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/*
* This file shows how to use "system prior"-type prior restrictions as in
* Michal Andrle/Miroslav Plašil (2018): "Econometrics with system priors",
* Economics Letters, 172, pp. 134-137 during estimation based on
* the baseline New Keynesian model of Jordi Galí (2015): Monetary Policy, Inflation,
* and the Business Cycle, Princeton University Press, Second Edition, Chapter 3
*
* THIS MOD-FILE REQUIRES DYNARE 4.5 OR HIGHER
*
* Notes:
* - The estimation will automatically take the gali_2015_prior_restrictions.m into
* account, which has the required name and format
* - Estimation is based on simulated data
* - The file also shows how to use a prior/posterior-function
*
* This implementation was written by Johannes Pfeifer. In case you spot mistakes,
* email me at jpfeifer@gmx.de
*
* Please note that the following copyright notice only applies to this Dynare
*/
/*
* Copyright © 2021-2023 Dynare Team
*
* This file is part of Dynare.
*
* Dynare 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.
*
* Dynare 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.
*
* You should have received a copy of the GNU General Public License
* along with Dynare. If not, see <http://www.gnu.org/licenses/>.
*/
var pi ${\pi}$ (long_name='inflation')
y_gap ${\tilde y}$ (long_name='output gap')
y_nat ${y^{nat}}$ (long_name='natural output') //(in contrast to the textbook defined in deviation from steady state)
y ${y}$ (long_name='output')
yhat ${\hat y}$ (long_name='output deviation from steady state')
r_nat ${r^{nat}}$ (long_name='natural interest rate')
i ${i}$ (long_name='nominal interrst rate')
n ${n}$ (long_name='hours worked')
nu ${\nu}$ (long_name='AR(1) monetary policy shock process')
a ${a}$ (long_name='AR(1) technology shock process')
z ${z}$ (long_name='AR(1) preference shock process')
p ${p}$ (long_name='price level')
w ${w}$ (long_name='nominal wage')
c ${c}$ (long_name='consumption')
;
varexo eps_a ${\varepsilon_a}$ (long_name='technology shock')
eps_nu ${\varepsilon_\nu}$ (long_name='monetary policy shock')
eps_z ${\varepsilon_z}$ (long_name='preference shock innovation')
;
parameters alppha ${\alpha}$ (long_name='capital share')
betta ${\beta}$ (long_name='discount factor')
rho_a ${\rho_a}$ (long_name='autocorrelation technology shock')
rho_nu ${\rho_{\nu}}$ (long_name='autocorrelation monetary policy shock')
rho_z ${\rho_{z}}$ (long_name='autocorrelation monetary demand shock')
siggma ${\sigma}$ (long_name='inverse EIS')
varphi ${\varphi}$ (long_name='inverse Frisch elasticity')
phi_pi ${\phi_{\pi}}$ (long_name='inflation feedback Taylor Rule')
phi_y ${\phi_{y}}$ (long_name='output feedback Taylor Rule')
eta ${\eta}$ (long_name='semi-elasticity of money demand')
epsilon ${\epsilon}$ (long_name='demand elasticity')
theta ${\theta}$ (long_name='Calvo parameter')
;
%----------------------------------------------------------------
% Parametrization, p. 67 and p. 113-115
%----------------------------------------------------------------
siggma = 1;
varphi = 5;
phi_pi = 1.5;
phi_y = 0.125;
theta = 3/4;
rho_nu =0.5;
rho_z = 0.5;
rho_a = 0.9;
betta = 0.99;
eta = 3.77; %footnote 11, p. 115
alppha = 1/4;
epsilon = 9;
%----------------------------------------------------------------
% First Order Conditions
%----------------------------------------------------------------
model(linear);
//Composite parameters
