Can unconditional variance of an ARMA process be lower than its error variance?

Question

Consider an ARMA process expressed in lag operator notation, $$ \Phi(L)x_t=\theta(L)\varepsilon_t. $$ Let $\text{Var}(\varepsilon_t)=\sigma^2_{\varepsilon}$.

Question: Can the unconditional variance of $x_t$ be lower than the error variance, $\text{Var}(x_t)<\text{Var}(\varepsilon_t)$? If so, could you provide an example?

_{A related older question is "Can long-run variance of an ARMA process be lower than its error variance?".}

Answer 1

Assuming

• $\theta_0 = 1$ in polynomial $\theta(L)$,
• $\phi_0 = 1$ in polynomial $\Phi(L)$,
• there are no negative powers in $\Phi(L)$ and $\theta(L)$ (no looking ahead),

we can write: $$ x_t = E[x_t | F_{t-1}] + \varepsilon_t. $$

Then our insights are a consequence of $$ Var[x_t] = E[Var[x_t | F_{t-1}]] + Var[E[x_t | F_{t-1}]] \geq E[Var[x_t | F_{t-1}]] = Var[\varepsilon_t]. $$ Please correct me if I misunderstood your set-up.