Discuss the properties of the auto covariance matrices

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Reference no: EM131441291

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

2143_Equation 2.jpg

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.

2202_Equation 4.jpg

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

1562_Equation 5.jpg

(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.

Reference no: EM131441291

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