Lag Operator - Difference Operator

Difference Operator

In time series analysis, the first difference operator Δ is a special case of lag polynomial.

$\begin{array}{lcr} \Delta X_t & = X_t - X_{t-1} \\ \Delta X_t & = (1-L)X_t ~. \end{array}$

Similarly, the second difference operator works as follows:

$\begin{align} \Delta ( \Delta X_t ) & = \Delta X_t - \Delta X_{t-1} \\ \Delta^2 X_t & = (1-L)\Delta X_t \\ \Delta^2 X_t & = (1-L)(1-L)X_t \\ \Delta^2 X_t & = (1-L)^2 X_t ~. \end{align}$

The above approach generalises to the ith difference operator

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