Where at is a white noise series with positive-definite

Question

Question: Suppose that zt is a k-dimensional weakly stationary, zero-mean time series following the VARMA(2,q) model

where {at} is a white noise series with positive-definite covariance matrix Σa. Let Γj be the lag-j autocovariance matrix of zt. (a) Consider the VAR(2) fitting

Show that the ordinary LS estimates of Φ_i^{(0) (i=1,2) satisfy the system of equations}

(b) Let a_t⁽⁰⁾ be the residual of the fitted VAR(2) model via the ordinary LS method. Discuss The properties of the autocovariance  matrices of a_t⁽⁰⁾ when q = 0 and when q = 1.

(c) Consider the model

Show that the ordinary LS estimates of Φ_i(0) satisfy the systems of equations

 

(d) Let a(1)t be the residual of the LS fit of the model in part (c).

Discuss the properties of the auto covariance matrices of a(1)t when q = 0, 1, and 2.