ARMA (1,1) Variance Calculation

Question

I really need help with this problem and would appreciate any input:

When $t$ denotes the time-period, terms $\alpha$, $\phi_1$, and $\theta_1$ are constants, $a_t$ represents error-terms that are NID(0, $\sigma^2$) if a variable r is modeled as ARMA(1,1) process,

$$r_t=\alpha + \phi_1 r_{t-1} + \theta_1 a_{t-1} + a_{t}$$

What is variance of the $r_t$?

Answer 1

It is better to write your model like this: $$r_t=\alpha + \phi_1 r_{t-1} + \theta_1 a_{t-1} + a_{t}$$ where, {$a_t$} is white noise series.

First, multiply the model by $a_t$ and take expectation: \begin{align} E(r_t a_t)&=\alpha E(a_t)+ \phi_1 E(r_{t-1}a_t)+\theta_1 E(a_{t-1}a_t)+ E(a_t^2)\\ &=E(a_t^2)=\sigma^2 \end{align}

Here, we make use of the fact that $E(a_t r_{t-1})=0$, $E(a_t)=0$ and $E(a_{t-1}a_t)=0$.

Taking the variance of original equation, we have \begin{align} Var(r_t)&=\phi_1^2 Var(r_{t-1})+\theta_1^2\sigma^2+\sigma^2+2 \phi_1 \theta_1 E(r_{t-1} a_{t-1}) \\ &=\phi_1^2 Var(r_{t-1})+\theta_1^2\sigma^2+\sigma^2+2 \phi_1 \theta_1 \sigma^2 \end{align} If $r_t$ is weakly stationary, then $Var(r_t)=Var(r_{t-1})$, so $$Var(r_t)=\frac{(1+2\phi_1 \theta_1 + \theta_1^2)\sigma^2}{1-\phi_1^2}$$