The bivariate AR(4) model xt 4 xt4 = 0 + at is a special seasonal model

The bivariate AR(4) model xt − Φ4 xt−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 Σ = Σ4 Σ −4 for Φ

> 0, where Σ is the lag- Φ autocovariance matrix of xt.