ARMA(1,1) Causal Representation

$$X_t = \phi X_{t-1} + Z_t + \theta Z_{t-1}, \qquad \{Z_t\} \sim \text{WN}(0, \sigma^2), \quad |\phi| < 1$$

$$X_t = Z_t + (\phi + \theta)\sum_{j=1}^{\infty} \phi^{j-1} Z_{t-j}$$

Derivation: From $\phi(B)X_t = \theta(B)Z_t$, i.e., $(1 - \phi B)X_t = (1 + \theta B)Z_t$:

$$X_t = \frac{1 + \theta B}{1 - \phi B} Z_t = \left(1 + \theta B\right)\sum_{j=0}^{\infty}\phi^j B^j Z_t$$

$$= Z_t + \sum_{j=1}^{\infty}\phi^j Z_{t-j} + \theta \sum_{j=0}^{\infty}\phi^j Z_{t-1-j}$$

$$= Z_t + \sum_{j=1}^{\infty}\phi^j Z_{t-j} + \theta \sum_{j=1}^{\infty}\phi^{j-1} Z_{t-j}$$

$$= Z_t + \sum_{j=1}^{\infty}(\phi^j + \theta\phi^{j-1}) Z_{t-j} = Z_t + (\phi + \theta)\sum_{j=1}^{\infty}\phi^{j-1} Z_{t-j}$$

The coefficients are $\psi_0 = 1$ and $\psi_j = (\phi + \theta)\phi^{j-1}$ for $j \geq 1$.

ARMA(1,1) Causal Representation#