I am regressing per capita health expenditure on per capita GDP. I have 3 data columns (health expenditure, gdp, population)

So my regression function is healthexpenditure/population = b0 + b1.(gdp/population)

1) My question will be what are the possible problems of this model? Is dividing dependent and independent variable with population number problematic? This is a theoritical question, I want to explore possible biases in my simple model.

2) Suppose that I have plotted my data and it looks like this. Do you see a problem that are related? enter image description here

  • $\begingroup$ Regarding your first question, if you divide both sides of an equation by the same constant, the relationship will not change. So the slope of the relationship between your dependent and independent variables will not change. What will change is the value of the intercept term, if you have one in the regression equation. $\endgroup$
    – AlexK
    May 2, 2019 at 7:09
  • $\begingroup$ hmmm. suppose that i have two models, suppose that my first model is on per capita basis (y and x are divived by population), my second model is not on per capita basis (y and x are not divided by population). which is a better model to use in terms of economic intuition? $\endgroup$
    – Ramsay
    May 2, 2019 at 9:19

1 Answer 1


There could be a question of whether your model and graph should be based on a linear relationship or a log-log relationship

The slightly widening residuals apparent in your linear graph seems to understate the real issue illustrated in a linear-axis graph from Gapminder. The green outlier at the top is the USA while the small red outliers on the right are rich but small Gulf and SEAsian countries, and the presence of both of these could exert excessive leverage on your regression coefficients

enter image description here

This left-right disparity in residuals does not look so bad with the same Gapminder data plotted with log-log scales. It also spreads out the big cluster of developing countries in the bottom-left corner between those which are extremely poor and those which are not so poor and so allows their relative difference to influence your model estimates

enter image description here

  • $\begingroup$ thanks for the answer. i am wandering that, in theory, what are possible outcomes of dividing your dependent and independent regression variables with another data set (such as population)? so you claim that if the relationship seems to be appropriate (whether it is a linear or log-log), there is no problem? my second theoritical question, how about dividing all of multiple regression with X1? $\endgroup$
    – Ramsay
    May 2, 2019 at 9:11
  • $\begingroup$ @Ramsay My graphs already divide by population on both axes $\endgroup$
    – Henry
    May 2, 2019 at 9:51
  • $\begingroup$ Channelling George Box: All regressions have problems; the question is how critical these are $\endgroup$
    – Henry
    May 2, 2019 at 9:59

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