From b7c60ddf599b3ab83fcf005c0ebf2c23d9f546e6 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?St=C3=A9phane=20Adjemia=20=28Scylla=29?= <stepan@dynare.org>
Date: Mon, 4 Feb 2019 10:00:16 +0100
Subject: [PATCH] Added a routine for writing the problem to be solved to
compute the BGP of a model.
- Only works with backward models.
- Probably doesn't work if the model includes auxiliary variables.
- Assumes that the trends are multiplicative.
---
matlab/+bgp/write.m | 168 ++++++++++++++++++++++++++++++++++++
tests/Makefile.am | 3 +-
tests/bgp/solow-1/solow.mod | 80 +++++++++++++++++
3 files changed, 250 insertions(+), 1 deletion(-)
create mode 100644 matlab/+bgp/write.m
create mode 100644 tests/bgp/solow-1/solow.mod
diff --git a/matlab/+bgp/write.m b/matlab/+bgp/write.m
new file mode 100644
index 000000000..35909154f
--- /dev/null
+++ b/matlab/+bgp/write.m
@@ -0,0 +1,168 @@
+function write(DynareModel)
+
+% Writes the nonlinear problem to be solved for computing the growth
+% rates and levels along the Balanced Growth Path. Note that for
+% the variables growing along the BGP, the identified levels are
+% meaningless, only the relative levels of the growing variables
+% sharing the same trend(s) are relevant.
+%
+% INPUTS
+% - DynareModel [struct] Dynare generated stucture describing the model (M_).
+%
+% OUTPUTS
+% None
+%
+% REMARKS
+% - The trends are assumed to be multiplicative.
+
+% Copyright (C) 2019 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/>.
+
+i0 = transpose(DynareModel.lead_lag_incidence(1,:)); % Indices of the lagged variables.
+i1 = transpose(DynareModel.lead_lag_incidence(2,:)); % Indices of the current variables.
+n0 = length(find(i0)); % Number of lagged variables.
+n1 = length(find(i1)); % Number of current variables.
+
+% Create function in mod namespace.
+fid = fopen(sprintf('+%s/bgpfun.m', DynareModel.fname), 'w');
+
+% Write header.
+fprintf(fid, 'function [F, JAC] = bgpfun(z)\n\n');
+fprintf(fid, '%% This file has been generated by dynare (%s).\n\n', datestr(now));
+
+% The function admits a unique vector as input argument. The first
+% half of the elements are for the levels of the endogenous
+% variables, the second half for the growth factors.
+fprintf(fid, 'y = z(1:%u);\n\n', DynareModel.endo_nbr);
+fprintf(fid, 'g = z(%u:%u);\n', DynareModel.endo_nbr+1, 2*DynareModel.endo_nbr);
+
+% Define the point where the dynamic model is to be evaluated.
+fprintf(fid, 'Y = zeros(%u, 1);\n', 2*(n0+n1));
+for i=1:length(i0) % period t equations, lagged variables.
+ if i0(i)
+ fprintf(fid, 'Y(%u) = y(%u);\n', i0(i), i);
+ end
+end
+for i=1:length(i1) % period t equations, current variables.
+ if i1(i)
+ fprintf(fid, 'Y(%u) = y(%u)*g(%u);\n', i1(i), i, i);
+ end
+end
+for i=1:length(i0) % period t+1 equaions lagged variables.
+ if i0(i)
+ fprintf(fid, 'Y(%u) = y(%u)*g(%u);\n', n0+n1+i0(i), i, i);
+ end
+end
+for i=1:length(i1) % period t+1 equations current variables.
+ if i1(i)
+ fprintf(fid, 'Y(%u) = y(%u)*g(%u)*g(%u);\n', n0+n1+i1(i), i, i, i);
+ end
+end
+fprintf(fid, '\n');
+
+% Define the vector of parameters.
+fprintf(fid, 'p = zeros(%u, 1);\n', DynareModel.param_nbr);
+for i = 1:DynareModel.param_nbr
+ fprintf(fid, 'p(%u) = %16.12f;\n', i, DynareModel.params(i));
+end
+fprintf(fid, '\n');
+
+% Initialize the vector holding the residuals over the two periods.
+fprintf(fid, 'F = NaN(%u, 1);\n', 2*DynareModel.endo_nbr);
+
+% Set vector of exogenous variables to 0.
+fprintf(fid, 'x = zeros(1, %u);\n\n', DynareModel.exo_nbr);
+
+% Evaluate the residuals and jacobian of the dynamic model in periods t and t+1.
