Solved – Must a time series be stationary if it has no unit root

Question

Must a time series be stationary if it has no unit root? I am not quite sure.

Answer 1

No, roots outside the unit circle gives asymptotic stationarity, not strict stationarity: For a standard ARMA time-series model, the recursive formula for a model locks in the autocorrelation of the process, but does not lock in the marginal/joint distribution for the process. To get the latter you also need to specify the marginal distribution at some point in time, and this specification then locks in the full joint distribution of the process. (Strictly, for an ARMA model, you need to specify the joint distribution of $p$ elements, where $p$ is the degree of the autoregressive characteristic polynomial.)

If all of the roots of an AR process are outside the unit circle (no unit roots and no explosive roots), and the error terms are IID, then the process will have mean-reverting behaviour and will converge asymptotically to a stationary distribution. This asymptotic convergence to a stationary distribution is a weaker property than strict stationarity. It is possible for such a process to be non-stationary by specifying an "anchoring distribution" that is not the asymptotic stationary distribution.

Stationarity for an AR(1) process: Consider a standard first-order auto-regressive process defined by the recursive equation:

$$\begin{matrix} X_{t+1} = \mu + \alpha (X_t - \mu) + \varepsilon_t & & \varepsilon_t \sim \text{ IID N}(0, \sigma^2). \end{matrix}$$

Suppose that $| \alpha | <1$ so that this has a single root $1/\alpha$ outside the unit circle. This model form defines a broad set of possible time-series processes --- i.e., all processes that obey the requisite recursive equation and have the specified noise distribution. However, with this recursive equation you have not yet specified an "anchoring distribution" for any particular point in the series, and so this presently encompasses some stationary and some non-stationary processes. Now, it can be shown that if $\sigma>0$ then (regardless of the anchoring distribution) this process has the asymptotic stationary distribution:

$$X_{\infty} \sim \text{N} \Big( \mu, \frac{\sigma^2}{1 - \alpha^2} \Big).$$

In order to form a strictly stationary process you would need to impose the requirement that the marginal distribution at some arbitrary time is equal to this anchoring distribution. Alternatively, if you want to specify a non-stationary process, you could impose a different marginal distribution at a particular time (perhaps with a different mean or variance, or maybe even a different distributional form).

Non-stationary AR(1) model (with root still inside the unit circle): Suppose you specify an "anchoring distribution" for the element $X_0$ that is still a normal distribution, but with some arbitrary mean and variance:

$$X_0 \sim \text{N} ( \mu_0, \sigma_0^2 ).$$

With this "anchoring distribution" it can be shown that the series of marginal distributions is:

$$X_t \sim \text{N} \Big( (1- \alpha^t) \mu + \alpha^t \mu_0, \alpha^{2t} \sigma_0^2 +\frac{1-\alpha^{2t}}{1-\alpha^2} \sigma^2 \Big).$$

For $\mu_0 \neq \mu$ this series has non-stationary mean, though it converges to the asymptotic mean $\mu$. For $\sigma_0 \neq \sigma$ this series has non-stationary variance, though it converges to the asymptotic variance $\sigma^2 / (1-\alpha^2)$. In this case the process is non-stationary, but it gets closer and closer to being stationary the further you get from the prescribed "anchoring point" of the process.