From 4560a1b76e1f6a3e259699d758b6bc2ddb64ee1a Mon Sep 17 00:00:00 2001
From: Johannes Pfeifer <jpfeifer@gmx,de>
Date: Wed, 9 Apr 2014 10:44:37 +0200
Subject: [PATCH] Document computation of theoretical mean at second order
(cherry picked from commit 7084913382dc892fa4385fb7783e5daec41449f7)
(cherry picked from commit fbdfaf640a0873fdb176fa8b62edde3025400b87)
---
matlab/th_autocovariances.m | 24 +++++++++++++++++++++---
1 file changed, 21 insertions(+), 3 deletions(-)
diff --git a/matlab/th_autocovariances.m b/matlab/th_autocovariances.m
index 759d8a4b8..e257b4ee6 100644
--- a/matlab/th_autocovariances.m
+++ b/matlab/th_autocovariances.m
@@ -2,7 +2,7 @@ function [Gamma_y,stationary_vars] = th_autocovariances(dr,ivar,M_,options_,node
% Computes the theoretical auto-covariances, Gamma_y, for an AR(p) process
% with coefficients dr.ghx and dr.ghu and shock variances Sigma_e_
% for a subset of variables ivar (indices in lgy_)
-% Theoretical HPfiltering is available as an option
+% Theoretical HP-filtering is available as an option
%
% INPUTS
% dr: [structure] Reduced form solution of the DSGE model (decisions rules)
@@ -23,8 +23,26 @@ function [Gamma_y,stationary_vars] = th_autocovariances(dr,ivar,M_,options_,node
%
% SPECIAL REQUIREMENTS
%
-
-% Copyright (C) 2001-2012 Dynare Team
+% Algorithms
+% The means at order=2 are based on the pruned state space as
+% in Kim, Kim, Schaumburg, Sims (2008): Calculating and using second-order accurate
+% solutions of discrete time dynamic equilibrium models.
+% The solution at second order can be written as:
+% \[
+% \hat x_t = g_x \hat x_{t - 1} + g_u u_t + \frac{1}{2}\left( g_{\sigma\sigma} \sigma^2 + g_{xx}\hat x_t^2 + g_{uu} u_t^2 \right)
+% \]
+% Taking expectations on both sides requires to compute E(x^2)=Var(x), which
+% can be obtained up to second order from the first order solution
+% \[
+% \hat x_t = g_x \hat x_{t - 1} + g_u u_t
+% \]
+% by solving the corresponding Lyapunov equation.
+% Given Var(x), the above equation can be solved for E(x_t) as
+% \[
+% E(x_t) = (I - {g_x}\right)^{- 1} 0.5\left( g_{\sigma\sigma} \sigma^2 + g_{xx} Var(\hat x_t) + g_{uu} Var(u_t) \right)
+% \]
+%
+% Copyright (C) 2001-2014 Dynare Team
%
% This file is part of Dynare.
%
--
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