10.1 Definition

When a process is non-stationary, the non-stationary sources of variation must be removed. A common approach, which goes back to Box and Jenkins, is to difference the series. If in the ARMA model \[ X_t = \alpha _1X_{t-1} + \dots + \alpha _pX_{t-p} + Z_t + \beta _1Z_{t-1} + \dots + \beta _qZ_{t-q} \] \(X_t\) is replaced by its \(d\)th difference \(\nabla ^dX_t\), the model is capable of describing a certain type of non-stationary series. Writing \[ W_t = \nabla ^dX_t = (1 - B)^dX_t , \] the model is the ARMA process \[ W_t = \alpha _1W_{t-1} + \dots + \alpha _pW_{t-p} + Z_t + \beta _1Z_{t-1} + \dots + \beta _qZ_{t-q} , \] or \(\phi (B)W_t = \theta (B)Z_t\), or, in terms of the original series, \[ \phi (B)(1 - B)^dX_t = \theta (B)Z_t . \] Thus \(W_t\) is a stationary ARMA(\(p, q\)) process, while \(X_t\), described through its \(d\)th difference, is an autoregressive integrated moving average process of order \((p, d, q)\), ARIMA(\(p, d, q\)). The name “integrated” is explained by the case \(d = 1\): then \(X_t = X_0 + W_1 + W_2 + \dots + W_t\), a sum (the discrete analogue of an integral) of the stationary \(W\)’s.

The model for \(X_t\) is non-stationary: its generalised AR operator \(\phi (B)(1 - B)^d\) has \(d\) roots on the unit circle, at \(B = 1\), rather than outside it. In practice \(d\) is usually \(1\), occasionally \(2\). The second difference is \begin {align*} \nabla ^2X_t &= \nabla (X_t - X_{t-1}) \\ &= (X_t - X_{t-1}) - (X_{t-1} - X_{t-2}) \\ &= X_t - 2X_{t-1} + X_{t-2} , \end {align*}

which is \((1 - B)^2X_t = (1 - 2B + B^2)X_t\), as the binomial expansion says. Special cases are familiar: ARIMA(0,1,0) is the random walk of Section 5, and ARIMA(0,0,\(q\)) and ARIMA(\(p\),0,0) are the MA and AR processes.

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