Trend-stationary process

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Trend-stationary processes are time series that are non-stationary only because they follow a time-based trend. If you remove a function of time f(t) from the series Yt, the remaining part e_t is stationary. The trend can be linear, quadratic, exponential, or any smooth function.

Difference-stationary vs trend-stationary: If you must take differences to make the series stationary, it’s called difference-stationary and involves unit roots. These ideas are related but different. A series can be non-stationary without a unit root and still be trend-stationary.

How to tell if a series is trend-stationary: Fit a model Y_t ≈ f(t) + e_t, where f(t) is a trend function (e.g., t for linear, t and t^2 for quadratic, or a log form for exponential growth). Estimate the trend and look at the residuals e_t. If the residuals are stationary, the original series is trend-stationary, and the detrended data are the residuals.

Examples:
- Linear or exponential growth: If GDP grows roughly at a constant rate, you can model log GDP_t = a t + u_t. If u_t is stationary, the detrended log GDP is stationary.
- Quadratic trends: If Y_t has a quadratic trend, regress on t and t^2; if the residuals are stationary, the series is trend-stationary.

What this means in practice: The trend can cause the mean to rise or fall over time, but after removing the trend, the remaining fluctuations are around a stable mean. A shock may be temporary and the series reverts toward its trend; in contrast, a unit-root (difference-stationary) process can have a permanent change in the level.