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Add bandpass filter to th_autocovariances.m 

Also adds documentation
time-shift
Johannes Pfeifer 2015-08-01 18:43:12 +02:00
parent e9f2be0c1a
commit 25df325899
1 changed files with 43 additions and 40 deletions
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83
matlab/th_autocovariances.m
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@ -1,8 +1,8 @@
function [Gamma_y,stationary_vars] = th_autocovariances(dr,ivar,M_,options_,nodecomposition)
% 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 HP-filtering is available as an option
% with coefficients dr.ghx and dr.ghu and shock variances Sigma_e
% for a subset of variables ivar.
% Theoretical HP-filtering and band-pass filtering is available as an option
%
% INPUTS
% dr: [structure] Reduced form solution of the DSGE model (decisions rules)
@ -156,7 +156,7 @@ if options_.order == 2 || options_.hp_filter == 0
end
end
end
if options_.hp_filter == 0
if options_.hp_filter == 0 && ~options_.bandpass.indicator
v = NaN*ones(nvar,nvar);
v(stationary_vars,stationary_vars) = aa*vx*aa'+ bb*M_.Sigma_e*bb';
k = find(abs(v) < 1e-12);
@ -207,37 +207,44 @@ if options_.hp_filter == 0
end
end
end
else% ==> Theoretical HP filter.
else% ==> Theoretical filters.
% By construction, all variables are stationary when HP filtered
iky = inv_order_var(ivar);
stationary_vars = (1:length(ivar))';
aa = ghx(iky,:);
bb = ghu(iky,:);
aa = ghx(iky,:); %R in Uhlig (2001)
bb = ghu(iky,:); %S in Uhlig (2001)
lambda = options_.hp_filter;
ngrid = options_.hp_ngrid;
freqs = 0 : ((2*pi)/ngrid) : (2*pi*(1 - .5/ngrid));
tpos = exp( sqrt(-1)*freqs);
tneg = exp(-sqrt(-1)*freqs);
hp1 = 4*lambda*(1 - cos(freqs)).^2 ./ (1 + 4*lambda*(1 - cos(freqs)).^2);
mathp_col = [];
freqs = 0 : ((2*pi)/ngrid) : (2*pi*(1 - .5/ngrid)); %[0,2*pi)
tpos = exp( sqrt(-1)*freqs); %positive frequencies
tneg = exp(-sqrt(-1)*freqs); %negative frequencies
if options_.bandpass.indicator
filter_gain = zeros(1,ngrid);
lowest_periodicity=options_.bandpass.passband(2);
highest_periodicity=options_.bandpass.passband(1);
highest_periodicity=max(2,highest_periodicity); % restrict to upper bound of pi
filter_gain(freqs>=2*pi/lowest_periodicity & freqs<=2*pi/highest_periodicity)=1;
filter_gain(freqs<=-2*pi/lowest_periodicity+2*pi & freqs>=-2*pi/highest_periodicity+2*pi)=1;
else
filter_gain = 4*lambda*(1 - cos(freqs)).^2 ./ (1 + 4*lambda*(1 - cos(freqs)).^2); %HP transfer function
end
mathp_col = NaN(ngrid,length(ivar)^2);
IA = eye(size(A,1));
IE = eye(M_.exo_nbr);
for ig = 1:ngrid
if hp1(ig)==0,
if filter_gain(ig)==0,
f_hp = zeros(length(ivar),length(ivar));
else
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*ghu1;IE]...
*M_.Sigma_e*[ghu1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) bb]*f_omega*[aa'*tpos(ig); bb']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
f_omega =(1/(2*pi))*([(IA-A*tneg(ig))\ghu1;IE]...
*M_.Sigma_e*[ghu1'/(IA-A'*tpos(ig)) IE]); % spectral density of state variables; top formula Uhlig (2001), p. 20 with N=0
g_omega = [aa*tneg(ig) bb]*f_omega*[aa'*tpos(ig); bb']; % spectral density of selected variables; middle formula Uhlig (2001), p. 20; only middle block, i.e. y_t'
f_hp = filter_gain(ig)^2*g_omega; % spectral density of selected filtered series; top formula Uhlig (2001), p. 21;
end
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
mathp_col(ig,:) = (f_hp(:))'; % store as matrix row for ifft
end;
% Covariance of filtered series
imathp_col = real(ifft(mathp_col))*(2*pi);
imathp_col = real(ifft(mathp_col))*(2*pi); % Inverse Fast Fourier Transformation; middle formula Uhlig (2001), p. 21;
Gamma_y{1} = reshape(imathp_col(1,:),nvar,nvar);
% Autocorrelations
if nar > 0
@ -253,44 +260,40 @@ else% ==> Theoretical HP filter.
Gamma_y{nar+2} = ones(nvar,1);
else
Gamma_y{nar+2} = zeros(nvar,M_.exo_nbr);
SS(exo_names_orig_ord,exo_names_orig_ord) = M_.Sigma_e+1e-14*eye(M_.exo_nbr);
SS(exo_names_orig_ord,exo_names_orig_ord) = M_.Sigma_e+1e-14*eye(M_.exo_nbr); %make sure Covariance matrix is positive definite
cs = chol(SS)';
SS = cs*cs';
b1(:,exo_names_orig_ord) = ghu1;
b2(:,exo_names_orig_ord) = ghu(iky,:);
mathp_col = [];
mathp_col = NaN(ngrid,length(ivar)^2);
IA = eye(size(A,1));
IE = eye(M_.exo_nbr);
for ig = 1:ngrid
if hp1(ig)==0,
if filter_gain(ig)==0,
f_hp = zeros(length(ivar),length(ivar));
else
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]...
*SS*[b1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
f_omega =(1/(2*pi))*( [(IA-A*tneg(ig))\b1;IE]...
*SS*[b1'/(IA-A'*tpos(ig)) IE]); % spectral density of state variables; top formula Uhlig (2001), p. 20 with N=0
g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % spectral density of selected variables; middle formula Uhlig (2001), p. 20; only middle block, i.e. y_t'
f_hp = filter_gain(ig)^2*g_omega; % spectral density of selected filtered series; top formula Uhlig (2001), p. 21;
end
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
mathp_col(ig,:) = (f_hp(:))'; % store as matrix row for ifft
end;
imathp_col = real(ifft(mathp_col))*(2*pi);
vv = diag(reshape(imathp_col(1,:),nvar,nvar));
for i=1:M_.exo_nbr
mathp_col = [];
mathp_col = NaN(ngrid,length(ivar)^2);
SSi = cs(:,i)*cs(:,i)';
for ig = 1:ngrid
if hp1(ig)==0,
if filter_gain(ig)==0,
f_hp = zeros(length(ivar),length(ivar));
else
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]...
*SSi*[b1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
f_omega =(1/(2*pi))*( [(IA-A*tneg(ig))\b1;IE]...
*SSi*[b1'/(IA-A'*tpos(ig)) IE]); % spectral density of state variables; top formula Uhlig (2001), p. 20 with N=0
g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % spectral density of selected variables; middle formula Uhlig (2001), p. 20; only middle block, i.e. y_t'
f_hp = filter_gain(ig)^2*g_omega; % spectral density of selected filtered series; top formula Uhlig (2001), p. 21;
end
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
mathp_col(ig,:) = (f_hp(:))'; % store as matrix row for ifft
end;
imathp_col = real(ifft(mathp_col))*(2*pi);
Gamma_y{nar+2}(:,i) = abs(diag(reshape(imathp_col(1,:),nvar,nvar)))./vv;