Generates a symmetric \(p \times p\) innovation covariance matrix \(\Sigma_{\boldsymbol{\nu}}\) for the VAR(1) component in the proposed model. The diagonal elements are fixed at 0.5, and a specified number of off-diagonal elements are randomly assigned nonzero values to introduce cross-correlation between variables.
random_Signu(p, num_nonzero)A numeric symmetric matrix of dimension \(p \times p\) representing \(\Sigma_{\boldsymbol{\nu}}\) with diagonal 0.5 and approximately `num_nonzero` nonzero off-diagonal entries.
Integer. Dimension of the covariance matrix (\(p\) variables).
Integer. Target number of nonzero off-diagonal entries (counted individually; both upper and lower triangles are included). Since nonzero values are inserted in symmetric pairs, an even value is recommended. The maximum meaningful value is \(p(p-1)\).
Each nonzero off-diagonal entry is placed in symmetric pairs \((i,j)\) and \((j,i)\) to ensure symmetry of the matrix. The magnitudes of the nonzero entries are randomly drawn from the set \(\{0.1, 0.2\}\) with randomly assigned signs. The diagonal entries are fixed at 0.5 to maintain a moderate level of innovation variance.
In the full model, \(\Sigma_{\boldsymbol{\nu}}\) governs the variability of the VAR(1) innovation term \(\boldsymbol{\nu}_t\) in \(\boldsymbol{\epsilon}_t = \Phi \boldsymbol{\epsilon}_{t-1} + \boldsymbol{\nu}_t\).