Trend stationary

In the statistical analysis of time series, a is trend stationary if an underlying trend (function solely of time) can be removed, leaving a stationary process. The trend does not have to be linear.

Contrarily, if the process requires one or more differencing to be made stationary, then it is called difference stationary and possesses one or more unit roots. Those two concepts may sometimes be confused, but while they share many properties, they are different in many aspects. It is possible for a time series to be non-stationary, have no unit root yet be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing or decreasing over time; however, in the presence of a shock, trend-stationary processes are mean-reverting (i.e. transitory, the time serie will converge again towards the growing mean, which was not affected by the shock) while unit-root processes have a permanent impact on the mean (i.e. no convergence over time).

A process {Y} is said to be trend stationary if

where t is time, f is any function mapping from the reals to the reals, and {e} is a stationary process. The value ${\displaystyle f(t)}$ is said to be the trend value of the process at time t.

Suppose the variable Y evolves according to

where t is time and e_t is the error term, which is hypothesized to be white noise or more generally to have been generated by any stationary process. Then one can uselinear regression to obtain an estimate ${\displaystyle {\hat {a}}}$ of the true underlying trend slope ${\displaystyle a}$ and an estimate ${\displaystyle {\hat {b}}}$ of the underlying intercept term b; if the estimate ${\displaystyle {\hat {a}}}$ is significantly different from zero, this is sufficient to show with high confidence that the variable Y is non-stationary. The residuals from this regression are given by

...
Wikipedia