Consider the AR(2) series yt = 1yt 1 + 2yt 2 + t with t N(0v).

Following Section 2.1.2, rewrite the model in the standard DLM form yt = Fxt and xt =Gxt 1+Ft where F= 1 0

xt = yt yt 1 G= 1 2 1

0 We know that this implies that, for any given t and over k 0 the forecast function is E(yt+kxt) = FGkxt

(a) Show that the eigenvalues of G denoted by 1 and 2 are the roots of the quadratic in given by 2 1 2 =0Deducethat 1 = 1+ 2 and 2 = 1 2

(b) Suppose that the eigenvalues 1 2 are distinct, whether they be real or a pair of complex conjugates. De ne

= 1 0 0

2 and E= 1 2 1

1 for any nonzero Note that E is nonsingular since 1= 2 Verify that GE = E , so that G = E E1 that is, E has columns that are eigenvectors of G corresponding to eigenvalues ( 1 2)

(c) We can take = 1 with no loss of generality as cancels in the identity G = E E1; do so from here on. Show that kE1 = 1

( 1 2)

k 1

k 2

k 1 2 1 k

(d) Deduce that E(yt+kxt) = akyt +bkyt 1 with lagged coe cients ak = ( k+1 1 k+1 2 )
( 1 2) and bk= ( k+1 1 2 + 1 k+1 2 )
( 1 2)

(e) Verify that this resulting expression E(yt+kxt) = akyt +bkyt 1 gives the known results in terms of 1 2 when k = 0 and k = 1

(f) Consider now the special case of complex eigenvalues 1 = rei and 2 =re i for some real-valued modulus r > 0 and argument > 0 Show that the lagged coe cients ak bk become ak = rksin((k +1) ) sin( ) and bk = rk+1sin(k ) sin( )
(g) Continuing in the case of complex eigenvalues, use simple trigono metric identities to show that the forecast function can be reduced to E(yt+kxt) = rkhtcos(k +gt)
k =01 a damped cosine form in k (in stationary models with 0 < r < 1).
Give explicit expressions for the time-dependent amplitude ht > 0 and phase gt in terms of and yt 1 yt