How do you tell a trend-stationary series from a difference-stationary one?

## Two processes that look the same Draw a trend-stationary series and a random walk with drift over 200 points and most people cannot tell them apart: both wander upward. The difference is not in the picture but in the mechanism, and the mechanism determines what happens to a shock. **Trend stationary:** `X_t = a + b*t + e_t`, with `e_t` stationary. The expected value moves — it is `a + b*t` — but it moves *deterministically*. Deviations from that line are stationary and temporary. The series has a fixed centre at every time point, and after a shock it comes back to the line. Long-horizon forecasts converge on the extrapolated line, and forecast uncertainty settles at the variance of the deviations rather than growing without limit. **Difference stationary:** `X_t = mu + X_{t-1} + e_t`. The level is a unit-root process. There is no line to return to; the series has stochastic drift as well as deterministic drift, and every shock is added permanently to the running total. Long-horizon forecast intervals widen without bound, because the accumulated uncertainty never stops accumulating. ## The operational question The most reliable practical framing is: **are shocks to this quantity permanent or temporary?** A one-off supply disruption that gets made up in the following quarters points toward mean reversion around a path. A one-off capacity addition that shifts the baseline and stays points toward a unit root. This substantive reasoning often decides better than any test, because the tests are weak precisely where the distinction is hardest. ## What the tests offer The augmented Dickey-Fuller test with a trend term included has as its null a unit root and as its alternative trend stationarity, so a rejection there is the affirmative evidence for the deterministic-trend model. Pair it with the KPSS test in its trend-stationary form, whose null is the opposite. Two crucial cautions: run both with the **same** deterministic specification, since a trend in one and not the other manufactures apparent contradictions; and remember that the power to separate a coefficient of 0.97 from exactly 1.00 over a few hundred observations is genuinely poor. A non-rejection settles very little. ## The cost of detrending a unit-root series Regress a difference-stationary series on time and you get a fitted line with a respectable-looking slope, but the residuals remain highly persistent — they are not the stationary deviations the model assumes. Consequences: standard errors and significance from that regression are not trustworthy, the extrapolated line has no basis in the data-generating process, and long-horizon forecasts are pulled toward a trajectory the series has no tendency to follow. Worst of all, the forecast intervals are far too narrow, because the model assumes uncertainty settles down when in reality it grows with the horizon. This is the more dangerous of the two errors: it produces confident forecasts that are wrong. ## The cost of differencing a trend-stationary series Take the trend-stationary process and difference it: `X_t - X_{t-1} = b + (e_t - e_{t-1})` The deterministic trend collapses into the constant `b`, which is fine. The error term, however, is now a difference of the original errors. If `e_t` was white noise with variance `sigma^2`, the differenced error has variance `2*sigma^2` — twice the noise — and it takes a moving-average form with a coefficient of exactly -1, which is **non-invertible**. Non-invertibility on the moving-average side pushes parameter estimates to a boundary and makes estimation ill-behaved. This is over-differencing, and it costs efficiency: wider intervals, noisier parameter estimates, an observation lost. It is a real cost but a more forgiving one than the opposite error, because the resulting forecasts are honest about their uncertainty rather than falsely confident. ## How to decide in practice 1. Plot the series and ask whether visible shocks were worked off or absorbed into the baseline. 2. Run the Dickey-Fuller test with the trend term included and the KPSS test in its trend form, on the same specification, and read the pair rather than one p-value. 3. Bring domain knowledge to bear on whether the quantity has a mechanism that pulls it back to a path — a physical capacity, a policy target, a competitive equilibrium — or not. 4. Where the evidence genuinely will not separate the two, choose the treatment whose failure mode you can tolerate at the horizon you forecast. For long horizons that usually means erring toward the difference-stationary treatment, so the uncertainty you report grows rather than being understated. 5. Whatever you choose, check the residuals of the fitted model afterwards. Leftover persistence in a detrended model is the tell that you picked the wrong one.