I am a beginner at econometrics and I have to build a model explaining scholing (as dependent variable) by migration (independent variable) and a couple of covariables. I'm going to use fixed effects to do this.

Now my question is, where and why should I use a LOG? I already read a lot of questions and answers here but it didn't help me any further.

Should I log migration? Government expenditure on scholing? GDP/Capita? Literacy? Ratio of scholing?

• For your examples: probably not (zero and negative values may be meaningful); possibly; possibly; probably not but consider logit if $0$ and $100\%$ cannot occur; it depends on what "ratio of schooling" means Aug 3 '16 at 14:35
• Thank you very much for helping me out here. Why is logit not a good idea of zero or 100% occur? Ratio of schooling is the number of kids who enrol in school divided through the number of children who are in the age group (so who could enrol in scholing). i also has fertility (number of children/women) as a covariable. should i log this? Thank you very much in advance! Aug 3 '16 at 15:06
• You cannot sensibly take the logarithm of a negative, zero or infinite number, but logarithms can make sense for values which are positive and subject to multiplicative growth. Proportions do not meet this (you cannot double 98% literacy), but sometimes logit makes sense (going from 1% to 2% literacy may be as substantial as going from 98% to 99%), so long as you do not apply it to the extremes of 0% or 100% Aug 3 '16 at 15:17
• I think I kind of understand what you mean but I'm still not really sure how to apply this to my model. For example my independent variable (migration) is in the range between 63 and 72000000. So maybe this does make sense to log in this case? Because then I can say 'a 1 percent increase in migration changes the ratio of scholing this much'? But if I understand what you're saying the log of a percent isn't a good idea? For example my dependent variable is the ratio of scholing (between about 50 and 105 percent) and loging this wouldn't make any sense. Aug 3 '16 at 15:33
• So taking the log of migration of women and migration of men and population density (not a percent) and fertility and gdp per capita would be useful because then I could see a percentage change. But logging variables that already are percentages wouldn't make any sense like the literacy rate, the scholing ratio and the government expenditure on scholing as a percent of total government expenditure Aug 3 '16 at 15:34

The use of the natural $\log$ permits to obtain approximations of percentage changes instead of unit changes (assuming linear regression). For instance, the parameter associated with $\log$ of GDP per capita would give you the changes in education (in the unit you measured it, e.g., years) as a result of an increase of 1% in GDP per capita. Notice that if your dependent variable is also a $\log$ transformation, you'll obtain elasticities. To conclude, it should be straightforward for you to understand that there is no reason to apply a logarithm on unordered variables, or on binary variables. Its use on ordered variables is up to you, as the economic signification of choice of the estimates is dependent on your interests.

• Thank you for your response. My dependent variable is a percentage. Does this change the interpretation? Or should I take a log of my dependent variable to interpret as elasticities? Aug 3 '16 at 16:44
• Let $y$ denote the dependent variable (a percentage, between $0$ and $100$) and $x$, the independent variable. If the model is of the form $y=\log x \cdot \beta$, then an increase of $1$ percent in $x$ causes an increase of approximately $\beta$ percentage points (which differ from percents) in $y$. A $\log - \log$ model would have a rather unintuitive meaning in this case (as $y$ is already a percentage).
– rsm
Aug 3 '16 at 17:06
• So it seems that you have something like "percentage of migrants" on as your independent variable. To get a nice elasticity interpretation in a potential log-log model, I would use the total number of immigrants as independent variable. Here the log would make sense. Aug 4 '16 at 6:08