Does unit root stationary imply mean stationary and variance stationary?

Question

Newbie question. I am reading about stationary series and understand that it has many forms:

• mean stationary
• variance stationary
• covariance stationary

My question is does unit root stationary imply all of the above? If I run an augmented dicky fuller test and find that the series is unit root stationarity is that sufficient to say that the series is stationary in all respects?

Answer 1

Consider the following AR(1) process with i.i.d. normal errors that have zero mean and finite variance $\sigma^2>0$,

$$ x_t = (1-\rho)\mu + \rho x_{t-1} + \epsilon _t$$

Now suppose $ \rho = 1/2$ and $\mu = 1$. This process does not have a unit root, and it is not mean stationary. At any point in time, the process has finite variance, although as time diverges to infinity, the level of $x_t$ diverges to infinity as well.

Answer 2

Write the series in the answer as

$(x_t - \mu) = \rho (x_{t-1} - \mu) + \varepsilon_t$

then if $\rho=.5$ and $\varepsilon_t$ is $N(0,\sigma^2)$, $(x_t - \mu)$ is stationary with mean $0$ and variance $\frac{\sigma^2}{1-\rho}$.

A time series process can have a deterministic part and a pure random part. The definition of stationarity (strict or strong or second order or \dots) refers only to the pure random part. If the deterministic part is a trend and the random part is stationary the series is trend stationary.