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Document computation of theoretical mean at second order 

(cherry picked from commit 7084913382dc892fa4385fb7783e5daec41449f7)
time-shift
Johannes Pfeifer 2014-04-09 10:44:37 +02:00 committed by Stéphane Adjemian (Scylla)
parent 7ccaf3e3fa
commit fbdfaf640a
1 changed files with 21 additions and 3 deletions
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24
matlab/th_autocovariances.m
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@ -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.
%