For a univariate series yt consider the simple rst-order polynomial (lo cally constant) DLM with local level t at time t. The p = 1 dimensional state is t = t, while Ft = 1 and Gt = 1 for all t Also, assume a con stant, known observation variance v

(a) Show that the usual updating equations for mtCt can be written in the alternative forms mt =Ct(R1 t mt 1 +v 1yt)

with C 1 t =R1 t +v 1

(b) Suppose that Rt = Ct 1 for some discount factor that C 1 t

=v 1+ v 1+ 2v 1+ + tC 1 0

(c) Deduce that, as t

(01] Show the variance Ct has the limiting form Ct

(1

)v Comment on this result in connection with the amount of information arising for inference on the local level at t after observing the data for several time points.

(d) Show that the implied limiting form of the usual updating equation for the posterior mean mt is, as t

, mt mt 1+(1 )yt and comment on this form.

(e) Assuming t is large enough so that this limiting form of mt is accurate, what is the contribution of a past observation yt k to the value of mt?