Consider the bivariate VMA(1) process ???????? = (???? − ????????)????????, with

???? =

[ 0.4 0.3

−0.5 0.8

]

???? =

[4 1 1 2]

(a) Find the lag 0 and lag 1 autocorrelations and cross-correlations of ????????; that is, find the matrices ????(0), ????(1), and ????(0), ????(1).

(b) Find the individual univariate MA(1) models for ????1???? and ????2????; that is, in the models ???????????? = (1 − ????????????)????????????, ???? = 1, 2, find the values of the parameters ???????? and

????2

???????? = var[????????????], from ????(0) and ????(1).

(c) Show that the bivariate VMA(1) model above is invertible, state the matrix difference equation satisfied by the matrix weights ???????? in the infinite AR form of the VMA(1) model, and explicitly evaluate the ???????? for ???? = 1, 2, 3, 4.

(d) It follows from Section 14.5.2 that the diagonal elements of ???? represent the one-step-ahead forecast error variances for the two series when each series is forecast from the past history of both series, that is, when each series is forecast based on the bivariate model. Compare these one-step forecast error variances in the bivariate model with the one-step forecast error variances ????2 ???????? based on the individual univariate models in (b)..