0

I notice that after normalizing a variable by firm size (to control for firm size), the variance of that variable goes up. I was expecting the variance to go down in fact. Does this mean that I should not normalize by firm size?

kjetil b halvorsen
  • 77,844
singhalc
  • 13
  • Normalizing by firm size seems a strange thing to do! You should better state your real modeling goal, as this could really be a version of XY-problem. Just maybe https://stats.stackexchange.com/questions/142338/goodness-of-fit-and-which-model-to-choose-linear-regression-or-poisson/142353#142353 could help! – kjetil b halvorsen Aug 08 '19 at 08:08

1 Answers1

0

Let me restate your question, to make sure I understand correctly and make my notation clear. It looks like you measure some random variable for each firm (let's call it $X$), as well as the size of each firm (let's call it $S$). You note that you calculated $Var(X)$, but when you calculated $Var(\frac{X}{S})$, you found that $Var(\frac{X}{S}) > Var(X)$.

It's worth noting that in the general case, this is not very strange. The source at [0] gives an approximation for the general case of $Var(\frac{X}{S})$, in formula (20), which helps us see that. If the variance of $S$ is high and the covariance between $X$ and $S$ is small, then it could simply be that $Var(\frac{X}{S}) > Var(X)$.

As to your question of whether or not you should use this value, that depends on your use case. If the quantity $Var(\frac{X}{S})$ has an important meaning for your business, then there's not really much to be done, unfortunately. But there's nothing obviously "wrong" with the fact that the variance increases, as I understand you (if you'd like to provide more information about your thinking, perhaps we could dig further into it).

[0] http://www.stat.cmu.edu/~hseltman/files/ratio.pdf

Louis Cialdella
  • 1,567
  • I have one more question - If I am normalizing (dividing by firm size) all of my independent variables, then do I need to do so for the dependent variable too? – singhalc Jun 08 '17 at 15:54
  • There's no hard and fast rule there - it depends on whether the normalized or unnormalized version of the dependent variable will be easier to interpret in your analysis. – Louis Cialdella Jun 08 '17 at 21:01