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

question 39

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

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Definitions:

Economic Costs

The comprehensive expense incurred from selecting one option over another, covering both direct and indirect costs.

Own Capital

The personal resources that an investor or business owner invests into a business venture.

Economic Costs

Total expenses incurred in the production of goods and services, including both explicit and implicit costs.

Own Capital

The funds or assets owned by a business or individual, invested to start or operate a business.