Converting results from percentage point to percent

[Joe Zonda]

Hello STATALISTERS,
I am estimating the effects of a program using the fixed effects model in which treatment effect is given as a = Y(I)-Y(0), where Y(I) and Y(0) are respectively the observed and unobserved outcomes.
In my specification, the outcome variable is in log.
Question:
(i). Would it be correct to interpret the treatment effect, a, as the percentage points>
(ii). If yes to (i), how can convert it to percent?

My question may be pretty basic, but it is crucial for me to be sure I am doing the right thing.
Thank you!

[Clyde Schechter]

You need to be very clear in your use of language here. When you say the outcome variable is "in log," I suppose you mean that Y, the dependent variable in your model, is the log of some other variable Z. Y = log(Z).

The effect a = Y(1)-Y(0) is then the absolute difference in Y. It is dimensionless, just as Y itself is because logarithms are inherently dimensionless: in particular, a is not percentage points. Now, since Y = log(Z), a = log(Z(1)) - log(Z(0)). Recall from high school algebra that the difference of logarithms is the logarithm of the ratio. So a = log(Z(1)/Z(0)). So if what you are interested in is the ratio Z(1)/Z(0), the relative change, then that is estimated by exp(a). If you want to express that ratio is a percent difference, then the percent difference would be 100*(exp(a)-1).

Now, there is a widely used approximation that goes like this: if a is close to zero, then (exp(a)-1) is approximately equal to a. So people often take the difference a and just multiply it by 100. I can't emphasize enough, however, that this is only a reasonable approximation if a is close to zero. For most practical purposes, this means |a| < 0.1. If the magnitude of a is larger than that, you should not use the approximation but should go through the 100*(exp(a)-1) calculation to get the correct value. (And if you need more than one decimal place accuracy, even |a| < 0.1 isn't quite good enough.)

[Joe Zonda]

Clyde Schechter you are right, I was not very clear in my explanation as to the meaning of "outcome variable being in log".
So basically, it is log(Y_it) =δ_i +ξ_t +β X_it +ε_it , i.e I take the log of Y.

[Clyde Schechter]

OK, so, what I said in #2 applies, it's just that what I called Z is itself Y, and what I referred to as Y is really log Y.

So, the relative change in Y (ratio Y(1)/Y(0)) is given by exp(a), and the percent difference is 100*(exp(a)-1), which you can approximate as 100*a if a is close to zero.

[Joe Zonda]

Perfect. Thank you so much Clyde Schechter for your vivid explanation.