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get_empirical_IRFs.m
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get_empirical_IRFs.m
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function [IRF_mat,IRF_weighting,IRF_quantiles]=get_empirical_IRFs(IRF_length,parametric_dummy)
% function [IRF_mat,IRF_weighting,IRF_quantiles]=get_empirical_IRFs(IRF_length)
% Estimates the IRFs to a goverment spending shock following
% Blanchard/Perotti (2002), employing a residual bootstrap to derive the
% confidence bands; as in Blanchard/Perotti, the confidence bands are
% symmetric around the mean response, assuming a normal distribution. Alternatively, the
% respective quantiles of the distribution can be used. In principle, one
% may additionally use a bias correction (e.g. Kilian (1998)).
% Unstable draws are discarded in the bootstrapping.
%
% Inputs:
% IRF_length [scalar] IRF horizon
% IRF_target [nperiods by nvars] matrix of target IRFs
% parametric_dummy [scalar] 1: use symmetric bands based on normal distribution
% 0: use quantiles
% Outputs:
% IRF_mat [nperiods by nvars] matrix of empirical IRFs
% IRF_weighting [nperiods*nvars by nperiods*nvars] matrix of weights for IRF matching
% (a matrix with the inverse of the
% variances of the IRFS on the diagonal)
% IRF_quantiles [nvars by nperiods by 2] IRF quantiles from bootstrapping
%
% Copyright (C) 2016-17 Benjamin Born and Johannes Pfeifer
%
% This is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% It is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% For a copy of the GNU General Public License, see <http://www.gnu.org/licenses/>.
if nargin<2
parametric_dummy=1;
end
%use 4 lags, constant, and deterministic trend
number_of_lags = 4;
constant_dummy =1;
trend_dummy = 1;
%% Data preparation
real_gdp = xlsread('BP.xlsx',1,'B8:B285');
gdp_deflator = xlsread('BP.xlsx',1,'D8:D285');
nom_gov_cons_inv = xlsread('BP.xlsx',1,'F8:F285');
nom_priv_cons_nd = xlsread('BP.xlsx',1,'J8:J285');
nom_priv_cons_serv = xlsread('BP.xlsx',1,'L8:L285');
nipa_pop=xlsread('BP.xlsx','N8:N285');
% construct real per capita values
Y = real_gdp./nipa_pop;
G = nom_gov_cons_inv./nipa_pop./gdp_deflator;
C = (nom_priv_cons_nd+nom_priv_cons_serv)./nipa_pop./gdp_deflator;
timeline = (1947:0.25:2016.25)';
gov_spend_to_gdp_share=nom_gov_cons_inv./(real_gdp.*gdp_deflator/100);
gov_spend_to_gdp_share_mean=mean(gov_spend_to_gdp_share(timeline>1953.75 & timeline<2008,:));
cons_to_gdp_share=(nom_priv_cons_nd+nom_priv_cons_serv)./(real_gdp.*gdp_deflator/100);
cons_to_gdp_share_mean=mean(cons_to_gdp_share(timeline>1953.75 & timeline<2008,:));
x =[log(G) log(Y) log(C)];
var_names={'G','Y','C'};
G_pos=1;
Y_pos=2;
C_pos=3;
%restrict to sample before financial crisis
x=x(timeline>1953.75 & timeline<2008,:);
%create regressor matrix of lagged variables
Z=[];
for lagnumber=1:number_of_lags
tempmat = lagmatrix(x,lagnumber);
Z = [Z tempmat(number_of_lags+1:end,:)];
end
[T, nvars] =size(x);
%add trend and constant
if trend_dummy==1
Z = [(1:T-number_of_lags)' Z];
end
if constant_dummy==1
Z = [ones(T-number_of_lags,1) Z];
end
%get OLS estimates using the Kronecker formula (e.g. Luetkepohl (2005))
Z = Z';
Y=x(number_of_lags+1:end,:)';
T=size(Z,2);
bhat = kron((Z*Z')\Z,eye(nvars))*Y(:);
betta_OLS=reshape(bhat,nvars,nvars*number_of_lags+constant_dummy+trend_dummy);
% get residuals and compute covariance matrix
resids=(eye(T)-Z'/(Z*Z')*Z)*Y';
L_OLS=chol(cov(resids),'lower');
%construct IRFs using the companion form, i.e. transforming the model into a
%VAR(1)
[F,Q]=build_companionform(betta_OLS(:,constant_dummy+trend_dummy+1:end),number_of_lags,...
