VARMASIM Call

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Changes and Enhancements

VARMASIM Call

generates a VARMA(p,q) time series

CALL VARMASIM( series, phi, theta, mu, sigma, n <, p, q, initial, seed>);

The inputs to the VARMASIM subroutine are as follows:

phi: specifies a km_p ×k matrix containing the autoregressive coefficient matrices, where m_p is the number of the elements in the subset of the AR order and $k\geq 2$ is the number of variables. You must specify either phi or theta.
theta: specifies a km_q ×k matrix containing the moving-average coefficient matrices, where m_q is the number of the elements in the subset of the MA order. You must specify either phi or theta.
mu: specifies a k ×1 (or 1 ×k) mean vector of the series. If mu is not specified, a zero vector is used.
sigma: specifies a k ×k covariance matrix of the innovation series. If sigma is not specified, an identity matrix is used.
n: specifies the length of the series. If n is not specified, n=100 is used.
p: specifies the subset of the AR order. See the VARMACOV subroutine.
q: specifies the subset of the MA order. See the VARMACOV subroutine.
initial: specifies the initial values of random variables. If initial=a₀, then y_-p+1, ... ,y₀ and ${\epsilon}_{-q+1}, ... , {\epsilon}_{0}$ all take the same value a₀. If the initial option is not specified, the initial values are estimated for the stationary vector time series; the initial values are assumed as zero for the nonstationary vector time series.
seed: specifies the random number seed. See the VNORMAL subroutine.

The VARMASIM subroutine returns the following value:

series: is an n×k matrix containing the generated VARMA(p,q) time series. When either the initial option is specified or zero initial values are used, these initial values are not included in series.

To generate a bivariate (k=2) stationary VARMA(1,1) time series

$y_t - {\mu}=\Phi ( y_{t-1} - {\mu} ) + {\epsilon}_t - \Theta {\epsilon}_{t-1}$

where

$\Phi=[\matrix{1.2 & -0.5 \cr 0.6 & 0.3 \cr }] \Theta=[\matrix{-0.6 & 0.3 \cr ... ... {\mu}=[\matrix{10 \cr 20 \cr}] \Sigma=[\matrix{1.0 & 0.5 \cr 0.5 & 1.25\cr }]$

you can specify

  phi  = { 1.2 -0.5, 0.6 0.3 };
  theta= {-0.6  0.3, 0.3 0.6 };
  mu   = { 10, 20 };
  sigma= { 1.0  0.5, 0.5 1.25};
  call varmasim(yt, phi, theta, mu, sigma, 100);

To generate a bivariate (k=2) nonstationary VARMA(1,1) time series with the same ${\mu}$ , $\Sigma$ , and $\Theta$ in the previous example and the AR coefficient

$\Phi=[\matrix{1.0 & 0 \cr 0 & 0.3 \cr }]$

you can specify

  phi  = { 1.0 0.0, 0.0 0.3 };
  call varmasim(yt, phi, theta, mu, sigma, 100) initial=3;

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