(Requires Appendix Material) Define the Difference Operator $\Delta = ( 1 - L )$

Essay

(Requires Appendix Material) Define the difference operator $\Delta = ( 1 - L )$ , where $L$ is the lag operator, such that $L ^ { j } Y _ { t } = Y _ { t - j }$ . In general, $\Delta _ { j } ^ { i } = \left( 1 - L ^ { j } \right) ^ { i }$ , where $i$ and $j$ are typically omitted when they take the value of 1. Show the expressions in $Y$ only when applying the difference operator to the following expressions, and give the resulting expression an economic interpretation, assuming that you are working with quarterly data: (a) $\Delta _ { 4 } Y _ { t }$