9.1 Definition

The MA and AR models can be combined. A mixed process with \(p\) AR terms and \(q\) MA terms is an ARMA process of order \((p, q)\), written ARMA(\(p, q\)): \[ X_t = \alpha _1X_{t-1} + \alpha _2X_{t-2} + \dots + \alpha _pX_{t-p} + Z_t + \beta _1Z_{t-1} + \dots + \beta _qZ_{t-q} , \] or, with the backward shift operator, \[ \phi (B)X_t = \theta (B)Z_t , \] where \(\phi (B)\) and \(\theta (B)\) are polynomials of order \(p\) and \(q\): \begin {align*} \phi (B) &= 1 - \alpha _1B - \alpha _2B^2 - \dots - \alpha _pB^p , \\ \theta (B) &= 1 + \beta _1B + \beta _2B^2 + \dots + \beta _qB^q . \end {align*}

The conditions on the two parts are those found separately in Sections 8 and 7:

• the process is stationary if the roots of \(\phi (B) = 0\) lie outside the unit circle;
• it is invertible if the roots of \(\theta (B) = 0\) lie outside the unit circle.

ARMA processes are important because a stationary series can often be described by an ARMA model with fewer parameters than a pure MA or pure AR model would need: the principle of parsimony.

One condition is easily overlooked: \(\phi (B)\) and \(\theta (B)\) must have no common factor. If they share a factor, it cancels from both sides of \(\phi (B)X_t = \theta (B)Z_t\), and the process is really of lower order. For example, \(X_t = 0.4X_{t-1} + Z_t - 0.4Z_{t-1}\) is \((1 - 0.4B)X_t = (1 - 0.4B)Z_t\), which is just \(X_t = Z_t\), white noise. Fitting such an over-specified model gives unstable estimates, since any common factor fits equally well.

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