Consider the ARMA(1,1) time series yt de_ned by yt 􀀀 yt􀀀1 = "t + _ "t􀀀1; with _2 = Var("t) = 1 and let _ = (_; ). (a) Show that a state space representation of yt is given by xt+1 = _ xt + "t yt =...

Consider the ARMA(1,1) time series yt de_ned by

yt 􀀀 yt􀀀1 = "t + _ "t􀀀1;

with _2 = Var("t) = 1 and let _ = (_; ).

(a) Show that a state space representation of yt is given by

xt+1 = _ xt + "t

yt = xt + "t;

and _nd .

(b) For which values of _ the observed process is stationary?

(c) Plot a trajectory of yt for di_erent values of _.

Verify if the time series

yt = 0:20yt􀀀1 + 0:63yt􀀀2 + _t + 2:3_t􀀀1 􀀀 0:50_t􀀀2

where _t _ WN(0; _2) is invertible and stationary.