(Requires Appendix Material)Define the Difference Operator Δ = (1 - $\Delta _ { \mathrm { j } } ^ { \mathrm { i } }$

Essay

(Requires Appendix material)Define the difference operator Δ = (1 - L)where L is the lag operator, such that L^jYt = Yt-j. In general, $\Delta _ { \mathrm { j } } ^ { \mathrm { i } }$ = (1- L^j)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)Δ₄Y_t
(b) $\Delta ^ { 2 }$ Y_t
(c) $\Delta _ { 1 }$ $\Delta 4$ Y_t
(d) $\Delta _ { 4 } ^ { 2 }$ Y_t