L_OLS);
n_augmented_vars=size(F,1);
IRFs_point=zeros(n_augmented_vars,IRF_length); %initialize IRF matrix
%set impulse to 1 percent of GDP
shock_vector=zeros(n_augmented_vars,1);
shock_pos=G_pos;
shock_vector(shock_pos)=1/Q(shock_pos,shock_pos); %set to 1 percent of G
% generate IRFs
for boot_iter=1:IRF_length
IRFs_point(:,boot_iter)=F^(boot_iter-1)*Q*shock_vector;
end
%select IRFs for matching
IRF_mat=[IRFs_point(G_pos,:)' IRFs_point(Y_pos,:)'];
%% get uncertainty bands
nbootstraps = 100;
generatedseries=NaN(T+number_of_lags,nvars); %initialize matrix
IRFs=zeros(n_augmented_vars,IRF_length,nbootstraps);
generatedseries(1:number_of_lags,:)=x(1:number_of_lags,:); %start with true starting values
determmatrix = Z(1:constant_dummy+trend_dummy,:); %select constant and trend if necessary
for boot_iter=1:nbootstraps
unstable_indicator=1;
iter=1;
while unstable_indicator==1 && iter<1000
iter=iter+1;
indices=ceil(rand(T,1)*T); %randomly generate index series to pick out residuals; scale the [0,1]-distribution up with nobs
bootresids=resids(indices,:); %select residuals
for jj=number_of_lags+1:T+number_of_lags %recursively generate new time series
ylagvectemp=generatedseries(jj-number_of_lags:jj-1,:); %stack endogenous variables in format compatible with matrix multiplication
ylagvectemp=ylagvectemp(number_of_lags:-1:1,:)'; %reverse order for vectorization
ylagvector=ylagvectemp(:);
generatedseries(jj,:) = determmatrix(:,jj-number_of_lags)'*betta_OLS(:,1:constant_dummy+trend_dummy)' + (betta_OLS(:,1+constant_dummy+trend_dummy:end)*ylagvector)'+ bootresids(jj-number_of_lags,:);
end
% construct lagged regressor matrix
Z_generated=[];
for lagnumber=1:number_of_lags
tempmat = lagmatrix(generatedseries,lagnumber);
Z_generated = [Z_generated tempmat(number_of_lags+1:end,:)];
end
Z_reg_mat = [determmatrix; Z_generated']; %add deterministics
Y_generated=generatedseries(number_of_lags+1:end,:)'; %generate dependent variables
% reestimate coefficients
bhat_boot = kron((Z_reg_mat*Z_reg_mat')\Z_reg_mat,eye(nvars))*Y_generated(:);
betta_boot=reshape(bhat_boot,nvars,nvars*number_of_lags+constant_dummy+trend_dummy);
resids_boot=(eye(T)-Z_reg_mat'/(Z_reg_mat*Z_reg_mat')*Z_reg_mat)*Y_generated';
L_boot=chol(cov(resids_boot),'lower');
%generate IRFs
[F_boot,Q_boot]=build_companionform(betta_boot(:,constant_dummy+trend_dummy+1:end),number_of_lags,L_boot);
if max(abs(eig(F_boot)))>0.999 %discard unstable draws
unstable_indicator=1;
continue
else
unstable_indicator=0;
end
shock_vector=zeros(size(F_boot,1),1);
shock_vector(shock_pos)=1/Q_boot(shock_pos,shock_pos); %set to 1 percent of government spending
for irf_iter=1:IRF_length
IRFs(:,irf_iter,boot_iter)=F_boot^(irf_iter-1)*Q_boot*shock_vector;
end
end
end
if iter==1000
error('Was not able to generate a stable draw in 1000 tries')
end
%get pointwise variance of IRFs across draws and compute weighting matrix
IRF_variances=var(IRFs([G_pos,Y_pos],:,:),0,3)';
IRF_weighting=inv(diag(IRF_variances(2:end)));
% compute confidence bands for plots
if parametric_dummy
IRF_std=std(IRFs,0,3);
IRF_quantiles(:,:,1)=IRFs_point-1.96*IRF_std;
IRF_quantiles(:,:,2)=IRFs_point+1.96*IRF_std;
else
IRF_quantiles = quantile(IRFs(1:nvars,:,:),[0.025 0.975],3);
end
% plot results, measured in percent of GDP
figure
subplot(1,3,1);
plot(1:IRF_length,IRF_quantiles(1,:,1),'r--',1:IRF_length,IRF_quantiles(1,:,2),'r--',1:IRF_length,IRFs_point(1,:),'b-')
title(var_names(G_pos));
ylim([-1 2.5])
subplot(1,3,2);
plot(1:IRF_length,1/gov_spend_to_gdp_share_mean*IRF_quantiles(2,:,1),'r--',1:IRF_length,1/gov_spend_to_gdp_share_mean*IRF_quantiles(2,:,2),'r--',1:IRF_length,1/gov_spend_to_gdp_share_mean*IRFs_point(2,:),'b-')
title(var_names(Y_pos));
ylim([-1 2.5])
subplot(1,3,3);
plot(1:IRF_length,cons_to_gdp_share_mean/gov_spend_to_gdp_share_mean*IRF_quantiles(3,:,1),'r--',1:IRF_length,cons_to_gdp_share_mean/gov_spend_to_gdp_share_mean*IRF_quantiles(3,:,2),'r--',1:IRF_length,cons_to_gdp_share_mean/gov_spend_to_gdp_share_mean*IRFs_point(3,:),'b-')
title(var_names(C_pos));
ylim([-1 2.5])
if nargout>2
IRF_quantiles=[IRF_quantiles([G_pos, Y_pos],:,:)];
end