+fprintf(fid, 'if nargout>1\n');
+fprintf(fid, ' J = zeros(%u, %u);\n', 2*DynareModel.endo_nbr, n0+n1+DynareModel.endo_nbr);
+fprintf(fid, ' [F(1:%u), tmp] = %s.dynamic(Y(1:%u), x, p, y, 1);\n', DynareModel.endo_nbr, DynareModel.fname, n1+n0);
+fprintf(fid, ' J(1:%u,1:%u) = tmp(:,1:%u);\n', DynareModel.endo_nbr, n0+n1, n0+n1);
+fprintf(fid, ' [F(%u:%u), tmp] = %s.dynamic(Y(1+%u:%u), x, p, y, 1);\n', DynareModel.endo_nbr+1, 2*DynareModel.endo_nbr, DynareModel.fname, n1+n0, 2*(n1+n0));
+fprintf(fid, ' J(%u:%u,1:%u) = tmp(:,1:%u);\n', DynareModel.endo_nbr+1, 2*DynareModel.endo_nbr, n0+n1, n0+n1);
+fprintf(fid, 'else\n');
+fprintf(fid, ' F(1:%u) = %s.dynamic(Y(1:%u), x, p, y, 1);\n', DynareModel.endo_nbr, DynareModel.fname, n1+n0);
+fprintf(fid, ' F(%u:%u) = %s.dynamic(Y(1+%u:%u), x, p, y, 1);\n', DynareModel.endo_nbr+1, 2*DynareModel.endo_nbr, DynareModel.fname, n1+n0, 2*(n1+n0));
+fprintf(fid, 'end\n\n');
+
+% Compute the jacobian if required.
+fprintf(fid, 'if nargout>1\n');
+fprintf(fid, ' JAC = zeros(%u,%u);\n', 2*DynareModel.endo_nbr, 2*DynareModel.endo_nbr);
+
+% Compute the derivatives of the first block of equations (period t)
+% with respect to the endogenous variables.
+for i=1:DynareModel.eq_nbr
+ for j=1:DynareModel.endo_nbr
+ if i1(j)
+ if i0(j)
+ fprintf(fid, ' JAC(%u,%u) = J(%u,%u)+J(%u,%u)*g(%u);\n', i, j, i, i0(j), i, i1(j), j);
+ else
+ fprintf(fid, ' JAC(%u,%u) = J(%u,%u)*g(%u);\n', i, j, i, i1(j), j);
+ end
+ else
+ if i0(j)
+ fprintf(fid, ' JAC(%u,%u) = J(%u,%u);\n', i, j, i, i0(j));
+ end
+ end
+ end
+end
+
+% Compute the derivatives of the second block of equations (period t+1)
+% with respect to the endogenous variables.
+for i=DynareModel.eq_nbr+1:2*DynareModel.eq_nbr
+ for j=1:DynareModel.endo_nbr
+ if i1(j)
+ if i0(j)
+ fprintf(fid, ' JAC(%u,%u) = J(%u,%u)*g(%u)+J(%u,%u)*g(%u)*g(%u);\n', i, j, i, i0(j), j, i, i1(j), j, j);
+ else
+ fprintf(fid, ' JAC(%u,%u) = J(%u,%u)*g(%u)*g(%u);\n', i, j, i, i1(j), j, j);
+ end
+ else
+ if i0(j)
+ fprintf(fid, ' JAC(%u, %u) = J(%u, %u)*g(%u);\n', i, j, i, i0(j), j);
+ end
+ end
+ end
+end
+
+% Compute the derivatives of the first block of equations (period t)
+% with respect to the growth factors.
+for i=1:DynareModel.eq_nbr
+ for j=1:DynareModel.endo_nbr
+ if i1(j)
+ fprintf(fid, ' J(%u,%u) = J(%u,%u)*y(%u);\n', i, n0+n1+j, i, i1(j), j);
+ end
+ end
+end
+
+% Compute the derivatives of the second block of equations (period t+1)
+% with respect to the endogenous variables.
+for i=DynareModel.eq_nbr+1:2*DynareModel.eq_nbr
+ for j=1:DynareModel.endo_nbr
+ if i0(j)
+ fprintf(fid, ' J(%u,%u) = J(%u,%u)+J(%u,%u)*y(%u);\n', i, n0+n1+j, i, n0+n1+j, i, i0(j), j);
+ end
+ if i1(j)
+ fprintf(fid, ' J(%u,%u) = J(%u,%u)+2*J(%u,%u)*y(%u)*g(%u);\n', i, n0+n1+j, i, n0+n1+j, i, i1(j), j, j);
+ end
+ end
+end
+fprintf(fid, ' JAC(:,%u:%u) = J(:,%u:%u);\n', DynareModel.endo_nbr+1, 2*DynareModel.endo_nbr, n0+n1+1, n0+n1+DynareModel.endo_nbr);
+fprintf(fid,'end\n');
+
+% Close file.
