A DLM for the univariate series yt is given by yt = F t + t where t

N(0v), and t = Gt 1+ twhere t N(0vW)withtheusual conditional independence assumptions. All model parameters FvGW are known and constant over time. The modeler speci es the model such that:

G has p real and distinct eigenvalues i i = 1 p with i < 1 for each i; and at t = 0 the state distribution 0 D0 N(m0 vC0) where m0 = 0 and C0 Csatis es the equation C = GCG+W. It can be shown that there is a unique variance matrix C satisfying this equation when i <1 as is true in this exercise.

(a) Show that the t step ahead prior distribution for future state vectors p( t D0) is given by t D0 N(0vC) for all t 0

(b) For any time point t and k 0 show that C( t+k t D0) = vGkC

(c) Show that the t step ahead forecast distribution p(yt D0) = N(0vs)

for some constant s > 0, and give the expression for s in terms of FGC.

(d) For any time point t and k 1 show that p(yt+k yt D0) is bivari ate normal with covariance that depends on k but not t Give an expression for this covariance in terms of k and model parameters.

(e) Deduce that yt is a stationary time series.

(f) Describe the qualitative form of the implied autocorrelation function

(k) as a function of lag k

(g) Comment on the connections with a stationary AR(p) model for yt