Commit dfab8d0e authored by Michel Juillard's avatar Michel Juillard
Browse files

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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