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## VARMACOV Call

computes the theoritical auto-cross covariance matrices for stationary VARMA(p,q) model

CALL VARMACOV( cov, phi, theta, sigma <, p, q, lag>);

The inputs to the VARMACOV subroutine are as follows:
phi
specifies to a kp ×k matrix containing the vector autoregressive coefficient matrices. All the roots of are greater than one in absolute value.

theta
specifies to a kq ×k matrix containing the vector moving-average coefficient matrices. You must specify either phi or theta.

sigma
specifies a k ×k symmetric positive-definite covariance matrix of the innovation series. By default, sigma is an identity matrix with dimension k.

p
specifies the order of AR. You can also specify the subset of the order of AR. By default, let ,
For example, consider a 4 dimensional vector time series, if is 4 ×4 matrix and p=1, the VARMACOV subroutine computes the theoritical auto-cross covariance matrices of VAR(1) as follows
If is 4×4 matrix and p=2, the VARMACOV subroutine computes the theoritical auto-cross covariance matrices of VAR(2) as follows
If is 8×4 matrix and p = {1,3 }, the VARMACOV subroutine computes the theoritical auto-cross covariance matrices of VAR(3) as follows

q
specifies the order of MA. You can specify the subset of the order of MA. By default, let ,
The usage of q is the same as that of p.

lag
specifies the length of lags, which must be a positive number. If lag = h, the VARMACOV computes the auto-cross covariance matrices from at lag zero to at lag h. By default, lag = 12.

The VARMACOV subroutine returns the following value:
cov
refers an (k*lagk matrices the theoritical auto-cross covariance VARMA(p,q) series. In case of VMA(q) when p=0, the VARMACOV computes the auto-cross covariance matrices from at lag zero to at lag q.

To compute the theoritical auto-cross covariance matrices of a bivariate (k=2) VARMA(1,1) model
with ,where
you can specify
  call varmacov(cov, phi, theta, sigma) lag=5;


The VARMACOV subroutine computes theoritical auto-cross covariance matrices for the VARMA(p,q) model when AR coefficient matrices , MA coefficient matrices , and an inovation covariance matrix are known. Auto-cross covariance matrices are
where satisfy
with , , and for j < 0.

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