+fclose(fid);
\ No newline at end of file
diff --git a/tests/Makefile.am b/tests/Makefile.am
index ca4c30634..fceb5fc65 100644
--- a/tests/Makefile.am
+++ b/tests/Makefile.am
@@ -357,7 +357,8 @@ MODFILES = \
observation_trends_and_prefiltering/calib_smoother/Tr_prefilter_loglin_calib_smoother.mod \
observation_trends_and_prefiltering/calib_smoother/Tr_prefil_f_obs_loglin_cal_smoother.mod \
observation_trends_and_prefiltering/ML/Trend_no_prefilter_selected_var.mod \
- dynare-command-options/ramst.mod
+ bgp/solow-1/solow.mod \
+ dynare-command-options/ramst.mod
PARTICLEFILES = \
particle/dsge_base2.mod \
diff --git a/tests/bgp/solow-1/solow.mod b/tests/bgp/solow-1/solow.mod
new file mode 100644
index 000000000..74e979296
--- /dev/null
+++ b/tests/bgp/solow-1/solow.mod
@@ -0,0 +1,80 @@
+// This mod file test the bgp.write function by characterizing the Balanced Growth Path of the Solow model. This is done 10000 times in a Monte Carlo loop
+// randomizing the initial guess of the nonlinear solver.
+
+var Efficiency
+ EfficiencyGrowth
+ Population
+ PopulationGrowth
+ Output
+ PhysicalCapitalStock ;
+
+varexo e_x
+ e_n ;
+
+parameters alpha
+ delta
+ s
+ rho_x
+ rho_n
+ EfficiencyGrowth_ss
+ PopulationGrowth_ss ;
+
+alpha = .33;
+delta = .02;
+s = .20;
+rho_x = .90;
+rho_n = .95;
+EfficiencyGrowth_ss = 1.01;
+PopulationGrowth_ss = 1.01;
+
+model;
+ Efficiency = EfficiencyGrowth*Efficiency(-1);
+ EfficiencyGrowth/EfficiencyGrowth_ss = (EfficiencyGrowth(-1)/EfficiencyGrowth_ss)^(rho_x)*exp(e_x);
+ Population = PopulationGrowth*Population(-1);
+ PopulationGrowth/PopulationGrowth_ss = (PopulationGrowth(-1)/PopulationGrowth_ss)^(rho_n)*exp(e_n);
+ Output = PhysicalCapitalStock(-1)^alpha*(Efficiency*Population)^(1-alpha);
+ PhysicalCapitalStock = (1-delta)*PhysicalCapitalStock(-1) + s*Output;
+end;
+
+
+verbatim;
+
+ bgp.write(M_);
+ MC = 10000;
+ KY = NaN(MC,1);
+ GY = NaN(MC,1);
+ GK = NaN(MC,1);
+ EG = NaN(MC,1);
+ options = optimoptions('fsolve','Display','off','Algorithm','levenberg-marquardt','MaxFunctionEvaluations',1000000,'MaxIterations',100000,'SpecifyObjectiveGradient',true,'FunctionTolerance',1e-8,'StepTolerance',1e-8);
+ for i=1:MC
+ y = 1+(rand(6,1)-.5)*.2;
+ g = ones(6,1);
+ [y, fval, exitflag] = fsolve(@solow.bgpfun, [y;g], options);
+ if exitflag>0
+ KY(i) = y(6)/y(5);
+ GY(i) = y(11);
+ GK(i) = y(12);
+ EG(i) = y(2);
+ end
+ end
+ % Compute the physical capital stock over output ratio along the BGP as
+ % a function of the deep parameters...
+ theoretical_long_run_ratio = s*EfficiencyGrowth_ss*PopulationGrowth_ss/(EfficiencyGrowth_ss*PopulationGrowth_ss-1+delta);
+ % ... and compare to the average ratio in the Monte Carlo
+ mu = mean(KY(~isnan(KY)));
+ s2 = var(KY(~isnan(KY)));
+ nn = sum(isnan(KY))/MC;
+ disp(sprintf('Theoretical K/Y BGP ratio = %s.', num2str(theoretical_long_run_ratio)));
+ disp(sprintf('mean(K/Y) = %s.', num2str(mu)));
+ disp(sprintf('var(K/Y) = %s.', num2str(s2)));
+ disp(sprintf('Number of failures: %u (over %u problems).', sum(isnan(KY)), MC));
+ assert(abs(mu-theoretical_long_run_ratio)<1e-8);
+ assert(s2<1e-16);
+ assert(abs(mean(GY(~isnan(GY)))-EfficiencyGrowth_ss*PopulationGrowth_ss)<1e-8)
+ assert(var(GY(~isnan(GY)))<1e-16);
+ assert(abs(mean(GK(~isnan(GK)))-EfficiencyGrowth_ss*PopulationGrowth_ss)<1e-8)
+ assert(var(GK(~isnan(GK)))<1e-16);
+ assert(abs(mean(EG(~isnan(EG)))-EfficiencyGrowth_ss)<1e-8)
+ assert(var(EG(~isnan(EG)))<1e-16);
+
+end;
\ No newline at end of file
--
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