#Omega=(1-alppha)/(1-alppha+alppha*epsilon); %defined on page 60
#psi_n_ya=(1+varphi)/(siggma*(1-alppha)+varphi+alppha); %defined on page 62
#lambda=(1-theta)*(1-betta*theta)/theta*Omega; %defined on page 61
#kappa=lambda*(siggma+(varphi+alppha)/(1-alppha)); %defined on page 63
[name='New Keynesian Phillips Curve eq. (22)']
pi=betta*pi(+1)+kappa*y_gap;
[name='Dynamic IS Curve eq. (23)']
y_gap=-1/siggma*(i-pi(+1)-r_nat)+y_gap(+1);
[name='Interest Rate Rule eq. (26)']
i=phi_pi*pi+phi_y*yhat+nu;
[name='Definition natural rate of interest eq. (24)']
r_nat=-siggma*psi_n_ya*(1-rho_a)*a+(1-rho_z)*z;
[name='Definition natural output, eq. (20)']
y_nat=psi_n_ya*a;
[name='Definition output gap']
y_gap=y-y_nat;
[name='Monetary policy shock']
nu=rho_nu*nu(-1)+eps_nu;
[name='TFP shock']
a=rho_a*a(-1)+eps_a;
[name='Production function (eq. 14)']
y=a+(1-alppha)*n;
[name='Preference shock, p. 54']
z=rho_z*z(-1) - eps_z;
[name='Output deviation from steady state']
yhat=y-steady_state(y);
[name='Definition price level']
pi=p-p(-1);
[name='resource constraint, eq. (12)']
y=c;
[name='FOC labor, eq. (2)']
w-p=siggma*c+varphi*n;
end;
shocks;
var eps_nu = 0.0025^2; //1 standard deviation shock of 25 basis points, i.e. 1 percentage point annualized
end;
% simulate data
stoch_simul(periods=100,drop=0,irf=0) yhat;
% save data
datatomfile('sim_data',{'yhat'});
estimated_params;
theta,0.75,beta_pdf,0.5,0.1;
betta, beta_pdf, 0.993, 0.002;
alppha, beta_pdf, 0.25, 0.02;
end;
varobs yhat;
% Run prior function to get prior slope of the Phillips curve based on independent priors
hh_fig=figure('Name','Slope of the Phillips Curve');
prior_function(function='gali_2015_phillips_curve_slope');
phillips_curve_slope_vec=cell2mat(oo_.prior_function_results(:,1));
optimal_bandwidth = mh_optimal_bandwidth(phillips_curve_slope_vec,length(phillips_curve_slope_vec),0,'gaussian');
[density(:,1),density(:,2)] = kernel_density_estimate(phillips_curve_slope_vec,512,length(phillips_curve_slope_vec),optimal_bandwidth,'gaussian');
figure(hh_fig)
subplot(3,1,1)
plot(density(:,1),density(:,2));
title('Prior')
% Run estimation with 1 observation to show effect of _prior_restriction .m
% on independent prior
estimation(datafile='sim_data',mode_compute=5,mh_replic=2001,mh_nblocks=1,diffuse_filter,nobs=1,mh_jscale=0.8);
posterior_function(function='gali_2015_phillips_curve_slope');
phillips_curve_slope_vec=cell2mat(oo_.posterior_function_results(:,1));
optimal_bandwidth = mh_optimal_bandwidth(phillips_curve_slope_vec,length(phillips_curve_slope_vec),0,'gaussian');
[density(:,1),density(:,2)] = kernel_density_estimate(phillips_curve_slope_vec,512,length(phillips_curve_slope_vec),optimal_bandwidth,'gaussian');
figure(hh_fig)
subplot(3,1,2)
plot(density(:,1),density(:,2));
title('Updated Prior')
% Run estimation with full observations
estimation(datafile='sim_data',mode_compute=5,mh_replic=2001,mh_nblocks=1,diffuse_filter,nobs=100,mh_jscale=0.8);
posterior_function(function='gali_2015_phillips_curve_slope');
phillips_curve_slope_vec=cell2mat(oo_.posterior_function_results(:,1));
optimal_bandwidth = mh_optimal_bandwidth(phillips_curve_slope_vec,length(phillips_curve_slope_vec),0,'gaussian');
[density(:,1),density(:,2)] = kernel_density_estimate(phillips_curve_slope_vec,512,length(phillips_curve_slope_vec),optimal_bandwidth,'gaussian');
figure(hh_fig)
subplot(3,1,3)
plot(density(:,1),density(:,2));
title('Posterior')
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