Formula for weighted covariance in Stata (correlate command)

[Hammad Shaikh]

I have come across research papers where they compute the weighted covariance between two variables. For example covariance between X and Y in year t is weighted by the population size. When I look at the Stata code, the authors would use:

correlate X Y [aweight=w], covariance

The stata correlate manual has a example with real data as well:

webuse census13
correlate mrgrate dvcrate [aweight=pop], covariance

I wanted to know what the formula for the weighted covariance is in Stata. I think the weighted covariance formula is:

sum(w_t(x_it - xbar)(y_it = ybar)) where w_t = w/sum(w),

and xbar, ybar are also weighted using w [so for example xbar = sum(w_t x_it)/sum(w_t)]. I have tested out some examples and this formula closely matches what is obtained from Stata. I wanted to ask whether this is what Stata implements or is it some variant?

[John Mullahy]

Hammad: Have you looked at the "Methods and Formulas" section of the pdf documentation for correlate ? I think this will give you the information you are seeking.

[Hammad Shaikh]

John, thanks for the reply. I just checked it, the formula for the weighted correlation is indeed there. I had missed this before, thanks I know what the weighted covariance formula is now. My initial guess was correct.