Population weighted vs. Population as a regressor

[Ariel Karlinsky]

Suppose you have country-year (panel) data. In each country there's some outcome variable Y, some time varying treatment variable T and some time-varying control variables X.

In some papers, I see that the estimated model for the question at hand includes population as one of the control variables. In some, variables are transformed to a per capita basis, and in others - each country's population are used as weights in the regression.

I was unable to find some succinct explanation for the differences and assumptions underlying each of these options.

[Phil Bromiley]

There are literatures on normalizing and per capita in sociology and political science.

I believe Robert Wiseman at Michigan State wrote on normalizations - his paper might give you an entre to the literature. Wiseman, R.M. (2009). On the use and misuse of ratio variables in strategic management research (pp. 75-110). In D. Ketchen & D. Berg (eds.), Research methodology in strategy and management, vol 5. San Diego: Elsevier JAI Press.

Some suggest that you should normalize everything if you're going to normalize (Kennedy or Verbeek's text says this I think). Like some apparently harmless choices, moving from controlling for population to per capita can dramatically change the results.