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

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.

• – mkt Jul 26 '17 at 13:40
• A few points in no particular order. 1: subsetting to require DC > 0 does NOT introduce bias. It's just a specific sub region in the population. 2: use the old "add 1" trick is only valid when very few observations are = 0. Certainly you have far to many observations at 0, and so in using the trick you get something which is very difficult to interpret. So, my advice is, don't use the log transformation for this one specific variable - you will loose exactly nothing. And you will save yourself a lot explaining in the final panel – Repmat Oct 8 '17 at 18:10

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.
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).