From fbdfaf640a0873fdb176fa8b62edde3025400b87 Mon Sep 17 00:00:00 2001 From: Johannes Pfeifer Date: Wed, 9 Apr 2014 10:44:37 +0200 Subject: [PATCH] Document computation of theoretical mean at second order (cherry picked from commit 7084913382dc892fa4385fb7783e5daec41449f7) --- 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. %