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

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?".

## 1 Answer

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.

• I think you understood it exactly right. – Richard Hardy Feb 2 at 10:01