Commit dfab8d0e by Michel Juillard

### internal documentation: updating th_autocovariances.m

parent 9aa79c61
 ... ... @@ -116,17 +116,25 @@ At 1st order, the approximated solution of the model takes the form: $y_t = \bar y + g_y (s_{t-1}-\bar s)+g_u u_t$ $\Sigma_y$, the covariance matrix of $y_t$ must satisfy where $s_t$ is the subset of variables that enter the state of the system. $\Sigma_y = g_y\Sigma_y g_y' + g_u \Sigma_u g_u' s_t = \bar s + g_y^{(s)} (s_{t-1}-\bar s)+g_u^{(s)} u_t$ $\Sigma_s$, the covariance matrix of $s_t$ must satisfy $\Sigma_s = g_y^{(s)}\Sigma_s g_y_{(s)}' + g_u^{(s)} \Sigma_u g_u^{(s)}'$ where $\Sigma_u$ is the covariance matrix of $u_t$. This requires that the eigenvalues of $g_y$ are smaller than 1 in modulus. the eigenvalues of $g_y^{(s)}$ are smaller than 1 in modulus. The full covariance matrix for $y_t$, $\Sigma_y$, or any subpart of it, can then be obtained as $\Sigma_ = g_y^{(s)}\Sigma_s g_y_{(s)}' + g_u^{(s)} \Sigma_u g_u^{(s)}'$ The above equation is a Sylvester equation that is best solved by a specialized algorithm. Dynare, currently, uses [[lyapunov\_symm.m]]. In the actual implementation, we distinguish between state variables and non state variables. specialized algorithm. Dynare, currently, uses [[lyapunov\_symm.m]]. The vector of standard deviations $\sigma_y$ is obtained by taking the square root of the diagonal elements of ... ...
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