Stabilizing the variance of exponential growth

Question

A business metric grows exponentially over time, and its period-to-period swings are proportional to its level (the bigger the value, the bigger the wiggles).

What single transformation stabilizes the variance so the series can then be differenced to stationarity?

Accepted Answer

When the fluctuations scale with the level (a multiplicative, percentage-style pattern) the raw series has a variance that balloons as the level rises, and differencing alone will not fix that. The standard fix is the log transformation. Taking logs does two useful things: 1. It stabilizes the variance. If swings are a roughly constant percentage of the level, then in log units they become a roughly constant absolute size, so the variance stops depending on the level. 2. It linearizes exponential growth. An exponential trend c\,e^{gt} becomes the straight line \ln c + g t in logs. After the l…

Stabilizing the variance of exponential growth

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