Interpretation of a lin-log model with share dependent variables

[Ignacio Belloc]

Hi all!

I am running a linear regression model where the dependent variable refer to a proportion variable --so it ranges from 0 to 1-- and the main independent variable is log-transformed. How would you interpret the main coefficient of interest from the estimates? For instance, a coefficient equal to -0.21 of the log-transformed regressor would imply that a one percent increase in that variable is related to a decrease of 0.21 units in the proportion variable? So this would imply a reduction of 21 percentage points related to an increase of one percent in the independent variable?

Many thanks in advance!

[Clyde Schechter]

Let's call your untransformed independent variable x and your outcome y so we can write some equations.
If x is increased by 1%, that means x changes to 1.01*x. So log x changes to log(1.01*x) which = log 1.01 + log x. In other words, log x has increased by log 1.01. And log 1.01 - 0.00995 (to 5 decimal places).

With a coefficient of -0.21, it follows that y will decrease by 0.21 * .00995 = 0.0021 (to four decimal places). So this is a decrease of 0.0021 units in the proportion variable, or, in the proportion metric, it is a decrease of 0.21 percentage points.

[George Ford]

A 1% change in X means a 0.0021 change in the level of the DV.

If constrained to the unit interval, then use fracreg logit.

clear all
sysuse auto, clear

summ mpg price
replace mpg = mpg/41  // set to unit interval
g lprice = ln(price)
reg mpg lprice
margins, at(lprice = (8.7266851 8.7166851)) post  //a 1% change
di "Diff = " e(b)[1,2] - e(b)[1,1]

fracreg logit mpg lprice
margins, at(lprice = (8.7266851 8.7166851)) post 
di "Diff = " e(b)[1,2] - e(b)[1,1]

[George Ford]

Clyde beat me to it, but his answers are better anyway.

[Ignacio Belloc]

Thank you very much, Clyde Schechter and George Ford, for your quick and helpful responses.

I have two additional questions:

1) If the dependent variable is a percentage rather than a share/proportion (i.e., it ranges from 0 to 100) and we get a coefficient for a log-transformed variable equal to -21 (note that this coefficient is 100 times the previous coefficient in my initial example), how do I interpret this coefficient? Y will decrease by 21 * 0.00995 (log1.01) = 0.2089 units. So a decrease of 20.89 percentage points in the percentage variable. However, this conclusion is not similar to what you previously mentioned. Should I divide the coefficient -0.2089 by 100 to interpret it in terms of unit changes in Y? And then it would imply a reduction of 0.21 percentage points.

2) For dummy variables in models where the dependent variable is a proportion (a continuous variable from 0 to 1), can I interpret the estimates in terms of units of changes in the proportion variable?

[Clyde Schechter]

However, this conclusion is not similar to what you previously mentioned.

Yes it is. A Y decrease from, say 52.2089% to 52.0000% is a decrease of .2089 percentage points, not 20.89 percentage points.

2) For dummy variables in models where the dependent variable is a proportion (a continuous variable from 0 to 1), can I interpret the estimates in terms of units of changes in the proportion variable?

I'm not entirely sure what you're asking here. If your independent variable is dichotomous, coded 0/1, then there will be no log transformation possible (can't take log of 0). If, in addition, the dependent variable is a proportion, you have a simple linear-linear model. So the coefficient of the dichotomous independent variable is equal to the expected difference in the outcome proportion between the two levels of the dichotomous variable. If the coefficient is -.21 it means that the expected outcome difference bewteen the two categories of the dichotomous variable is .21 (with the higher outcome value in the 0 category of the independent variable).

[Ignacio Belloc]

Dear Clyde Schechter,

Thank you very much for your responses.

Best regards,
Ignacio