# How to interpret a regression coefficient of an independent variable expressed as a percentage?

I fitted a continuous dependent variable against two explanatory variables. However, one of the explanatory variables is expressed as a percentage. Consider the regression model: $$y = 0.05 + 0.032x_1 + 0.024x_2,$$ where $$x_2$$ is expressed as a percentage. How can one interpret $$\hat{\beta}_2=0.024$$?

The interpretation is not very different from the normal regression interpretation: a 1 unit (in this case 1 percentage point) increase in $$x_2$$ is associated with a 0.024 unit increase in $$y$$.
This is easy to get wrong. A $$1$$ percentage point ($$1\%p$$) increase in $$X_2$$ is associated with a $$0.024$$ unit increase in $$Y$$. That is, if two observations have equal values of $$X_1$$ while their $$X_2$$ values differ by $$1\%p$$, the estimated expected difference in their $$Y$$ values is $$0.024$$.
Note the subtle difference between percent and percentage points: values $$50\%$$ and $$51\%$$ differ by $$1\%p$$, yet at the same time the latter is $$2\%$$ larger than the former $$(\frac{51\%-50\%}{50\%}=0.02=2\%)$$. Here is a related thread.
• @dipetkov, after mkt's edit, the answers say the same thing. When I posted my answer, they were saying different things. Regarding how natural something is, I think it varies from person to person. I have encountered plenty of cases where people were mistaken due to missing the subtle difference between $\%$ and $\%p$. Commented Aug 29, 2022 at 12:08