3. Impulse response of an ARMA(1, 1) process Let « follow ARMA (1, 1) process-that is, the sum of an AR(1) and MA(1) process:
2+=02t-1+Et+01Et-1
where {E} is white noise.
(a) Write this in the form (L)xt = O(L)E for polynomials $(L) and O(L), being sure
to specify the polynomials.
(b) Assuming the roots of the polynomial () lie outside the unit circle, the process is covariance stationary, and the series can be represented by a square summable
sequence 21 = 29-0 W;Er-;. Using tr = $(L) -‘O (L)Er, rearrange terms to solve for
{W;}. You should be able to find the pattern via algebraic rearrangement.
(c) The mapping t -> + is the impulse response function. Plot the coefficients {v? for
0 = 0.2 and (1) either p = 0.9 or p = -0.9. How do the responses compare?