Don’t detrend random walks with linear trends

Leads to spurious dynamics, as the linear trend isn’t the real “trend,” it’s merely what happened to take place. It could have been quite different and things wouldn’t look so linear if random shocks had played out differently.

One good way to determine the plausibility of trend stationarity around a linear trend is to examine the spectral density of the first difference of the time series around frequency zero. Per @cochraneHowBigRandom1988, trend stationary series will have a spectral density of zero at frequency zero. More realistically, trend stationary series will have low power spectrum at frequency zero, while series with significant random walk components will have a high power spectrum at frequency zero.

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References

Spurious Periodicity in Inappropriately Detrended Time Series

An Exploration of Trend-Cycle Decomposition Methodologies in Simulated Data

Beats and Misses Are Forever

The Universal Law of SaaS Growth

@cochraneHowBigRandom1988

any test for trend stationarity (${\sigma }_{\Delta z}^2=0$ or $S_{\Delta y}(e^{-i0})=0$)

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