9 Integrated Arma

When a process is non- stationary it is necessary to remove non-stationary sources of variation. In the process \[X_t + \alpha _1 X_{t-1} + \alpha _2 X_{t-2} + \cdots \cdots + \alpha _p X_{t-p} + Z_t + \beta _1 Z_{t-1} + \cdots \cdots + \beta _q Z_{t-q}\] \(X_t\) is replaced by \(\nabla ^d X_t\) then the model is capable of describing a certain type of non-stationary series. This model is called an integrated model, written as \(W_t= \nabla ^d X_t=(1 - B)^d X_t\) then the ARMA process is of the form \[W_t= \alpha _1 W_{t-1} +\alpha _2 W_{t-2} +\cdots \cdots +\alpha _p W_{t-p} +Z_t +\beta _1 Z_{t-1} +\cdots \cdots +\beta _q Z_{t-q}\]

or
\(\phi (B) W_t =\theta (B) Z_t\)
or
\(\phi (B) (1-B)^dX_t=\theta (B) Z_t\)

Thus we have an ARMA of order \((p,q)\) for \(W_t\). While the equation \[\phi (B) (1-B)^dX_t =\theta (B) Z_t\] describes the \(d^{th}\) difference of \(X_t\) it is an ARIMA process of order \((p,d,q)\). The \(X_t\) model is non-stationary as the AR operator \(\phi (B) (1-B)^d\) has d roots outside the unit circle. In particular the value of \(d\) is 1. \begin {align*} \nabla ^2 X_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*}

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