# How can I interpret intercept when my dependent variable is in log form?

My model consists of both log independent and dependent variables, and percentage share, as well as number of people and etc. My dependent variable is log(GPD per capita in USD), my statistic package output said "7.9" I would like to ask in this case, how can I interpret the constant term? because it is significant at 1 percent, although I remember asking one of my instructor about necessity of interpreting intercept for linear model. I was told that it is not necessary to do so, however, since I saw that it is highly significant and only a few of my independent variables give significant results. I thought interpreting constant is helpful to know if all independent being set to zero, how much the base GPD per capita in USD will be. Could anyone who is acknowledged about converting this number into absolute USD ? Thank you!

An intercept can be interpreted as the average value of your outcome (dependent variable, in your terms) for the subgroup of observations defined such that your covariates (independent variables, in your terms) are equal to 0. In your case, the outcome is the average of the log GDP per capita, and the estimate of the intercept is 7.9. The corresponding value in levels is exp(7.9), which is roughly equal to 2700 USD per capita. Note that, because the average of the log is not the log of the average, you cannot say that the average GDP per capita on this subgroup is equal to 2700 USD per capita.

Interpreting this intercept is only desirable when you choose your covariates in a way that setting them to zero makes sense. For instance, it makes sense to set to zero a covariate that is a dummy for the country being landlocked. It would make less sense to set to zero a covariate like life expectancy. If you want to interpret the value of your intercept, I would advise to choose your covariates with care, or to transform them (by subtracting the mean, for instance) in a way that setting them to zero results in a sensible situation.

Significance tests the equality of the coefficient to 0. A very significant estimate means that you can reject the null hypothesis that this coefficient is equal to 0. In your case, this is not very interesting, because you would expect that some average of log GDP per capita should always be greater than 0.

• "This converts into exp(7.9)" isn't really right. The average value of the log response may be 7.9, but that implies the average value of the response (for the same settings of the explanatory variables) must be greater than exp(7.9). It would also help to clarify that by "covariates ... equal to 0" you mean the logarithms of the original independent variables are 0; that is, the original variables are all set to $1.$ – whuber Aug 19 '18 at 14:54
• Sorry if this point was unclear. We agree that the average of the log is not the log of the average: I edited the answer to avoid potential confusion. – Roland Aug 19 '18 at 17:38

As this page describes, the intercept for a linear regression model is the predicted value of the dependent variable when all predictor (independent) variables are 0 (for continuous predictors) or at their reference levels (categorical predictors). But these are the values after transformations were applied to predictors in their original scale. For log-transformed predictors, this is the situation when their log-transformed values are 0 and thus when their original untransformed values are 1.

It thus seems unlikely that the intercept represents any value of practical interest in this case. Its “significance” simply means that the log-transformed value of US GDP at those baseline predictor values happens to be significantly different from 0 with the way that you structured the regression. Different choices of reference levels and pre-scaling or transformations of continuous predictors could give a non-significant intercept for a model of the same structure and that gives the same prediction for variables n their original scales.

The value for the intercept in the original scale, should you nevertheless choose to calculate it, is the anti-log of your value of 7.9 in whatever log base you used: for natural log, exp(7.9); for log10, 10^7.9; for log2, 2^7.9 .