Interpreting log-log regression results where the original values of one IV have all been increased by 100%

Question

Sorry for the very cryptic title, but I do not know how to describe my problem in another way and I'm sure the solution is rather straightforward.

I am currently running a regression where the dependent and independent variables are transformed to natural logarithms. The problem I have is that one IV, DC-ratio, has a lot of values of 0% and log(0) is undefined and thus results in a missing value in Stata. If I use this value, Stata will drop the missing values and my sample would be biased since it only uses the funds which actually have a DC-ratio >0%. This is an issue as more than half of my dataset have a DC-ratio of 0%. My idea was to turn all values of 0% DC-ratio to 0.0000001%.

My councelors solution, which according to him is often used in research, is to add 1 to all values of the DC-ratio, turning 0% DC-ratio in 100% and 30% in 130%. I have followed his solution.

Besides the fact that I am wondering is this is at all possible when in the original values a DC-ratio of 100% is the maximum, I can't figure out how to interpret the results. Do I need to subtract the +1 somewhere?

I am familiair with interpreting the log-log results, but I can't figure out how to deal with this adjustment of original values. In the picture (regression results), regression number 9 is preferred and I know that a 1% increase in the DC-ratio will lead to an increase of 1.01^7.126= 1.0735-1= 7.35% increase in the geometric mean of the dependent variable. But I can tell you right now that this does not make any sense in an economic context.

I hope someone could shed some light on this problem.

Many thanks in advance!

Answer 1

To answer your specific question, you just undo the transform, just as you did with variables you take a log of. If one of the regression variables $X_1$ was log transformed, and you get a good linear fit with the coefficient a, then $e^a$ is the 'real' value. In other words, if the regression $Y = a*ln(X_1) + c$ gives $a=2$ it means that $Y = ln(X_1)$ increases additively by 2 units for each increase of $ln(X_1)$ by 1. This means that Y grows by multiplying it by $e^2$ each time $X_1$ grows by one unit. (Note I am doing a semi-log model, not a log-log one here.)

Interpreting adding something and taking a log is no different, just not intuitive. It means $Y = ln(X_1+k)$ grows by adding a for each unit growth of $X_1$, so $X_1+k$ really grows by multiplying it by $e^a$ for each unit growth in $X_1$ (since a unit growth in $X_1+k$ is really just a unit growth in $X_1$, since k is a constant). But I am not sure what that means, for exactly the reason you state. Log transforms are done to make ratios into differences, and adding before taking the log destroys that relationship. I would try your idea of using .00001, because that, in a sense, has some linkage to reality and avoids the complexity here.

However - and I am making an educated guess about what you are doing with this regression - it might make more sense to have (or also have) a dummy variable for the data points with zero value. Sometimes such a degenerate value is the result of a 'corner condition', not part of a linear (or log-linear) regime. For example, if you were doing this with stock data and one of the variables is dividends, there are a good number of companies that pay no dividends - and.

Of course, there are no negative dividends, so looking for a linear relationship directly or after log transforms misses this. Most companies that pay no dividend do it because they choose not to pay dividends, not because they only have a few pennies of cash and thus couldn't pay any dividends.

In an ideal world, it would reduce the relevance or impact of the $ln(DC-ratio)$ so assumptions there don't matter (or don't matter as much).