Necessary Condition for Autoregressive Process to be Stationary

Theorem

Let $S$ be a stochastic process based on an equispaced time series.

Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$

Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:

$\tilde z_t = z_t - \mu$

Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$

Let $M$ be an autoregessive model on $S$ of order $p$:

$\map \phi B \tilde z_t = a_t$

where $\map \phi B := 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$ is the autoregressive operator of order $p$.

Consider the polynomial equation in $B$ of degree $p$:

$(1): \quad \map \phi B = 0$

Let $\map R \phi \subseteq \C$ denote the set of roots of $(1)$, considered as a polynomial of degree $p$.

It is noted that the elements of $\map R \phi$ may be real or complex.

For $S$ modelled by $M$ to be a stationary process, it is necessary that the elements of $\map R \phi$ have a complex modulus greater than $1$:

$\forall z \in \map R \phi: \size z > 1$

Proof

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Sources

• 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):

$1$: Introduction:

$1.2$ Stochastic and Deterministic Dynamic Mathematical Models

$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Autoregressive models