Unit Roots in Short Horizon

Question

I have a series that is stationary in the long run. However, in the model development sample - which is a short horizon - the same series is trending. Now, should I consider this series as non-stationary because it is trending in the model development sample? Looking for some reference articles.

Answer 1

Ornstein Uhlenbek type of process produces non-stationary in short term and mean-reverting in long run series. A simple analog in discrete time is AR(1) process: $$x_t=\phi_1 x_{t-1}+\varepsilon_t\\var(\varepsilon_t)=\sigma_\varepsilon^2$$ When the autoregressive coefficient $\phi_1=1-\lambda$ close to 1 i.e. $\lambda\to 0$, we have something that looks like a random walk in short range: $$\Delta x_t\equiv x_t-x_{t-1}=-\lambda x_{t-1}+\varepsilon_t\\ \Delta x_t\approx\varepsilon_t$$

In a long range it's still a stationary process with mean zero and variance: $$var(x_t)=\frac{\sigma_\varepsilon^2}{\lambda}$$ If this was unit root ($\lambda=0$) then the variance would not be bounded: $$var(x_t)=\sigma_\varepsilon^2 t$$