Measuring Change with Log Differences

Measuring change over time can be complicated.  One of the most common methods in use is “percent change”, which subtracts one number from another, and then divides the result by the quantity chosen as the base.  For example, the percent change between the quantity from a given time period and the quantity from the prior period may be calculated as follows:

Unfortunately, this method is not “symmetrical”, which means that calculating the percent change going forward from a prior period to a subsequent period is not the same as calculating the percent change going backward from the subsequent period to the prior period.  To illustrate, suppose that the value of the prior period is 100 and the value of the subsequent period is 125.  The percent change from the prior period to the subsequent period is 25% (i.e. the quantity increases by 25% from its base value of 100), but the percent change from the subsequent period to the prior period is -20% (i.e. the quantity decreases by 20% from its base value of 125).  This lack of symmetry means that extra care must be taken to ensure the appropriate base is chosen for the calculation of this metric.

An alternative to using percent change is “log differences”.  In this method, the logarithm of one quantity is subtracted from the logarithm of another quantity. The advantage of measuring change in this way is that the calculations are symmetrical going forward and backward.  The log difference in one direction is simply the additive inverse of the log difference in the other direction. (A number added to its additive inverse equals zero.)  Furthermore, the sum of the log differences over an entire series of numbers is the same as the log difference between the last observation and the first. 

Consider the following table, which offers the example of a series with five observations of x and their corresponding natural logarithms, along with calculations of change, percent change, and log difference from prior period:

Here we see that the log difference calculated in one direction (i.e. between x5 and x1) is the additive inverse of the log difference calculated in the opposite direction (i.e. between x1 and x5), so the measure of change is symmetrical.  Also, the sum of log differences of the entire series {x1, x2, x3, x4, x5} is the same as the log difference between the last and the first observation, which means that we don’t need to calculate each individual log difference if we are only interested in the net change across the whole series of observations.

Reference:  https://en.wikipedia.org/wiki/Logarithm