Added two examples (forecasts and IRFs in backward models).
Stéphane Adjemian (Scylla)
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#!/bin/sh
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rm -rf solow_irf solow_forecast
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rm -f *.m *.mat *.log
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close all
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var Efficiency $A$
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EfficiencyGrowth $X$
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Population $L$
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PopulationGrowth $N$
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Output $Y$
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PhysicalCapitalStock $K$ ;
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varexo e_x $\varepsilon_x$
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e_n $\varepsilon_n$;
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parameters alpha $\alpha$
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delta $\delta$
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s $s$
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rho_x $\rho_x$
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rho_n $\rho_n$
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EfficiencyGrowth_ss $X^{\star}$
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PopulationGrowth_ss $N^{\star}$ ;
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alpha = .33;
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delta = .02;
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s = .20;
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rho_x = .90;
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rho_n = .95;
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EfficiencyGrowth_ss = 1.02;
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PopulationGrowth_ss = 1.02;
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model;
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Efficiency = EfficiencyGrowth*Efficiency(-1);
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EfficiencyGrowth/EfficiencyGrowth_ss = (EfficiencyGrowth(-1)/EfficiencyGrowth_ss)^(rho_x)*exp(e_x);
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Population = PopulationGrowth*Population(-1);
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PopulationGrowth/PopulationGrowth_ss = (PopulationGrowth(-1)/PopulationGrowth_ss)^(rho_n)*exp(e_n);
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Output = PhysicalCapitalStock(-1)^alpha*(Efficiency*Population)^(1-alpha);
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PhysicalCapitalStock = (1-delta)*PhysicalCapitalStock(-1) + s*Output;
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end;
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shocks;
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var e_x = 0.005;
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var e_n = 0.001;
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end;
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// Set initial condition for the forecast (we simulate a sample, forecast will start at the end of the sample)
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histval;
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Efficiency(0) = 1;
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EfficiencyGrowth(0) = .5;
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Population(0) = 1;
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PopulationGrowth(0) = .5;
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PhysicalCapitalStock(0) = 1;
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end;
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oo__ = simul_backward_nonlinear_model([], 10, options_, M_, oo_);
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initialcondition = dseries(transpose(oo__.endo_simul(:,10)), 2017Q1, cellstr(M_.endo_names));
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/* REMARKS
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**
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** dseries object initialcondition may have more variable than required.
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*/
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// Define the variables for which we want to compute the IRFs
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listofvariables = {'Efficiency', 'Output'};
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/* REMARKS
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**
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** If possible do not forecasts all the endogenous variables (this is the default).
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*/
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// Compute the forecasts
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forecasts = backward_model_forecast(initialcondition, listofvariables, 16, true);
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/* REMARKS
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**
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** - The third argument is the number of periods for the forecast. Default is 8.
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** - The last argument is for computing (or not) the confidence bands. If false (default), only point forecast is produced.
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** - forecasts is a structure, each field is a dseries object for the point forecast (ie forecasts without innovations in the future),
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** the mean forecast, the median forecast, the standard deviation of the predictive distribution and the lower/upper bounds of the
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** interval containing 95% of the predictive distribution.
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*/
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// Plot forecast for output
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plot(forecasts.meanforecast.Output, '-k', 'linewidth', 2)
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hold on
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plot(forecasts.lb.Output,'-r')
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plot(forecasts.ub.Output,'-r')
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hold off
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/* REMARKS
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**
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** In this model there is no steady state (only a stable balanced growth paths), this explains the
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** shape of the forecast for Output.
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*/
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@ -0,0 +1,106 @@
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close all
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var Efficiency $A$
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EfficiencyGrowth $X$
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Population $L$
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PopulationGrowth $N$
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Output $Y$
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PhysicalCapitalStock $K$ ;
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varexo e_x $\varepsilon_x$
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e_n $\varepsilon_n$;
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parameters alpha $\alpha$
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delta $\delta$
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s $s$
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rho_x $\rho_x$
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rho_n $\rho_n$
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EfficiencyGrowth_ss $X^{\star}$
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PopulationGrowth_ss $N^{\star}$ ;
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alpha = .33;
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delta = .02;
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s = .20;
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rho_x = .90;
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rho_n = .95;
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EfficiencyGrowth_ss = 1.0;
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PopulationGrowth_ss = 1.0;
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model;
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Efficiency = EfficiencyGrowth*Efficiency(-1);
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EfficiencyGrowth/EfficiencyGrowth_ss = (EfficiencyGrowth(-1)/EfficiencyGrowth_ss)^(rho_x)*exp(e_x);
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Population = PopulationGrowth*Population(-1);
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PopulationGrowth/PopulationGrowth_ss = (PopulationGrowth(-1)/PopulationGrowth_ss)^(rho_n)*exp(e_n);
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Output = PhysicalCapitalStock(-1)^alpha*(Efficiency*Population)^(1-alpha);
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PhysicalCapitalStock = (1-delta)*PhysicalCapitalStock(-1) + s*Output;
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end;
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shocks;
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var e_x = 0.005;
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var e_n = 0.001;
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end;
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/*
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** FIRST APPROACH (PASS A DSERIES OBJECT FOR THE INITIAL CONDITION)
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*/
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// Define a dseries object.
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ds = dseries(repmat([1, .5, 1, .5, 0, 1], 10, 1), 1990Q1, cellstr(M_.endo_names), cellstr(M_.endo_names_tex));
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// Alternatively we could build this object with a stochastic simulation of the model.
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//oo_ = simul_backward_model(rand(6,1), 10, options_, M_, oo_);
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// Set the initial condition for the IRFs using observation 1991Q1 in ds.
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set_historical_values ds 1991Q1;
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// Define the shocks for which we want to compute the IRFs
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listofshocks = {'e_x', 'e_n'};
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// Define the variables for which we want to compute the IRFs
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listofvariables = {'Efficiency', 'Population', 'Output'};
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// Compute the IRFs
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irfs = backward_model_irf(1991Q1, listofshocks, listofvariables, 50); // 10 is the number of periods (default value is 40).
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// Plot an IRF (shock on technology)
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figure(1)
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plot(irfs.e_x.Output)
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legend('Output')
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title('IRF (shock on the growth rate of efficiency)')
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/*
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** SECOND APPROACH (USE THE HISTVAL BLOCK FOR SETTING THE INITIAL CONDITION)
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**
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** In this case the first argument of the backward_model_irf routine is a date object (for specifying
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** the period of the initial condition).
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*/
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histval;
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Efficiency(0) = 1;
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EfficiencyGrowth(0) = .5;
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Population(0) = 1;
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PopulationGrowth(0) = .5;
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PhysicalCapitalStock(0) = 1;
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end;
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// Define the shocks for which we want to compute the IRFs
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listofshocks = {'e_x', 'e_n'};
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// Define the variables for which we want to compute the IRFs
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listofvariables = {'Efficiency', 'Population', 'Output'};
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// Compute the IRFs
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irfs = backward_model_irf(1991Q1, listofshocks, listofvariables, 50); // 10 is the number of periods (default value is 40).
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// Plot an IRF (shock on technology)
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figure(2)
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plot(irfs.e_x.Output)
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legend('Output')
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title('IRF (shock on the growth rate of efficiency)')
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/* REMARK
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** ------
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**
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** This model is nonlinear and non stationary. For different initial conditions the IRFs may look quite different.
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*/
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