Does detrending fix a random walk?

Question

A pure random walk Xt = X{t-1} + \varepsilont (no drift) is fit with a straight line by regressing Xt on time t, and the fitted line is subtracted off. Does subtracting the fitted linear trend make the series stationary? Answer yes or no.

Accepted Answer

No. A random walk has a stochastic trend, not a deterministic one, so there is no fixed line to remove. In any single sample the random walk wanders in some direction, and least squares will happily fit a sloping line to that wandering, but the wandering is not a real trend, it is accumulated noise. After subtracting the fitted line, the residuals are still essentially a random walk: their variance still grows over time and they still fail to revert to a fixed mean. The correct fix for a random walk is differencing, which turns Xt into the stationary white noise \varepsilont. Detrending only …

Does detrending fix a random walk?

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