Simple and / or multiple regressions frequently incorporate logarithms into the equation in order to provide stability in the regressors, reduce outliers and establish different views of the estimate, among other applications.
The main utility of logarithms for econometric analysis is their ability to eliminate the effect of the units of the variables on the coefficients. A variation in the units would not imply a change in the slope coefficients of the regression. For example, if we treat prices as a dependent variable (Y) and noise pollution as an independent variable (X).
To see the above more clearly, let’s imagine that we have a variable in euros and another in kilos. If we pass the two variables to logarithms, we will have them measured in the same ‘units’ and therefore our model will have more stability.
We can find natural logarithms, (ln), where the base is e x , and logarithms of other bases, (log). In finance, the natural logarithm is used more by considering e x to capitalize continuous returns on an investment. In econometrics it is also common to use the natural logarithm.
Logarithm Considerations in Econometric Analysis
Another advantage of applying logarithms over Y is its ability to narrow the range of the variable by a smaller amount than the original. This effect reduces the sensitivity of the estimates to extreme or atypical observations, both for the independent and dependent variables. Outliers are data that, as a result of errors or due to being generated by a different model, are quite different from most other data. An extreme example would be a sample where the majority of observations are around 0.5 and there are a couple of observations with values of 2.5 or 4.
The main characteristic that we look for from the variables so that we can apply logarithms is that they are strictly positive quantities. The most typical examples are salaries, the number of sales of a company, the market value of companies, etc. We also include the variables that we can measure in years, for example, age, work experience, years of teaching, length of service in a company, etc.
Normally, in samples containing large whole numbers of elements, logarithms have already been applied and are presented transformed to facilitate their interpretation. Some examples of variables where we can apply logarithms would be the number of students enrolled in educational institutions, Spanish intra-community citrus exports, population of the European Union, etc.
Variables that are represented by proportions or percentages can appear in both ways interchangeably, although there is a generalized preference for use in their original state (linear form). This is because the regressor will have a different interpretation depending on whether or not logarithms have been applied to the regression variables. An example would be the annual growth of the consumer price index in Spain. The adjacent table lists the different interpretations of the regressor, in this case, a simple regression.
Interpretation of logarithms in econometrics
Here is a summary table of how logarithms are calculated and interpreted in an econometric regression model.
We are going to explain it in a simpler way, so that it is better understood.
- The Level-Level model represents the variables in their original form (regression in linear form). That is, a change of one unit in X affects Y by β 1 units.
- The Level-Log model is interpreted as an increase of 1% change in X is associated with a change in Y of 0.01 · β 1 .
- The Log-Level model is the least frequently used and is known as the semi-elasticity of Y with respect to X. It is interpreted as an increase of 1 unit in X is associated with a change in Y of (100 · β 1 )%.
- The Log-Log model is attributed to β 1 the elasticity of Y, with respect to X. It is interpreted as an increase of 1% in X is associated with a change in Y of B 1 %.