Derive the mean vector and covariance matrix of xt - derive

Question

The bivariate AR(4) model x_t - φ₄x_t-4 = φ₀ + at is a special seasonal model with periodicity 4, where {a_t} 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 x_t is weakly stationary. Derive the mean vector and covariance matrix of x_t.

(b) Derive the necessary and sufficient condition of weak stationarity for x_t.

(c) Show that Γ_l = φ₄Γ_l -4 for l > 0, where  is the lag- l auto covariance matrix of x_t.