From 1fe9682ce7963a50e3c1d02e19f87f581d3ebfad Mon Sep 17 00:00:00 2001 From: John Stachurski Date: Fri, 29 May 2026 08:16:20 +1000 Subject: [PATCH] [time_series_with_matrices] Replace tilde A indirection with A^{-1} Drops the relabeling block 'let \tilde A = A^{-1}' (it appeared twice) and uses A^{-1} directly in the population-moments and moving-average formulas. Co-Authored-By: Claude Opus 4.7 --- lectures/time_series_with_matrices.md | 16 +++------------- 1 file changed, 3 insertions(+), 13 deletions(-) diff --git a/lectures/time_series_with_matrices.md b/lectures/time_series_with_matrices.md index e32f0742..9817e1e1 100644 --- a/lectures/time_series_with_matrices.md +++ b/lectures/time_series_with_matrices.md @@ -370,14 +370,6 @@ $$ You can read about multivariate normal distributions in this lecture [Multivariate Normal Distribution](https://python.quantecon.org/multivariate_normal.html). -Let's write our model as - -$$ -y = \tilde A (b + u) -$$ - -where $\tilde A = A^{-1}$. - Because linear combinations of normal random variables are normal, we know that $$ @@ -387,13 +379,13 @@ $$ where $$ -\mu_y = \tilde A b +\mu_y = A^{-1} b $$ and $$ -\Sigma_y = \tilde A (\sigma_u^2 I_{T \times T} ) \tilde A^T +\Sigma_y = A^{-1} (\sigma_u^2 I_{T \times T} ) (A^{-1})^T $$ Let's write a Python class that computes the mean vector $\mu_y$ and covariance matrix $\Sigma_y$. @@ -566,12 +558,10 @@ Since $A^{-1}$ is lower triangular, each row represents $ y_t$ for a particul - a time-dependent function $A^{-1} b$ of the initial conditions incorporated in $b$, and - a weighted sum of current and past values of the IID shocks $\{u_t\}$. -Thus, let $\tilde{A}=A^{-1}$. - Evidently, for $t\geq0$, $$ -y_{t+1}=\sum_{i=1}^{t+1}\tilde{A}_{t+1,i}b_{i}+\sum_{i=1}^{t}\tilde{A}_{t+1,i}u_{i}+u_{t+1} +y_{t+1}=\sum_{i=1}^{t+1}(A^{-1})_{t+1,i}b_{i}+\sum_{i=1}^{t}(A^{-1})_{t+1,i}u_{i}+u_{t+1} $$ This is a **moving average** representation with time-varying coefficients.