Derive MA (Moving average) representation of a first-difference-process

I have a non-stationary AR(1)-process. After taking the first difference, how can I derive the MA representation of the resulting „difference process“ Delta_xt?

As an example, consider

xt = 1.5xt-1 + et

<=> (xt-xt-1=) Delta_xt = 0.5xt-1 + et.

How can I convert this process Delta_xt into an MA process?

• You don't do differencing. You do back substitution, So substitute 1.5 X(t-2) + e(t-1) for X(t-1) making X(t) = (1.5)$^2$ X(t-2) + 1.5 e(t-1) + e(t) and continue t substitute for X(t-2) and so forth in the same way to get the infinite moving average representation. The infinite series of the e(t) is divergent which shows that the process is non stationary – Michael Chernick Jan 20 '18 at 2:25
• But I am looking for the MA representation of the „difference-process“ Delta_xt. I don’t want the MA of the original process. – Abc123 Jan 20 '18 at 2:42
• That isn't what your title said and it wasn't explained in the question. – Michael Chernick Jan 20 '18 at 2:58
• I changed the text and hope that my question is clearer now. I would still appreciate any help – Abc123 Jan 20 '18 at 3:07