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There is a similar question posted but this is not what I am looking for: Interpreting coefficients when dependent variable is a fraction/proportion?

I was reading a paper that ran a difference-in-differences regression and the coefficient value was -0.036. The dependent variable is vote shares and none of the variables were logged. In this case, is the coefficient just a percentage? So the impact of X on Y is a decrease of 3.6% of vote shares?

In addition, for simple OLS regressions, are coefficients usually percentages? Sometimes people say percentages and sometimes percentage points.. and this has been giving me a headache.. Can anyone help me clarify please?

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Talking about percents always causes confusion. For instance, if the starting value of $10\%$ undergoes an increase of $10\%$, there are legitimate claims of saying that the ending value (after the increase) is $11\%$ and $20\%$. For the latter, I would prefer to describe it as an increase of β€œ10 percentage points,” but this is far from a universal practice.

A related question here on Cross Validated

With that out of the way, if the model is linear, the it is just a matter of taking partial derivatives and interpreting them as usual in calculus. If the derivative is $-0.0036$, this means that you expect a value $0.0036$ lower than it would have been for a predictor value one unit lower (the usual interpretation of a linear regression coefficient). In other words, this would be a decrease of 3.6 percentage points, to use my terminology (not to claim that I invented the terminology, but I do use it).

Don’t overthink it. The regression coefficients are telling you exactly what they always tell you.

(A drawback of this kind of model, however, is that it can result in predicted probabilities outside of $[0,1]$, but it is on the author to defend the decision to allow for this. Your interpretation of the coefficients does not depend on this.)

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  • $\begingroup$ Thank you for your comment! One other question I have is, if you look at this post (stats.stackexchange.com/questions/549830/…) the answer says that if a dependent variable is logged transform, then b is the % change in y when x changes by 1 unit. However, isn't that incorrect? I thought that we have to exponentiate the coefficient and subtract 1 and multiply by 100 to get the percentage.. $\endgroup$ Commented Oct 31, 2022 at 6:08
  • $\begingroup$ its approximately correct (for small coefficients). if we define $z$ as your log transformed value, then your proportion change is $\exp(z + b)/\exp(z) -1=\exp(b) -1\sim b$ by Taylor expansion of exponential. $\endgroup$ Commented Oct 31, 2022 at 11:09
  • $\begingroup$ @seanv507 Thanks for your comment! But I meant the raw b value. So if the beta value is 0.36, then you're saying that as long as the coefficient is small enough, I can interpret it as if x changes by 1 unit y changes by 0.36%? $\endgroup$ Commented Oct 31, 2022 at 17:27
  • $\begingroup$ @seanv507 Also, by looking at exp(𝑧+𝑏)/exp(𝑧)βˆ’1=exp(𝑏)βˆ’1βˆΌπ‘. it seems like you are talking about if the z value is small enough rather than b? which is the coefficient. (My math is rusty.. so sorry if I'm making dumb questions.. but really appreciate any help!) $\endgroup$ Commented Oct 31, 2022 at 17:27
  • $\begingroup$ y changes by 36% (not 0.36%) $\endgroup$ Commented Oct 31, 2022 at 18:05

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