8 ARMA Models
8.1 ARMA(p,q)
ARMA(\(p\), \(q\)) is a mixed autoregressive and moving average process.
\[ y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p} + \epsilon_t + \theta_1\epsilon_{t-1} + \dots + \theta_q\epsilon_{t-q}, \]
or
\[ \phi(L) y_t = \theta(L) \epsilon_t, \]
where \(\{\epsilon_{t}\} \sim \text{WN}(0, \sigma^2)\).
The MA part is always stationary as shown in Proposition 7.1. The stationarity of an ARMA process solely depends on the AR part. The condition is the same as Proposition 6.2.
Assume \(\phi^{-1}(L)\) exist, then the ARMA(\(p\),\(q\)) process can be reduce to MA(\(\infty\)) process:
\[ y_t = \phi^{-1}(L)\theta(L)\epsilon_t = \psi(L) \epsilon_t, \]
where \(\psi(L) = \phi^{-1}(L)\theta(L)\).
Compute the MA equivalence for ARMA(1,1).
8.2 ARIMA(p,d,q)
ARMA(\(p\),\(q\)) is used to model stationary time series. If \(y_t\) is not stationary, we can transform it to stationary and model it with an ARMA model. If the first-order difference \((1-L)y_t = y_t - y_{t-1}\) is stationary, then we say \(y_t\) is integrated of order 1. If it requires \(d\)-th order difference to be stationary, \((1-L)^dy_t\), we say it is integrated of order \(d\). The ARMA model involves integrated time series is called ARIMA model:
\[ \phi(L)(1-L)^d y_t = \theta(L)\epsilon_t. \]