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// --+ options: json=compute, stochastic +--
var y x z v;
varexo ex ey ez ;
parameters a_y_1 a_y_2 b_y_1 b_y_2 b_x_1 b_x_2 d_y; // VAR parameters
parameters beta e_c_m c_z_1 c_z_2; // PAC equation parameters
a_y_1 = .2;
a_y_2 = .3;
b_y_1 = .1;
b_y_2 = .4;
b_x_1 = -.1;
b_x_2 = -.2;
d_y = .5;
beta = .9;
e_c_m = .1;
c_z_1 = .7;
c_z_2 = -.3;
var_model(model_name=toto, eqtags=['eq:x', 'eq:y']);
pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
pac_target_info(pacman);
target v;
auxname_target_nonstationary vns;
component y;
auxname pv_y_;
kind ll;
component x;
growth diff(x(-1));
auxname pv_dx_;
kind dd;
end;
model;
[name='eq:y']
y = a_y_1*y(-1) + a_y_2*diff(x(-1)) + b_y_1*y(-2) + b_y_2*diff(x(-2)) + ey ;
[name='eq:x']
diff(x) = b_x_1*y(-2) + b_x_2*diff(x(-1)) + ex ;
[name='eq:v']
v = x + d_y*y ; // Composite target, no residuals here only variables defined in the auxiliary VAR model.
[name='zpac']
diff(z) = e_c_m*(pac_target_nonstationary(pacman)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman) + ez;
end;
shocks;
var ex = .10;
var ey = .15;
var ez = .05;
end;
// Initialize the PAC model (build the Companion VAR representation for the auxiliary model).
pac.initialize('pacman');
// Update the parameters of the PAC expectation model (h0 and h1 vectors).
pac.update.expectation('pacman');
/*
**
** Simulate artificial dataset
**
*/
// Set initial conditions to zero.
initialconditions = dseries(zeros(10, M_.endo_nbr+M_.exo_nbr), 2000Q1, vertcat(M_.endo_names,M_.exo_names));
// Simulate the model for 5000 periods
TrueData = simul_backward_model(initialconditions, 5000);
/*
**
** Estimate PAC equation (using the artificial data)
**
*/
// Provide initial conditions for the estimated parameters
clear eparams
eparams.e_c_m = .9;
eparams.c_z_1 = .5;
eparams.c_z_2 = .2;
edata = TrueData; // Set the dataset used for estimation
edata.ez = dseries(NaN, 2000Q1); // Remove residuals for the PAC equation from the database.
pac.estimate.nls('zpac', eparams, edata, 2005Q1:2005Q1+4000, 'fmincon'); // Should produce a table with the estimates (close to the calibration given in lines 21-23)
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