I keep getting confused when i go back and look at these. Can you explain how it goes from the first line: xt - 5xt-1 + 7xt-2 - 3xt-3 to the second line in brackets please in simpler terms? And also for the question at the top of the next page please.
Since we need to express the given expression in terms of difference operator that is ∆Xt= Xt - Xt-1, We need same coefficients of (Xt, Xt-1), (Xt-1,Xt-2) and (Xt-2, Xt-3). In this example, we need same coefficient of Xt-1 as that of Xt (which is 1). So, we can split 5Xt-1 into Xt-1 and 4Xt-1. Now, Xt - Xt-1 will can be written as ∆Xt and we are left with 4Xt-1 part. Again we need same coefficients for Xt-1 and Xt-2. Xt-1 has coefficient 4(from above result) and we can split 7Xt-2 into 4Xt-2 and 3Xt-2. Now, 4Xt-1 - 4Xt-2 would become 4∆Xt-1. We are left with 3Xt-2(from above result) and 3Xt-3 now, which can be written as 3∆Xt-2. Hope this helps!
