Example 3.1.1 An ARMA(1,1) Process Consider the ARMA(1,1) process (X,} satisfying the equa- tions Xt - 0.5X1-1 = Z+ + 0.471-1, (Z+) ~ WN (0, 02). (3.1.9) Since the autoregressive polynomial o(z) = 1 - 0.5z has a zero at z = 2, which is located outside the unit circle, we conclude from (3.1.4) and (3.1.6) that there exists a unique ARMA process satisfying (3.1.9) that is also causal. The coefficients { ;) in the MA(oo) representation of {X,} are found directly from (3.1.7): yo = 1, 1 = 0.4 + 0.5, 2 = 0.5(0.4 + 0.5), y; = 0.51-(0.4 + 0.5), j = 1, 2,.... The MA polynomial 0(z) = 1 + 0.4z has a zero at z = -1/0.4 = -2.5, which is also circle This implies that {X,} is invertible with coefficients {n;}