Interpreting coefficients (percentage points vs percentage)

[Jun Park]

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. The author wrote that the impact of X on Y is a decrease of 3.6% of vote shares.
Is this interpretation correct?.. I thought the coefficients are percentage points not actual percentages.. This is an article published at a top social science journal and I feel like I'm the one missing something here.

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?

[Hemanshu Kumar]

When the dependent variable is in logs, then the marginal effect is in terms of percentages; when it is linear then the marginal effect is in the units of the dependent variable. If the dependent variable is in percentage terms in a linear specification (e.g. the dependent variable is the interest rate or vote percentage), then the marginal effect is in "percentage points".

When the readers are expected to know this well, such as the target audience of a top social science journal, the authors will feel okay with not explicitly saying "percentage points". So a change of -3.6% in vote shares is taken to mean a change from say 20% to 16.4%.

[Jun Park]

Hemanshu Kumar

Thank you so much for your reply! One addition question I have is from the following paragraph from another post.

Only the dependent/response variable is log-transformed. Exponentiate the coefficient, subtract one from this number, and multiply by 100. This gives the percent increase (or decrease) in the response for every one-unit increase in the independent variable. Example: the coefficient is 0.198. (exp(0.198) – 1) * 100 = 21.9. For every one-unit increase in the independent variable, our dependent variable increases by about 22%

You mentioned above that when the dependent variable is in logs, the marginal effect is in terms of percentages. Did you mean by taking into account exponentiating the coefficient and subtracting 1 and multiplying by 100? From the above example, if we were to not exponentiate it, how could we interpret 0.198?

[Hemanshu Kumar]

Yes, this is the correct procedure. The raw coefficient is an approximation to the percentage increase, for small values.

Suppose the coefficient were 0.02. Then the percentage increase is approximately 2%. The more exact calculation would give you (exp(0.02)-1)*100 = 2.02%. So the approximation was fairly good.

For higher values, the approximation is worse. For 0.198, the true value as you found was 21.9%. For a coefficient of 0.80, the true value would be 122.55% instead of 80%.

[Jun Park]

Hemanshu Kumar

Oh.. That makes so much sense now... I was having a hard time when authors would use the raw version or the exponentiated version interchangeably..
So the only time when we can use the raw version is when the "Coefficient itself is small" right? There isn't any other criteria we can use? for example, the size of the observation.. etc
And also, is there a rule of thumb to argue whether the coefficient is small enough to use the raw value?

Thanks!

[Hemanshu Kumar]

Whether the approximation is good enough is for you to judge based on the application, the conventions in your field, etc. E.g. is the difference between 2% vs 2.02% likely to matter to someone?

In the areas I work in, below 0.10 is one oft-used threshold for the approximation being good enough. Your mileage will vary.