Why is asymmetry in percentage change a problem when analyzing time series?

Question

When analyzing time series, a benefit of log difference, $log(y_t) - log(y_{t-12})$, is that it is symmetric, unlike percentage change, $\frac{y_t-y_{t-12}}{y_{t-12}}$.

My question is —— why is asymmetry an issue? Or how is log difference helpful in ways that percentage change is not when analyzing a time series (more specifically, time series of economic indicators and stock index)?

To start, I looked into this paper but remained confused. https://www.jstor.org/stable/2683905?seq=1

"Avoiding mistakes" is not satisfying as a reason, and why is having an "additive identity" significant?

Thank you!

Answer 1

Let $y_t$ the $t^{\textrm{th}}$ value of the time series. Suppose that $\frac{y_{t+1}}{y_{t}}=c$. Then the relative change is $\frac{y_{t+1}-y_{t}}{y_{t}}=c-1$. Now suppose that $y_t$ is larger than $y_{t+1}$, with $\frac{y_{t+1}}{y_{t}}=\frac1{c}$. Here the time series is decreasing. In this case the relative change of is $\frac1{c}-1$, which is not the negative value of $c-1$.

We keep the assumtion that $\frac{y_{t+1}}{y_{t}}=c$. Then the change in terms of logs is $$\ln\left(\frac{y_{t+1}}{y_{t}}\right)=\ln\left(\frac{c}{1}\right)=\ln{(c)}-\ln{(1)}=\ln(c)$$.

In the case of an decreasing series we have the relation $\frac{y_{t+1}}{y_{t}}=\frac1{c}$. And the change in terms of logs becomes

$$\ln\left(\frac{y_{t+1}}{y_{t}}\right)=\ln\left(\frac{1}{c}\right)=\ln{(1)}-\ln{(c)}=-\ln(c)$$

In the case of logaritmiszed values the changes are symmetric in the sense, that the decreasing change rate is equal to the negative value of the increasing change rate.

Answer 2

A while back, I (hypothetically) bought into a highly volatile mutual fund. Last year, its price went up by a whopping 300%. But this year, it went down 75%.

The percentage change framing makes it seem at first glance that I came out ahead 225%, and got a 112.5% annual rate of return. Awesome!

But in reality, $(1 + 3.00)(1 - 0.75) = 4 \times \frac{1}{4} = 1$, and I'd end up with the same balance I started with.

Now, consider this alternate phrasing of the same numbers: Last year, the price increased (logarithmically) by 2 doubles, but this year, it decreased by 2 doubles. With this logarithmic framing, it's immediately obvious that the decrease cancels out the increase.

It's not as big of a deal for smaller changes. For example, a 5% increase followed by a 5% decrease is only a net 0.25% decrease. But for large changes, the error in naive arithmetic manipulation of percentages gets much larger, as in my first example.