4.6 Nonuniqueness of the mean for a CAR model. Let {Xt} be a stationary AR(1) process in one dimension, with mean ????, autoregression parameter 0

???? . Then {Xt} is also a CAR model satisfying the equations E[(Xt − ????)|X∖t] = ????

1 + ????2 {(Xt−1 − ????)+(Xt+1 − ????)}, var(Xt|X∖t} = ????2

????∕(1 + ????2

)

(see Exercise 4.4). Define a new process Yt = Xt + c????t

, with mean E(Yt) =

???? + c????t = ????t, say, where c is a scalar constant.

Show that {Yt} satisfies the same CAR equations E[(Yt − ????)|Y∖t] = ????

1 + ????2 {(Yt−1 − ????)+(Yt+1 − ????)}, var(Yt|Y∖t} = ????2

????∕(1 + ????2

).

Hence, deduce that if {Yt} is not assumed to be stationary, then the CAR equations do not determine the mean of {Yt}.