6.3

(a) Using the ideas from Sections 4.8.2 and 4.6.1, show that the AR(1) and .

CAR(1) models in Section 6.3 have the spectral densities given in (6.13)

and (6.17).

(b) Show that the spectral densities are the same as one another under the conditions in (6.18). Further show that ???? and ???? in (6.20) are related by

???? = ????∕

(

1 + ????2

)

.

(c) For z = exp(i????), show that z + z = 2 cos????,

(1 − ????z)(1 − ????z) = 1 + ????2 − 2???? cos????, 1 1 − ????2

{ 1 1 − ????z

+

1 1 − ????z − 1

}

= 1

(1 − ????z)(1 − ????z)

, and use the geometric and logarithmic series expansions 1

1 − ????z = ∑∞

j=0

????j zj

, −log(1 − ????z) = ∑∞

j=1 1

j

????j zj

.

to verify the Fourier expansions (6.14)–(6.16).

(d) Prove the identities (6.20) to express ???? and ???? in terms of the covariance function (6.19).