Random walk with drift, which transform?

Question

A series is Xt = 0.1 + X{t-1} + \varepsilont with white-noise shocks. Is it trend-stationary or difference-stationary, and what single transformation makes it stationary?

Accepted Answer

This is a random walk with drift: today equals yesterday plus a constant 0.1 plus a shock. It has a unit root, so its trend is stochastic, not a fixed line. That makes it difference-stationary, and the cure is to difference. Take one difference: \Delta Xt = Xt - X{t-1} = 0.1 + \varepsilont, which is white noise around the constant 0.1, stationary. The reason detrending fails here: although the series drifts upward on average, its deviations from any fitted line are themselves a random walk, so subtracting a straight line leaves nonstationary residuals. Only differencing removes the accumulate…

Random walk with drift, which transform?

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