. I have a vague sense of having seen this done in an archaeological context (how standardized were a given culture's clay pots?) but couldn't cite any sources.

• Thanks rolando2, I'll see what I can find in the archeology literature. Converting SE to std dev is a good suggestion.
– Argi
Sep 24 '12 at 16:14

I think you are going to run into some problems here. The wealthiest neighborhoods, almost by definition, are composed almost entirely of wealthy people; similarly, the poorest neighborhoods are going to be composed almost entirely of poor people. It seems that there will be, almost as a necessity, less mixing in those neighborhoods.

If you get around that, and you have data on the percentiles of income, you could use some variation on the GINI coefficient in each neighborhood. I think these are more understood and accepted than CV is.

Another possible problem with CV is that income is likely to be very skewed. How will CV behave under such circumstances? Might it make sense to take CV of log income? (It often does make sense to take log of income, for various purposes, I am not sure about whether this is one of them).

• Peter, I don't have individual data to take the CV of log income, but it would be interesting to see what that does! Your GINI coefficient idea makes a lot of sense; I'll check whether the percentile income data are available. Thanks a lot for your suggestion.
– Argi
Sep 24 '12 at 16:21