CovarianceMatrix2AutoregressiveModel

This program computes a VAR(p) model from empirical covariance matrices. The inputfileCovarianceMatrix represent the covariance structure of the process: the first file should contain the auto-covariance, the second the cross-covariance of lag one, the next cross-covariance of lag two and so on.

Cross-covariance matrices $\Sigma_{\Delta_k}$ are defined as the cross-covariance between epoch $t-k$ and $t$. If the process realizations $x_{t}$ are arrange by ascending time stamps ($\{\dots, x_{t-2}, x_{t-1}, x_{t}, x_{t+1}, x_{t+2},\dots\}$), the covariance structure of the (stationary) process is therefore given by \[ \begin{bmatrix} \Sigma & \Sigma_{\Delta_1} & \Sigma_{\Delta_2} & \cdots \\ \Sigma_{\Delta_1}^T & \Sigma & \Sigma_{\Delta_1} & \cdots \\ \Sigma_{\Delta_2}^T & \Sigma_{\Delta_1}^T & \Sigma & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix}. \] The estimate AR model is saved as single matrix outputfileAutoregressiveModel according to the GROOPS AR model conventions.