Consider the univariate state-space model given by state conditions x0 =

w0, xt = xt−1 + wt and observations yt = xt + vt, t = 1, 2, . . ., where wt and vt are independent, Gaussian, white noise processes with var(wt) = σ2 w and var(vt) = σ2 v.

(a) Show that yt follows an IMA(1,1) model, that is, ∇yt follows an MA(1)

model.

(b) Fit the model specified in part

(a) to the logarithm of the glacial varve series and compare the results to those presented in Example 3.32.