Suppose that { X t } is an ARIMA( p, d, q ) process satisfying the difference equations φ(B)( 1 − B) d X t = θ(B)Z t , { Z t } ∼ WN(0 ,σ 2 ) . Show that these difference equations are also satisfied...

Suppose that {X_t
} is an ARIMA(p, d, q) process satisfying the difference equations

φ(B)(1 −B)^dX_t
=θ(B)Z_t ,{Z_t
} ∼ WN(0,σ
²).

Show that these difference equations are also satisfied by the processW_t
=

X_t
+A
₀ +A
₁
t+ · · · +A<sub>d

−1</sub>
t<sup>d

−1</sup>, whereA
₀
, . . . , A<sub>d

−1</sub> are arbitrary random variables.