The bivariate AR(4) model xt - φ4xt-4 = φ0 + at is a special seasonal model with periodicity 4, where {at} is a sequence of independent and identically distributed normal random vectors with mean zero and covariance matrix Σ.
Such a seasonal model may be useful in studying quarterly earnings of a company.
(a) Assume that xt is weakly stationary. Derive the mean vector and covariance matrix of xt.
(b) Derive the necessary and sufficient condition of weak stationarity for xt.
(c) Show that Γl = φ4Γl -4 for l > 0, where is the lag- l auto covariance matrix of xt.