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If a time series has a unit root, that can be modeled as discretized geometric Brownian motion, then are there any ways to reduce the series to $\sim I(0)$? subject to the fact that no other time series $I(1)$ exists with which a linear combo of them would produce $\sim I(0)$ via cointegration?

Without having to 1st-difference the series?

I even tried detrending the processes by eliminating the drift, and almost every possible way except for 1st-differencing, and the final process still ends up with a unit root.

Aaron Hall
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Kevvy Kim
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2 Answers2

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If a process has a unit root (a stochastic trend) and you want to make it stationary, you need to difference it. In other words, if $x_t\sim I(1)$, then $\Delta x_t:=x_t-x_{t-1} \sim I(0)$.

Without differencing you will not get rid of the unit root. E.g. subtracting a deterministic trend will not help because a unit root produces a stochastic trend; so you might end up with a combination of a stochastic and a deterministic trend in the end (the latter being introduced by subtracting a deterministic trend).

If you did not difference and the process truly has a unit root, then no wonder that whatever you tried has failed to produce a stationary transformation.

Richard Hardy
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taking the difference of the time series is NOT the only way to detrend a time series and to remove the unit root. the cointegration should be able to serve the same purpose.

Wensui Liu
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    Your answer is generally relevant but not helpful in this particular case. Read sentence no. 2 of the original post. It explicitly says cointegration cannot be invoked. – Richard Hardy Dec 31 '18 at 08:11