Autoregressive model

Remarks

If $\phi(z) \ne 0$ for all complex $z$ with $\abs{z} = 1$, the equation has exactly one weakly stationary solution. If moreover $\phi$ has no root in the closed unit disk $\abs{z} \le 1$, this solution is causal: $X_t = \mu + \sum_{j \ge 0} \psi_j \varepsilon_{t-j}$ with $\sum_j \abs{\psi_j} < \infty$ and $\mu = c / (1 - \phi_1 - \cdots - \phi_p)$. Many texts include causality in the definition.

Example: for $p = 1$, $c = 0$ and $\abs{\phi_1} < 1$, the process $X_t = \sum_{j \ge 0} \phi_1^j \varepsilon_{t-j}$ converges in mean square and satisfies $X_t = \phi_1 X_{t-1} + \varepsilon_t$. Its variance is $\sigma^2 / (1 - \phi_1^2)$ and its autocorrelation at lag $h$ is $\phi_1^{\abs{h}}$.

Non-example: for $\phi_1 = 1$ and $c = 0$, the equation $X_t = X_{t-1} + \varepsilon_t$ defines a random walk, and it has no weakly stationary solution. Indeed $X_t - X_{t-n} = \varepsilon_{t-n+1} + \cdots + \varepsilon_t$ has variance $n \sigma^2$, which is unbounded in $n$, while for a stationary solution it is at most $4 \operatorname{Var}(X_0)$. Here $\phi(z) = 1 - z$ has the root $z = 1$ on the unit circle.