Diffrence in logs vs. a % changes in econometrics: why is the dif log approvimation almost always used when the exact quantity is easily available?

Question

I have observed that in econometrics work people almost always use the difference in logs rather than the actual percentage change. This makes no sense to me. I understand that the difference in logs is a pretty good approximation provided the (PERCENTAGE) change is small, and more or less why. But it is still only an approximation. And sometime your data includes large jumps that you might not know about. Given that your data is already entered into a computer, calculating the actual percentage changes is the work of seconds. It's not like anyone is out there doing econometrics with a slide rule. Is there some advantage in the diffrence in logs above the actual percentage changes that I do not know about?

Answer 1

I could not immediately find this question already answered, the closest being the comments here.

The key reasons why it's better to work in the log scale are outlined there:

1. log changes are additive, percent changes are not. +10% followed by -10% will not put you back where you started, whereas +0.095 followed by -0.095 in the natural log scale will.
2. Percent change is truncated at -100%, it is not possible to achieve anything lower. Linear regressions don't take this into account, they usually work on the normal distribution with infinite support, which will lead to issues with model fit as you approach the domain limit (e.g. confidence bounds or even estimates that are below it). This is also the reason why other bounded outcomes, such as counts (>=0) or proportions (0-1) are almost never fit in the original scale but log/logit-transformed instead.

Both of these tie to a deeper issue with interpreting percent change, especially the larger values you refer to, in that they have an inherent multiplicative symmetry: -50% ($\times 0.5$) is the inverse of +100% ($\times 2$), and going from the reference level to -90% ($\times 0.1$) is the same as going from -90% to -99% ($\times 0.1$). This is why the log scale is a very natural way to express it, and I shed a single tear every time I see percent change plotted on a linear axis.

In closing, it is the percent change which is approximated by the log difference when both are small; for larger values the log change you draw from a model will be much better behaved than the percent change. Nothing stops you from back-transforming these log estimates to fold (my preference) or percent change, but you have to keep in mind the above points on interpretation.

Answer 2

Many econometricians/statisticians view the "force of growth" as the most natural measure and therefore see the percentage change as the approximation

The reason for this usage is that logarithmic difference gets you the "force ofgrowth" under continuously compounded growth, which is perhaps the most natural way to measure growth in a quantity over time. (In different econometric contexts this may have a more specific name such as the force of interest, etc.)

If you have a nominal growth rate $r$ and you compound the growth $n$ times per time-unit for $t$ time units then the total growth $p_n$ over the time period satisfies:

$$1+p_n = \bigg( 1 + \frac{r}{n} \bigg)^{nt}.$$

Continuous compounding occurs when we take $n \rightarrow \infty$ which then gives:

$$1+p_\infty = \exp(rt),$$

and $r$ now represents the "force of growth". One of the nice things about measuring using the force of growth is that it is comparable for things growing over different time periods. The total force of growth over $t$ time-units is $rt$, which is a linear function of the amount of time elapsed so the rate $r$ has a natural meaning that is easily comparable across time periods of different lengths. If you start with an amount $A_0$ at some time and then this grows to $A_t$ after $t$ time-units then you have $A_t = A_0 (1+p_\infty) = A_0 \exp(rt)$, which then gives:

$$rt = \log(A_t) - \log(A_0).$$

Suppose you have a series of values $A_0,...,A_k$ over times $0,t_1,...,t_k$. One of the nice things about this result is that the total growth rate using the "force of growth" is additive:

$$r t_k = \log(A_k) - \log(A_0) = \sum_{i=1}^k [\log(A_i) - \log(A_{i-1})].$$

The fact that the total growth rate under the force of growth is additive and represents continuous compounding of growth makes it a natural measure for growth. There are many mathematical benefits to framing growth in terms of the "force of growth" (continuously compounded) instead of the percentage change occurring over a particular period. In particular, it leads to additive growth rates and it also covers the entire real number line without any lower bound.

Your question amounts to asking why we would look at $rt$ instead of looking at $p_n$, with the former asserted to be an approximation of the latter. However, it is actually somewhat more natural to see the force of growth as the important measure, such that the percentage change in $p_n$ is actually the approximation to the more important quantity.